Effet non linéaire d'auto-démodulation d'amplitude dans

HAL Id: tel-00004227https://tel.archives-ouvertes.fr/tel-00004227

Submitted on 20 Jan 2004

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Effet non linéaire d’auto-démodulation d’amplitude dansles milieux granulaires: théories et expériences

Vincent Tournat

To cite this version:Vincent Tournat. Effet non linéaire d’auto-démodulation d’amplitude dans les milieux granulaires:théories et expériences. Acoustique [physics.class-ph]. Université du Maine, 2003. Français. tel-00004227

https://tel.archives-ouvertes.fr/tel-00004227

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U<hA@óB<qAUf£¥@B<hAJó|?6@B<hAJ<¦?6@J<?AdqhAq<3?6C -?dÇm@?6ÁwWVB<<+-?7¢qè?6n@B<Ú<ÆÕ q<h¢nq<>?6H¢h<q< V¿?6@dRffq<@<n<+@B@nn@B3?6nq< KMA a)^dCKMAKasa

un =

√2RE

3m(1 − ν2)[δ0 − (un − un−1)]

3/2 − [δ0 − (un+1 − un)]3/2 Ð T g Ó67

un<hAÙ<J-?¢h<Á<@ <-? nn<

n?6M|?6 +A|? ÚAn@²f£¥Vnnnq<C

un®£ ?¢h¢h|?6@<H-?+nn<

nCm<

δ0-?4dA|?6@B¢h<+?6q<ÚA <@¿q< <hA <dëø¢h<@q<hAJ<Unn<hA?6Åq?¢h<@q<hA!

;Ñ£ ?6q3ëmnÁ7?6@\nn@B3?6nq<<&®£¥V?6@`Ð T g Ó<hAq4q<@¿B<B?6@BAw-?ÊnnÁwnq<ø<hAwÚ<nq<hAwÖÎÏÁ7?6@BAdÇ¿@?6ÁwWV¿B<hA |un − un−1| |δ0|

Gá¦§<h¢+¢h<q<4?6qhëdnÁ7?6@C @B<¢h@BA|?6@¿q<<+|?6<ÚJ<hAHq<hAqA|qA<hAÔq¦§h< ó?6<

K =

(∂δ0

∂F0

)−1

=3

4(RF0)

1/3

(4E

3(1 − ν2)

)2/3 Ð T a§Ó

9 9 ñ:81;=<*>6>*?81&(@A-0 #47%&B6?47*? #81;C')"0<D<E8F<D*>6?*?8G&H@.<*?') 5"JIK*>BL("<E"0-NM1283;O$QPR/3+&($B8

67F0<hAF-? Îpq¢h<ÔA|?6WV¿B<Ô?6nWVBh< Aq-?J¢|è?#_@B<C§<

δ0<Ô-?¢h<Á<@¿ÙA|?6WV¿B< ?AqAqm¢s¤H@B<é<hAnÁ7?6@

<-?$|?6<Kf£ @¢h@¿|?¢Ú

R = 0.5 cmCF0 = 10 N

CE = 6 1010 Pa

<ν = 0.21

@@B<K ' 5 106 N/m

CF¢h< VÔ¢hq<hAÚ@B @#Ámd<øf£ Ù@ó/< ff<h¢ ÎJ<E ' 5 108 Pa

CqùhAn@dÎp<¢h<nd1Á7?6q-?6#n Õ®ÁÁ<s;?q<-?6@#<dAÚ<qA@1<@¿q<<7@BÁ+q<&f£¥@B<

k<+-?A|?6@

ωB?6@BA¢h<q<?6q3ëmnÁ7?6@A3£¥<@¿J<@ABAn?6@¿@-?¢h<Á<@ < -?wÎÏÁ<un = Aeiωt−k2Rn B?6@BA®£ ?6q3ëmnÁ7?6@Ênn@B3?6nq<+<U®£¥V¿?6@#Ð T g Ó

ω = 2

√K

m| sin(kR) | Ð T T Ã§Ó

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k/kc

!""! #$ % &

ω/ω

c

9 ÉWq iL M!&.I/ NOo.6 h%TYM!hQ75Q0R7

;?JA|?6@7< ¢hq<H<Ô-?U¢qè?#_@B<CA|?6@Á7?×ëdnÁ+ÁR?67< <é-? VB<n<Ô<hAÑÁ¿<hAÙ?¢hBAqWVB<hA@ónBdn@?6dë¢h@BAqqhA <¦¿<@@B<@¿J¦?6@B<hAq¢h<@¿qA3C¦?6H?6qA3C

ωc = 2

√K

m

Ð T TsT Ó;?ÊÎ qV¿B<@B¢h<ø<ø¢hq<<²-?$¢|è?6n@B<?AqAqm¢h<C<ÖëmnÁh<<@aÎÏ@B¢@ <hAw?6|?6Áùq<hA&?¢h¢h<hAqAn<hA

<ÖëdÚnÁ<@|?6<Á<@U<hA @@Bh<>?6

fc =1

π

√K

m=

3F1/60

4π3/2ρ1/2R4/3

(4E

3(1 − ν2)

)1/3 Ð T T Â§ÓG ¢h<q<øÎ qV¿B<@B¢h<CÑ<@BÁ+q<f£¥@B<<¢hq<¢hq<hAÚ@B $-?$nnÁwnq<<ø-?$q<Áwùq< [3@B<<

êMnnn@<+¦?6kc = π

2R

¤H@B<w<hAnÁ7?6@#<w-?&ÎÏqVB<@B¢h<7<¢hq<7f£ @B<¢|è?#_@B<767R = 0.5 cm

CF0 = 10 N

Cρ = 4000 kg/m3 C E = 6 1010 Pa

<ν = 0.21

@@B<fc ' 16 kHz

â¿U-?=Bóq< T ÄCÚ<hAq<qhA|<@qh<-?Ú<@BB?6@B¢h<&<7-?øA|?6@aA<@BÁ+q<²f£¥@B<&B?6@BA+-?q<Áwùq< [3@B<&<7êMnnn@H;<|?6Ú

ω/k@@B<-?/¦¿nq<hA|Aq<<è?A|<<ø®£¥@B<$?¢hBAWV¿B<B?6@BA-?/¢qè?#_@B<<-?/|?6@ó§<@q< -?

¢hÚ<q<qhAq<@qh<wAJ-? Bóq< T Ä∂ω/∂k

@@B<-?¦mnq<hAqAq<<+óqÚ<sKM<hAJ¦¿nq<hAqA|<hAJA|@Uq<qhAq<@qh<hA<@²Îp@B¢@<>-?A ?6@A -? Bóq< T d0

;?¦mnq<hAqAq<<è?Aq<4?¢hBAWVB<4B?6@BAJ-?&¢qè?#_@B<CÚB?6@BAU-?7nnÁwnq<?A|Aq< ÎÏqV¿B<@B¢h<C<hAq<@$nBAÎp?6n<V¿B< -?7¦mnq<hAqAq<+<+q?6ó?6@/<hA@B<hA>?¢hBAWV¿B<hAUB?6@BAJ< Á7?6q-?6¢h@BAn?6@><hAnn<hAUdÎ®?6nJ<

Q07Y p '!#XV0 P0,.$% D4$?y0= T, 9 ;5@TT456QGJ 909

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ω/ωc

cg(ω)

&&"! #$ % &

c ω

=0)

cφ(ω)

9É S M1A"! 9 1+4UWV^$I!T5H$TAO*'4O P5/!4H PQ07Q0R07

-?q<-?6n¦§<Á<@Îp?6n< |?6<H<hA ¢h@|?¢qAJ¢hÁw?6qh<+?6Á7?6q-?6<hAÔnn<hA3CB< <n<+A3£¥h¢n

c1D = limk→0

ω

k= 2R

√K

m= Rωc =

3F1/60

2√

πρR1/3

(4E

3(1 − ν2)

)1/3 Ð T T Ä§Ó;?¦mnq<hAqAq<U<Uè?Aq<><U®£¥@B< ?¢hBAWV¿B< <@&Îp@B¢@<U-?Î qV¿B<@B¢h<>B?6@BAé-?4¢qè?#_@B<+Aq<>hdn <U-?

q<-?6@< dAÚ<qA@Ð T T ÃÓ cφ(ω) =

ω

k=

ωc

k| sin(kR) | Ð T T d¿Ó

G`-?Î qV¿B<@B¢h<>@n<C ¢h<q<>¦mnq<hAqAq<J¦?6H<@ cφ(0) = Rωc

Ð T T À§Óf£¥67@B< ?6q< <Öëdq<hAqA@n< Ô-?4¦mnq<hAqAq<U<>è?A|< <@²ÎÏ@B¢@Ê<>-?Î qV¿B<@B¢h<

cφ(ω) = cφ(0)ω/ωc

arcsin(ω/ωc)

Ð T T Y§Óá<hAnÁ7?6@Cf-?²¦mnq<hAqAq<

cφ(0)¦?6+<@¿¦¿nq@

500 m/sÐR = 0.5 cm

CF0 = 10 N

Cρ = 4000 kg/m3 C

E = 6 1010 Pa<

ν = 0.21Ó

;?4¦mnq<hAqAq<U<>óqÚ<C7V¿?6@ 4<n<C ?4-?ÎÏÁ<+An¦?6@¿q<

cg(ω) = cφ(0)

√1 − ω2

ω2c

Ð T T ^Ó;qA V¿B<

ωq<@B¦§<qA

0Cf<hA ¦mnq<hAqAq<hA <4óqÚ<7<<4è?Aq<Aq@¿ó?6<hA!;qA VB<

ωq<@B¦§<qA

ωc-?

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8 mmCÚÎÏq¢h<7A|?6WVB<<

30 NC

ÎÏqV¿B<@B¢h<Hf£¥<Öë¢n|[email protected] MHz

Ó m¡U?6@BAM¢h<q<J¢h@ZBó|?6@677nBA<qAM@óB<qAÔf£¥@B<hAÔ?¢hBAWV¿B<hAA|@¢h@q<@¿B<hAÙB?6@BA@wd-?6Áùq< <Mnn<C<hA@B<hA<ÔAÆÎ®?¢h<é<MEJ?hÇm<nóèAq@ó§@Bqh<hAÑ?6F¢h@¿¦§<qA@<Ám< ²¢qè? V¿B<w¢h@|?¢+<@¿q<4nn<hA!fnBA<qA>ÁjÇ§<@BA>Ú<Á<q<@+<wA3£ ?A|Aq<><-?&@?6q<w<4¢h<hA@B<hAjC¢hÁwÁ< -?4Aq<@BAnnnnq+dAnó@?6 ®£¥|?6<>AqÆÎp?¢h<><hAHnn<hA <>è?A|?6ó§<<

π/2<hAH@B<hA

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G è?6q<hAÎÏqV¿B<@B¢h<hAÑf£¥<Öëd¢n|?6@$Ðf fc

ÓÖCnBA<qA Ç¿Ú<hAÑ<éÁm<hAÑ<Ô¦mn|?6@7<hAnn<hAÚ<¦§<@?6?6|?#_q< ïH-?ÃÂ

• <hA Ám<hAH< EJ?hÇm<nóèÚ <hA V¿B<A @B<+@B<+<+AÆÎ®?¢h< Aq<>q?6ó§<?6q<+-?4nn<?6qA VB<< ¢h<@q<>q<hAqq<>nÁwÁn<s• ;<hAÔÁm<hA < ÔèAÚ<n@óó?6n<Ç • ;<hAÔÁm<hA <>q<hAn|?6@C7V¿¢hq<hAÚ@B<@ @B<>¦mn|?6@|?d-?6< <>-?nn<s;<hA>ÎÏqVB<@B¢h<hA<qhA|@?6@B¢h<7<7¢|è? VB<Ám<7?6n@BAQV¿B<<hA+@BÁ+q<hA4f£¥@B<?A|Aq¿¢hA+<¦§<@¿ q<

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2RC¿n <hAA|An<Hf£¥q<@nM<hAÙq<-?6@BAé<

Q07Y p '!#XV0 P0,.$% D4$?y0= T, 9 ;5@TT456QGJ 9Zq

dAÚ<qA@<w¢qè? V¿B<wÁ¿<w<4q?6ó?6@HFKM<hA>Ám<hA><4¦¿n|?6@@¿+<hAJÎÏqVB<@B¢h<hA <4qhA|@?6@B¢h<<@AÚ<q<hA -?ÎÏqVB<@B¢h<+<>¢hq<

fc<hAÔÁm<hAÔ@ónBdn@?6dë

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F1/60

<hA><@BA|<¦§h<C¢hÁwÁ<wqhdn ?6 ®£¥V¿?6@\Ð T T ÄÓ KM<Ú<@BB?6@3CqA V¿B<H<hAMÁ7?6q-?6dë²?3¦§<h¢<hAé|?6qAé<AqÆÎp?¢h<U?6nqqhA>Ð ?6M<Öë<Áw<H?6éhëdÇdB?6@ ÓÙA|@énnAqhA3Cm¢h<q<<@BB?6@B¢h<+A3£¥h¢3?6q< <

F1/60

BÒ f<@<hA <UÁÁ< B?6@BA -? [3@B<><>-?A¢nq <hAH¢h@|?¢qA!BH<hA nn<hA<U¦§<q<CB?6@BAÔ@B<Uó?6ÁwÁ< <JÎpq¢h<>A|?6WV¿B< ?6n-?6@H<

F0 = 44 N

F0 = 578 NCd@B<>Ú<@BB?6@B¢h< <U-?

¦¿nq<hA|Aq<4<4q?6ó?6@<@F

1/60

<hA><@/BAq<¦§h< KMA asa f;?&èBh<<4ï<\[<hAq @B¢4<@?B?6qh<B?6@BA ¢h< ¢3?A!

¡U<Îp?65h@/V¿n¦?6<@¿q<CÚ@B<w<@BB?6@B¢h<7<-?ÎÏqV¿B<@B¢h<<w¢hq<w<hAUÁm<hA>@ónBdn@?6dë/<-?¢qè?#_@B< Aq -?ÎÏq¢h<>Aq|?6WVB<

F0<hA BAq<¦§h<+<@

F1/60

C¢hÁwÁ< qhdnH?6Ô-?4èBh< < ïJ<\[ KMA aÀ 9ÉWqfÉ i É q î #"0j24J-04Z+*,".$B8F;!"6>* !"(;

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F0 = 0ÓÖCm¢h< V¿<hAqMnÁwÚAqAn<ÚM<hAMÁm<hAM@ónBdn@?6dë

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KÐ ¦§nU-?tBóq< T TsT Ó

;á?wq<-?6@Ki = µK

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F = KX = KiXi ,Ð T ÂÃ§Ó

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Xi =K

KiX =

X

µ X .

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NTÁ<@qAJ<hAH?6qA ó?6<

XT =

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X +

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N

1

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νr

µ

],

Ð T ÂÄ§Ó67

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σ = Eiε

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∑

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),

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Γ moyenn

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α =1 + νr/µ

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[1 + νr/µ]n,

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α4

β[1&,I k36N4 M3.6 5

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< VB<U< Á¿d< f£¥-?Aq¢nq>Á3Ç§<@Emoyen <hA ó?6 βE

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−5 −4 −3 −2 −1 0−5

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µ:d &,4_$ ,H)]!&lM17 r

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σ0 = −C(−ε0)3/2 ,

Ð T Â)^Ó67

C = ncE3π(1−ν2)

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â¿f@B<><?6@$?¢hBAWV¿B< <hAÔn@qmdnq<C¢h<q< q<-?6@q<@Bø-?ÎpÁ<σ = −C(−ε − ε0)

3/2 ,Ð T Â g Ó

Q07 ,.$% D4$F=)+5_4$2+V'?$>O5.A&C,.4_A )* @l i S

σ

ε

εε

ε

σ

σ σ

t

9 É>9 ir &?JKl@' Mt$%4,HK!@ 46yid !0!#'; '+=6t)+l/$2+V^?$%6'17

67εhAnó@B< -?>ÖÎÏÁ7?6@²?¢hBAWV¿B< <

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σ = −C

[(−ε0)

3/2 − 3

2(−ε0)

1/2ε +3

8(−ε0)

−1/2ε 2 + . . .

],

Ð T Âsa§Ó<@B¢hq<>Ú -??6< dÇm@?6ÁwWVB< < -?4¢h@|?6n@q<@WV¿B<Á<@3C

σ =3

2C(−ε0)

1/2ε

[1 − 1

4ε0ε + . . .

].

Ð T ÄÃ§ÓF?6?6@?6ó<?3¦§<h¢>@¦§<Ú<Á<@¢-?A|AWVB<+<@ø@B@dÕ®nn@B3?6nqhA<> Ç¿Ú<UAqA|?6@B¢h<hAAB¢h¢h<hAqAn¦§<hAjC

σ = Eε[1 + Γ2ε + Γ3ε

2 + . . .]

,Ð T Ä T Ó

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c ]q M&c FM)$"!*H3)BEi"TzB<)'zDODE,)L~)IH') L;$iLkOBIDL *QDB@PDER"'B',D* .-/ &= 3V ; 0 #o q p\[/o o ]]q c ] \ ef `c ?1; V^ F;) L#$AiLN!DL;$'-/&= 3V ;Fc W 7# Y o'[I Y q o ]+] c ]+] c 7dk78F = 0 74 3:: =&\)B<C"k@A"<D/ B/"'@A"'LFEA)L|/E.DECOBIDL *QDB@PDER"'B,D+* ,-/ &= 36V 6 0W . Z p+[ Z p Z o ]+]] c ] p\ 16wc 7EB<"'/$J/EB'K! E)L§iL|/E.DE,CUEST)$J@A"'LF/,)L;D*_OBIDL *QDBA@P)$"!*@P"/$,DJLjEi"UD+K!3"'L#C")IHH'B'C!E,)L+-/ = 3FV 6 0 # q q p\[ q p+ ] o ]+] p 3d%p q 5 ( 3d 4 ?1 !=&\"<CFDLCD*/ER"'B/"'/J|iL B<)CDEU*Q)S /EB<DJL D@?z#* JE F$&"'jDL#$ !"'J/@,CH'B<"/! "'L#C!."',- &= 3 0 Fr 7# q+ [ o ] p q : ` _ : + acv_8 c_M1 &b +7\DCJLOM)IHAC)@?z#B<"'/,K3*"OB<DLF F*QDBU@kDER"'B'D+* ,- &= 3V ;c ;F o Z q o [ q o +Y ++ +\ Mv 7v16_1 w8F_` 66F³aB<)/JL&O)IHk,$&"'LFE,C/D+*T')+* iE.DB'NSTD"'PiLDMCFDJL)IHU"!*QD/E,CGK"<D$,-/ &=3V ; 0 W n7o + o q + ]q 16# wD"hz#B<)zDODE,)LMJLUOB<DL *QDBD/!"'@PK3* ."',-/ &= 3V ; 0 ,Y ; YY [ Y o o ]]q q+] V^ 7F³)@fz;* ,DL#C'"%)IH%"!*QD/E,CK/)$,"'JLC/)LE.DC!Et-/#8F ;6l## ] [ Z o ]q+] o' ; 5 e ¬ 0 ;aK!3"'BDE,)L)HUL;)L* JL6"<DBGJLER"'BIDC!E,)L )IHND+C) /E,CNSTD"'GiLOB<DL *QDB^@PDER"'B,D+* ^v$"'@P)$ *QDE,)LzB<)C"'/.-/ &= 3#c X #n66o [+ Y +o ]+Z ;M6 #`%6M#8 1;#V^ ,F\w)B<C"k$J/EB,K! E,)LiLDOB<DLF *QDB^@A"<$J @?-/ &= 3V ; 02 ,Y Y o q [ Y o ] o ]+]Z 3\ 7 _ F`wM_8 16V% ,7+FEB/"'/PEB<DL/@J//,)LEiBI) +O\jEiB<"/"!$i@A"'L/,)L;D+*)B<$&"'B<"/$OB<DL *QDBDBB<D,- &= 3FV ; 02WW # Y o Y q + ]Z dk &=: cv &¢ <:&=&+>^)L#* JL;"/DBD"'B<)C"'/!"'JL C/) /E,C!/ e h9 = 3e o ]+]Z Z+Y +4 : \\B<)'zDODE,)LN)HL#)L#* iL6"<DBfC/)@?z#B<"'/)Lz F* !"'JLOBIDL *QDB@P"/$,D'-/8 + = 3 X q p\[ p o ]Z+Y t« ]+] 16 t« %`7 _e 6LjEi"^z;*QDEC!JEDL;$M"!*QD/E,C!JE)H$*QDE.DLFEOB<DL *QDBA@PDER"'B,D+* ,-/w8Fv = 3#1 66o Y+] p o ]+]] ] p 5 dk : ;f¢ V : % r : >)L* JL6"<DB L#$&"'B'STDER"'B C/) EC! 1 'b[.s /: o ]Z p b ]Z 0 _V^ b :;_8 5 d 0 5 [ 7`_Mw8 16V% \"'LF/JE% FC!E DE,)LPiLj,K!B<DER"<$OB<DL *QDB^@PDER"'B,D+* ,-/&= 3V ; 02 7o ] p&o![/o ]Z o ]]+Z ¢ + F¢ ++E.DE,! ""'EaB<¥<)+*Q)/O."^$w¤ LG@* ."' %OBIDL *QDiB/"^C)LL;¥R-/ ¬ 9 ?r9(4_V1 ¢ 3=&# q g ] 8F ` +F` _#1 0 5 != & +&xvy'zF"'B'i@A"'LE.D*ER"'/E)IH(Ei"%$ )L~D'zzB<)yJ@kDE,)LH3)B@ * E z#* ."<$P'C/DEER"'B<"/$!) L#$'-/ &=3FV ; 0§#X ,Y Y o [ Y o+o q o ]+]+ ] p 8F ` +` _1 _r'_8 3 aT1 F a _³v*QD//,C/D+*STD"z#B<)zDODE,)L£iL /EB<)L&O+* 'C/DEER"'B'iLOU@A"<$D'-/ &= X wl q [po o ]+] p ++\ 38F#8 5 '= 5 ;1 ![.V '=&+¨?ER"'B<"'/JJLN,K!BIDER"<$kOB<DL *QDB@A"<$D'-/ &= X Fw# Z pp\[ Z]+Z + V ] 4V \g ,M&8 8 [R8F 1;+V «6w)BIC'"$iEB,K! E,)LUJLU$&"'L!"fEST)+R$J@P"'L/,)L#D+*OB<DLF *QDBf//ER"'@,-/ = 3FV 6Fc 6 #+p q [+pp o ]+] V ]Z 4V \g 0 h tfM8 8F [8 hJ@k)$D+*CFDB<DC!ER"'B|)IH~/EB<"'EB<DL/@J//,)L JL OB<DLF *QDBzDCJL&O,- &= 3V 6c #F o o![ q o ]+]Z V 3+ __V 77xE F$&"G$&"*QDUO&¥<)@P¥'EB',"$&"A@* ."' +yGOBIDL *QDJB<"'P@P)$!*"'PEB,$i@A"'L/,)LL6"!* zDB/J@ *QDE,)LL @P¥!B,! "R- F¬ 9 V o79(4_V 1 Q¬ + Y g + V \= Z p\ V = 0 8 sF1 +F_z#ECD*O&"'L;"'B<DE)LMDL;$P$&"'ER"<C!E,)L)H? BiH3DC"^DC/) EC^STD"'()L~D,zF"'B<"R- &= 3c #X ,] #p [p Z o ]+Z p V #p Z ¢ %V : (16Fr';1 = 7¶"<)B<"'E,C/D+*H3) L#$DE,)LF%)IHL;)L* JL6"<DBDC/) /E,C!/ e 5 !b[s /: o ] p Z 1 Zq c61 b ?;8F #* EB<D')LF,C(z#B<)'zD3ODE,)L|iL|C3*Q)!"!QzDC"<$~$J')B<$"'B/"<$ ,zF"'L/,)L,-/w8 &=3# oo Y+] p&o![ Y+] p+p o ]Zq

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5f !"#%$%&'($)*+,&-. /*

Gc=9??9ACES?[]\^YZe~ACB9[TchJB@<X?9G=9ACEH=PbY4°ACB@IKGKpm«Aast®zRt

ρ0∂2Ui

∂t2=

∂σij

∂xj,

ka =M£Ai

ρ0Gc=@?PbYW[SGJLH=@<X?9T[aE±IV<XPX<>GJE"sgPd\mTcDMES<XPX<XeSB9GtN

Ui=9ACLF?VP>Gc=VhcACIVQ0At='YZLM?9Gc=|[aE±ijGchJ?9GJESB|[STJQSPbYChcGJIKGJLM? ~U

NGJ?σij

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ρ0∂2

∂t2

(div~U

)=

∂2σij

∂xi∂xj,

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?@<>ACLt

W = ρ0c20

[1

2

(div~U

)2− Γ2

3

(div~U

)3]

,ka^j£

Aic0

Gc=9?PbYki*<X?9Gc=9=9GK[aE='ACLGJ?Γ2

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σij = ρ0c20

[(div~U

)− Γ2

(div~U

)2]

δij .karyt£

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σHij = −B

(−div~UH

)κδij ,

ka^xj£YOijGch%PbYhcACLH=@?'YZLF?9G

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FACR.x À NFµRS<bx ¿ N* GJPbs ¿ z`NMGJ? PbYhcACLH=@?'YZLM?9GκhcACIVQSB@<>=9G,GJLM?@B9G

3/2GJ?

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P0=9ACLF?,[STchJB@<X?9Gc=:Q.YZB¶t

σinij = −B

(−div~U in

)κδij = −P0δij .

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hcACLM?@B'YZ<XLF?9G4=@?'YZ?@<>DFEHGtN.GJLhcACLH=@<>[STJB'YZLF?w[SG%QSPXEH=:ESLHG%Q0GJB@?@[email protected]?@<>ACLgYChcACEH=@?@<>DMEHGGc=@?:TchJB@<X?9G=9ACEH=PbY6°ACB@IKG t

σij = σHij − σin

ij ' κB1/κP(κ−1)/κ0

[(div~U

)− κ − 1

2B1/κP

−1/κ0

(div~U

)2]

,kanmj£

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κ = 3/2N

c0 ∝ P1/60

NΓ2 ∝ P

−2/30

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c0 ∝ P1/40

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κ = 2GJ?

Γ2 ∝ P−1/20


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[∂2

∂t2− c2

0∆ + L

](div~U ) = −c2

0Γ2∆(div~U )2 ,kanmm£

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GJ?,Q~ACESB ~Uω=9G%dYZ<X?:Q.YZB:IKAjGJLSL.YZjG=@ESBESLHGQ0TJB@<>AM[SG[SG2Pd\mACLH[SG;hgNHGJ?

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[∂2

∂t2− c2

0∆ + L

](div~Uω) = 0

kanmkj£[

∂2

∂t2− c2

0∆ + L

](div~UΩ) = −c2

0Γ2∆〈(div~Uω)2〉 .kanm ¿ £

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ρ0∆〈Wω〉

NAiWω

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[∂2

∂t2− c2

0

∂2

∂x2

]UΩ = −Γ2

ρ0

∂

∂x〈W ω〉 ,

kanm=M£Ai

UΩGc=@?PbYhcACIVQ0At='YZLM?9G|=9GJP>ACL

x[aE[STJQSPbYChcGJIKGJLM?6wg¸GJ? 〈W ω〉 = 2〈Wω〉 = ρ0〈

(∂Uω∂t

)2〉 ≡ ρ0〈v2ω〉

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UωGJ?

vω=9ACLM?%B9Gc=@Q~GchJ?@<XijGJIKGJLM?2P>G4[STJQSPbYChcGJIKGJLM?GJ?

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cg(ω)∂

∂x〈Wωb〉 +

∂

∂t〈Wωb〉 +

1

τ(ω)〈Wωb〉 = 0 ,

kanm À £Ai

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ωNGJ?

τ(ω)Gc=@?kP>G

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τa(ω)£2GJ?PbY[a<¢§~EH=@<>ACLº¤QSB@<>=9G|GJLhcACIVQS?9GfYcijGchfP>GV?9GJIVQH=

hOYZB'YChJ?9TJB@<>=@?@<>DMEHGτs(ω)

£oN YOijGchVPbYkB9GJPbYZ?@<>ACLτ−1(ω) = τ−1

a (ω) + τ−1s (ω)

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ωGc=@?Q0GJ?@<X?[SGJitYZLF?

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55 !"#%$%&'($)*+,&-. /*

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∂

∂t〈Wωd〉 = D(ω)

∂2

∂x2〈Wωd〉 −

1

τa(ω)〈Wωd〉 +

1

τs(ω)〈Wωd〉 ,

kanmOj£

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h'R.YZIVQYChcACEH=@?@<>DMEHGVhcACRHTJB9GJLF?;hgN~GJ?D(ω)

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D = cel∗/3

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D ≡ d c2g(ω) τs(ω) ,

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x = 0=9ACLF?5t

cg(ω)〈Wωb〉 = Iωf(t/τm) ,

Dω∂

∂x〈Wωd〉 = 0 ,

kanmOxj£

AiIω

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〈Wωd(x → ∞)〉→0 ,kanmOsj£

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〈Wωb〉 =Iω

cg(ω)f(Ω)e

h− 1

cg(ω)

“1

τ(ω)−iΩ

”i

,kak j£

〈Wωd〉 =Iω

cg(ω)D(ω)τs(ω)f(Ω)

1

1c2g(ω)

(1

τ(ω) − iΩ)2

− 1D(ω)

(1

τa(ω) − iΩ) ×

1cg(ω)

(1

τ(ω) − iΩ)

√1

D(ω)

(1

τa(ω) − iΩ)e

−

r1

D(ω)

“1

τa(ω)−iΩ

”x− e

− 1cg(ω)

“1

τ(ω)−iΩ

”x

,kak_m£

Ai

〈Wωb,d〉 ≡∫ +∞

−∞〈Wωb,d〉eiΩtdt ,

f(Ω) ≡∫ +∞

−∞f(t)eiΩtdt .

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〈W ω〉 = 〈Wωb〉 + 〈Wωd〉 .Y"='ACPXES?@<>ACL·[SGkPd\mTcDFE.YZ?@<>ACLkanm=*£='YZ?@<>=dYZ<>='YZLM?¸sWPbYWhcACLH[a<X?@<>ACL·PX<XIV<X?9GUΩ(x = 0) = 0

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UΩ ≡ UΩe−i Ω

c0x ≡ UΩb + UΩd

∼= kakkj£

∼= Γ2Iω

ρ0c20cg(ω)

f(Ω)

1cg(ω)

(1

τ(ω) − iΩ)

1c2g(ω)

(1

τ(ω) − iΩ)2

+ Ω2

c20

1 +

1

τs(ω)(

1τa(ω) + D(ω)Ω2

c20− iΩ

)

,

AiUΩ ≡

∫ +∞−∞ UΩeiΩtdt

~¥LghcACLH='TcDFEHGJLH=9GtNUΩ

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x ≤ maxcg(ω)τ,√

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UΩbGJ?

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UΩ = UΩb + UΩd = UΩb

1 +

1τs(ω)

1τa(ω) + D(ω)Ω2

c20− iΩ

.

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5 !"#%$%&'($)*+,&-. /*

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UωbNGonShJ<X?9TcGQ.YZB|[STJIKA*[aESPbYZ?@<>ACL½[SGc=|ACLH[SGc=|e.YZPX<>=@?@<>DMEHGc=K;ºgGc=@?

YZL.YZPXa=9TcGtSGc=LHAC?'YZ?@<>ACLH==@ES<XitYZLF?9Gc=,=9ACLF?<XLF?@B9A*[aES<X?9Gc=ON

τ± ≡ τ

(1 ± cg(ω)

c0

),

V0 ≡ Γ2Iω

2ρ0c20

,kak =M£

Q0ACESBWP>Gc=W?9GJIVQH="hOYZB'YChJ?9TJB@<>=@?@<>DMEHGc=GJ?WPd\^YZIVQSPX<X?@EH[SG²hOYZB'YChJ?9TJB@<>=@?@<>DMEHG²[SGPbY·i*<X?9Gc=9=9GQ.YZB@?@<>hJESPbYZ<XB9G²[SGPd\mACLH[SGYChcACEH=9?@<>DFEHGwg8G=@Q0GchJ?@B9G6[SG4PbY|i*<X?9Gc=9=9G4Q.YZB@?@<>hJESPbYZ<XB9G

VΩb ≡ −iΩUΩbGc=@?ON]YOijGch6hcGc=wLHAC?'YZ?@<>ACLH=ON]TchJB@<X?2=9ACEH=

PbY4°ACB@IKGxt

VΩb = V0f(Ω)τ(ω)

1

τ−(ω)

Ω(Ω + i 1

τ−(ω)

) +1

τ+(ω)

Ω(Ω + i 1

τ+(ω)

)

.

kak À £

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GJL[SGJEanW?9GJIVQH=hOYZB'YChJ?9TJB@<>=@?@<>DMEHGc=τ±(ω)

Gc=@?[aEHGÁs²PbY²QSB9Tc=9GJLHhcG[SGPbY·[a<>=@Q~GJB9=9<>ACL[SGiM<X?9Gc=9='G[SGQSB9ACQ.YZFYZ?@<>ACL[HYZLH=P>GIV<XPX<>GJE tB'YZLMESPbYZ<XB9G

cg(ω) 6=c0£±pm;%YC='stsaN gACE.stxZz`4¯%YZLH=hcGJ?@?9G½YZL.YZPXa=9GtNPd\ <XLHTJFYZPX<X?9T ∂cg(ω)

∂ω < 0Gc=9?=@ESQSQ0At=9TcG±ijTJB@<¢¡.TcGµ[HYZLH=?9ACES?9G

Pd\mTJ?9GJLH[aEHG|B9TcDFEHGJLM?@<>GJPXP>GlTJ?@EH[a<>TcGt¥L±YChchcACB9[ÝcijGchfPd\mTcDFE.YZ?@<>ACLkak =*£oNτ+(ω) ≥ τ−(ω)

PvGc=9?4=@ESQSQ0At=9T|DMEHGDME.YZPX<X?'YZ?@<XijGJIKGJLF?

τ+(ω)=9GB'YZQSQSB9A*h9RHG%[SG

τ−(ω)P>ACB9='DFEHG

ω='GB'YZQSQSB9A*h9RHG2[SGwPbY¤B9TcDMEHGJLHhcG%[SG,hcACESQSESB9G

ωcNMGJ?

DMEHGtNtQ.YZB hcACLH=9TcDFEHGJLM?ONcg(ω) → 0

M¦\^YZL.YZPX*='GGc=@?¦PX<XIV<X?9TcG%YZEanKIKA*[SGc=vYChcACEH=@?@<>DMEHGc=v;hgWQSB9ACQ.YZFYZ?@<¢°=wω < ωc

£oNP>Gc=:IKAM[SGc=:TJitYZLHGc=9hcGJLM?9=:LHG=9ACLM?Q.YC=hcACLH=@<>[STJB9Tc=<>hJ<d

¥L=@ESQSQ~At=YZLF?DFE\mGJLkjTJLHTJB'YZPdN*Pd\ <XLHTJFYZPX<X?9T 1τ+(ω) 1

τ−(ω)

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VΩb∼= V0f(Ω) ×

2(−iΩτ(ω))=@<

Ω 1τ+(ω) ≤ 1

τ−(ω) ,

(−iΩτ(ω))=@< 1

τ+(ω) Ω 1τ−(ω) ,

τ(ω)τ−(ω)

=@< 1τ+(ω) ≤ 1

τ−(ω) Ω .

kaktj£

GkB9TJt<XIKGk<XLF?9GJB@IKTc[a<bYZ<XB9G 1τ+(ω) Ω 1

τ−(ω)

Gc=@?|YZeH='GJLF?VP>ACB9=9DMEHGkPd\ <XLHTJFYZPX<X?9T 1τ+(ω) 1

τ−(ω)

L\mGc=@?VQ.YC=ijTJB@<¢¡.TcGt qM<

f(θ) ≡ f(t/τm)Gc=@?VESLHGlACLHhJ?@<>ACL½[SGkIKAM[aESPbYZ?@<>ACL·[SGl? MQ0GkFYZEH=9=@<>GJLSLHGACE±P>ACB9GJLM? >c<>GJLSLHGQ.YZB

GonSGJIVQSP>GtNH[]\ ESLHG[aESB9TcG2hOYZB'YChJ?9TJB@<>=9?@<>DFEHGτmN.YZP>ACB9=ONS[HYZLH=P>G[SACI|YZ<XLHG2?9GJIVQ~ACB9GJPdNHP>Gc=B9TJt<XIKGc=hOYZB'YChJ?9TJB@<>=@?@<>DMEHGc=

[SG%Pd\mTcDME.YZ?@<>ACLkaktF£Q~GJESijGJLM?J?@B9G[STchJB@<X?9=hcACIVIKG

VΩb∼= V0 ×

2( τ(ω)τm

)∂f∂θ

=@<τm τ+(ω) ≥ τ−(ω) ,

( τ(ω)τm

)∂f∂θ

=@<τ+(ω) τm τ−(ω) ,

τ(ω)τ−(ω)f

=@<τ+(ω) ≥ τ−(ω) τm .

kakjyt£

q*GJP>ACLPd\mTcDFE.YZ?@<>ACL±kakjyj£oN.P>GQSB9AZ¡HP[SGPd\ <XIVQSESP>=@<>ACLYChcACEH=@?@<>DMEHG[STJIKA*[aESP>TcG4wg²Gc=9?:°ACB@?9GJIKGJLF?%[STJQ~GJLH[HYZLM?%[SGPbYB9GJPbYZ?@<>ACL·GJLM?@B9GP>Gc=?9GJIVQH=KhOYZB'YChJ?9TJB@<>=@?@<>DMEHGc=|[aE±Q.YCDMEHGJ?K[]\mACLH[SG;hgN

τmGJ?

τ−(ω)¦GkQSB9AZ¡HP

VΩb(θ)[SG

PbYi*<X?9Gc=9=9GKQ.YZB@?@<>hJESPbYZ<XB9GYC='=9AMhJ<>TcG sPd\ <XIVQSESP>=@<>ACL± g B9GJQSB9AM[aES<X?Pd\mGJLMijGJP>ACQSQ~Gf(θ)

[aE´Q.YCDMEHGJ?6[]\mACLH[SGk;hg=@<τ−(ω) τm

NaGJ?Gc=9? QSB9ACQ0ACB@?@<>ACLSLHGJP s4PbY6[STJB@<XijTcG2[SG%Pd\mGJLFijGJP>ACQSQ0G ∂f(θ)∂θ

=@<τ−(ω) τm

aGw?@B9AC<>=@<>UJIKG%B9TJt<XIKG[SGPd\mTcDFE.YZ?@<>ACL±kakjyj£L\mGona<>=@?9GQ.YC==O\ <XP L\ YQ.YC=,[SG4[a<>=@Q0GJB9=@<>ACL[SGiM<X?9Gc=9='G|¤P>ACB9=9DMEHG

cg(ω) = c0Nτ−(ω) = 0

¹ #º¨ªµ ¦§¢5²´¢p¦5²« º¥¤B¢Ü©¦»¤B¥ ;«¯¸¢p¦ 5

GJ?Pd\ <XLHTJFYZPX<X?9Tτ−(ω) τm

L\mGc=@?9YZI|YZ<>=ijTJB@<¢¡.TcG£oIjw\mGc=@?P>GKhOYC=2Q0ACESBPd\^YZLM?9GJLSLHG|Q.YZB'YZIKTJ?@B@<>DMEHGf[HYZLH=Pd\mGOYZEGJLPd\^YZeH=9GJLHhcG[SGeSESPXP>Gc=[]\^YZ<XBpmuwAOiSxjy®z`HYZB,hcACLH=9TcDMEHGJLF?ON.P>Gc=:[STJQ0GJLH[HYZLHhcGc=,=@<XtLS<¢¡.hOYZ?@<XijGc=w[aEQSB9AZ¡HP [aE=@<XtL.YZP[STJIKAM[aESP>Tl=@ESB4P>Gc=Q.YZB'YZIKUJ?@B9Gc=V[aEIV<XPX<>GJEtB'YZLMESPbYZ<XB9Gk=9ACLF?6ESLHG|<XIVQ0ACB@?'YZLF?9GlI|YZLS<¢°Gc=@?'YZ?@<>ACL±[SGc=QSB9ACQSB@<>TJ?9Tc=[a<>=@Q0GJB9=@<XijGc=:[SG2hcG2IV<XPX<>GJE

¹,LHGlACeH='GJB@iCYZ?@<>ACL²GonaQ~TJB@<XIKGJLM?'YZP>G[SGkhcGJ?KGo§0GJ?Q0GJES?VJ?@B9GkES?@<XPX<>=9TcGkQ~ACESBVGona?@B'YZ<XB9G[SGc=<XLaACB@I|YZ?@<>ACLH=|=@ESBP>Gc=lQ.YZB'YZIKUJ?@B9Gc=[aE¸IV<XPX<>GJEºtB'YZLFESPbYZ<XB9Gt:Gc=hcACLH[a<X?@<>ACLH==@E_r|='YZLF?9Gc=Q~ACESBACeH=9GJB@ijGJBPbY?@B'YZLH=°ACB@I|YZ?@<>ACL[SGVΩb ∼ ∂f(θ)

∂θ

GJLVΩb ∼ f(θ)

Q0GJESijGJLF?J?@B9GACB@I4ESP>TcGc=hcACIVIKGg=9ES<X? t =@ESQSQ0At=9ACLH=|Q0ACESBhcACIVIKGJLHhcGJBDFEHGτa(ω) τs(ω)

2ijGchPd\^YZL.YZPX*='GtN%QSB9Tc=9GJLF?9TcG´Q.YZBWPbYµ=@ES<X?9GtN%[SGPbY·hcACLM?@B@<XeSES?@<>ACLVΩd

ijGJL.YZLF?[aE h'R.YZIVQYChcACEH=@?@<>DMEHG4[a<¢§0EH=@<¢¾=@ESBwP>G=@<XtL.YZP?9AC?'YZPdN~<XPYZQSQ.YZB'YZ<X?2DMEHGhcGJ?@?9G6hcACLF?@B@<XeSES?@<>ACLGc=9?,hcACIVQSP>UJ?9GJIKGJLF?2LHTJtPX<XjGOYZeSP>G |VΩd| |VΩb|

£bs?9ACES?9Gc=P>Gc=B9TcDFEHGJLHhcGc=,=9ACEH=PbYVhcACLH[a<X?@<>ACLτa(ω) τs(ω)

HACESB,ESLHG4YZL.YZPX*=9GDFE.YZPX<X?'YZ?@<XijGtNτs(ω)

Gc=9?YC=9='AMhJ<>T4sESLHGk[a<¢§0EH=@<>ACL[SGl? MQ0Gf«wYc*P>GJ<XtRτs(ω) ∼ 1

ω4

£|pmt<bYtstsCz`NGJ?4ESLHGk[STJQ0GJLH[HYZLHhcGlPX<XLHTOYZ<XB9GGJL"B9TcDFEHGJLHhcG B9TcDMEHGJLHhcGKhJ*hJPX<>DMEHG£%Gc=@?ES?@<XPX<>=9TcGVQ0ACESBPd\^YZeH=9ACB@QS?@<>ACL

τa(ω) ∼ 1ω

p µRS<bx ¿ Nq0YZ<bstsZeSz`¥LH=9ES<X?9GtNPd\ <XLHTJFYZPX<X?9T

τa(ωin) τs(ωin)Gc=@?=@ESQSQ~At='TcG,ijTJB@<¢¡.TcG%Q0ACESBESLHG2ACLH[SG2YChcACEH=@?@<>DFEHG2;hg=@<8PbY4B9TcDMEHGJLHhcG,<XLS<X?@<bYZP>G

ωinGc=@?=@E_r|='YZIVIKGJLM?¦TJP>AC<XtLHTcG [SG PbY:¤B9TcDMEHGJLHhcG[SGhcACESQSESB9G

ωcωin ωc

£oC²hcGJ?@?9GvB9TcDFEHGJLHhcGtNZPbY,[a<>=@Q0GJB9=@<>ACLGc=@?hcACLH=@<>[STJB9TcG°YZ<XeSP>G

cg(ω) ' c0GJ?w[SACLHhtN

τ−(ωin) ' τa(ωin)(1 − cg(ωin)

c0

) τa(ωin)

.¥Lgh9RHAC<>=@<>='='YZLF?,ESLLHACIeSB9Gw=@E_r|='YZIVIKGJLM? tB'YZLH[

n[SG,Q~TJB@<>A*[SGc=¾[]\mACLH[SGc=;hg[HYZLH=¾P>GQ.YCDFEHGJ?[]\mACLH[SGw;ºgW<XLS<X?@<bYZP

τ inm ≡ n

(2πωin

) N<XP2Gc=@?k?9ACECACESB9=kQ0At=9=@<XeSP>GW[SGgB9TOYZPX<>=9GJBPbYhcACLH[a<X?@<>ACL

τ inm τ−(ωin)

qM<YZP>ACB9=PbY¤B9TcDMEHGJLHhcG"[SGgPd\mACLH[SG"[SGQ~ACIVQ.YZjG

ωGc=9?6YZEStIKGJLF?9TcGGJL´FYZB9[HYZLM?

n¡SnaT£oN P>Gf?9GJIVQH=V[SG|IKA*[aESPbYZ?@<>ACL

τm[a<XIV<XLFEHGlQ~GJLH[HYZLM?VDFEHGfP>G

?9GJIVQH=hOYZB'YChJ?9TJB@<>=@?@<>DMEHGτ−(ω)

YZEStIKGJLM?9Gt®¥¦LdYZ<X?ONO=@<jQ0ACESB Gc=@?@<XI|YZ?@<>ACLN®PbYB9GJPbYZ?@<>ACL[SG¦[a<>=@Q0GJB9=@<>ACL4[]\ ESLHGvh9R.Y>LHGQ~TJB@<>A*[a<>DMEHGK[SGKeS<XPXP>Gc=6=@QSRHTJB@<>DMEHGc=4Gc=@?4ES?@<XPX<>=9TcGpm;%YC='sts®zw¤ijAC<XB6P>G|h'R.YZQS<X?@B9Gm£oNYZP>ACB9=

cg(ω) = c0

√1 −

(ωωc

)2

GJ? ∂τ−(ω)∂ω = τa(ω)

|ω|1r

1−“

ωωc

”2> 0

Q~ACESBw?9ACES?9Gc=wP>Gc=B9TcDMEHGJLHhcGc=O8 P GJLW[STchcACESP>GDFE\mGJL"YZEStIKGJLF?'YZLM?ωEH=9DME\s

ESLHGB9TcDFEHGJLHhcGK¡HL.YZP>Gωf

=@E_r|='YZIVIKGJLF?6R.YZES?9GtNPbYhcACLH[a<X?@<>ACLτfm = n

(2πωf

) τ−(ωf )

Gc=9?B9TOYZPX<>=9TcGtq*GJP>ACLPd\mTcDFE.YZ?@<>ACLWkakjyt£oNtPbY%?@B'YZLH=°ACB@I|YZ?@<>ACLk[aEKQSB9AZ¡HP8[STJIKA*[aESP>T:wg"[SG ∂f(θ)

∂θ

sf(θ)

[SGJiMB'YZ<X?¾J?@B9G,ACeH=9GJB@ijTcGYZES?9ACESB[SGPbYwB9TcDFEHGJLHhcG,hJB@<X?@<>DFEHG

ωcrNjAi

τm(ωcr) ∼ τ−(ωcr) ∼ τa(ωcr)(1 − cg(ωcr)

c0

) Gc=9?¦ijTJB@<¢¡.TcGtF¦\mACeH='GJB@iCYZ?@<>ACLk[SGhcGJ?@?9G?@B'YZLH=°ACB@I|YZ?@<>ACLVQSB9AMhJESB9G:[SGc=<XLa°ACB@I|YZ?@<>ACLH=¦=@ESBPd\^YZeH=9ACB@QS?@<>ACLfGJ?PbY,[a<>=9Q~GJB9=@<>ACLK[]\mACLH[SGc=vYChcACEH=@?@<>DFEHGc=¾;ºg`P dYZES?wLHAC?9GJB2DMEHGPbYf?@B'YZLH=°ACB@I|YZ?@<>ACL[aEWQSB9AZ¡HP¦wg·Gc=@?YChchcACIVQ.YZtLHTcGQ.YZB%ESLHGVYZEStIKGJLF?'YZ?@<>ACL=9<XtLS<¢¡.hOYZ?@<XijG[SG%Pd\^YZIVQSPX<X?@EH[SG[aE=9<XtL.YZPQSB9ACQ~ACB@?@<>ACLSLHGJPXP>G<s

τ(ωf )

τ−(ωf ).τm(ωin)

τ(ωin)=

1

1 − cg(ωin)c0

τm(ωin)

τ(ωin) 1 .

¯wEQ0AC<XLF?V[SGliMEHGfQSRFa=@<>DMEHGtNPd\ <XLF?9TJtB'YZ?@<>ACL½YC[S[a<X?@<>ACLSLHGJPXP>G[aE±=@<XtL.YZPwgN [HYZLH=P>GlhOYC=AiPbYg[a<>=@Q0GJB9=@<>ACL´[SGiM<X?9Gc='=9G|Gc=@?6<XIVQ0ACB@?'YZLF?9GtNGc=@?4PX<>TcG sPd\^YZeH=9GJLHhcGk[SGf=9MLHh'RSB9ACLS<>='YZ?@<>ACL²GJLF?@B9GlPbY=9ACESB9hcG|LHACL´PX<XLHTOYZ<XB9Gk[aE´=@<XtL.YZP[STJIKAM[aESP>T ∂

∂x〈Wωb〉ACLH[SGc= ;hgDMES<8=9G:QSB9ACQ.YZjGJLF?>s2PbYiM<X?9Gc='=9G

cg(ω)£oNMGJ?vP>G=@<XtL.YZP0[STJIKA*[aESP>TPXES<¢_dIKJIKG

VΩbDFES<*=9G¦QSB9ACQ.YZjGwsPbYi*<X?9Gc=9=9Gc0£o®¥LdYZ<X?ON

τ−(ω)Gc=9?ESLHGvIKGc=@ESB9G¾[SGvPbY:=9TJQ.YZB'YZ?@<>ACL?9GJIVQ0ACB9GJPXP>GvGJLM?@B9G P>Gc=B9ACLF?9=

[]\mACLH[SGc=YChcACEH=@?@<>DFEHGc=;hgGJ?vwgDFES<~Gc=@?vTJ?'YZeSPX<>G2YcitYZLF?Pd\^YZ?@?9TJLME.YZ?@<>ACL[aE|Q.YCDMEHGJ? []\mACLH[SG%;hg·ht\mGc=@?_s®_`[a<XB9G2YZEe~ACES?:[]\ ESL?9GJIVQH=[SG%Pd\mACB9[aB9G[aE?9GJIVQH=,hOYZB'YChJ?9TJB@<>=@?@<>DMEHG[]\^YZ?@?9TJLFE.YZ?@<>ACL

τ(ω)£o

qMESBPbYk¡HtESB9Ggka^j£Gc=@?QSB9Tc=9GJLM?9TcG|PbY?@B'YZLH=ACB@I|YZ?@<>ACL´[aE=@<XtL.YZP [STJIKAM[aESP>TfQSB9Tc[a<X?9GKQ.YZB4PbY=9ACPXES?@<>ACL[SGPd\mTcDFE.YZ?@<>ACLkak À £o*Gc=QSB9AZ¡HP>=ACeS?9GJLFEH=LFESIKTJB@<>DMEHGJIKGJLF?,='ACLF??@B'YChcTc=:GJLl°ACLHhJ?@<>ACL[aE?9GJIVQH=LHACB@I|YZPX<>=9T

t/τmQ~ACESBwhJ<XLHDliCYZP>GJESB9=w[SGPbYV¤B9TcDMEHGJLHhcGLHACB@I|YZPX<>='TcG4[SGPd\mACLH[SG4[SGQ~ACIVQ.YZjGtω/ωc = 0.01, 0.1, 0.2, 0.4, 0.8

YACLHhJ?@<>ACL[SGwIKA*[aESPbYZ?@<>ACL

f(θ)Gc=@? ESLHG A YZEH=9=@<>GJLSLHG%[SG%[aESB9TcGwhOYZB'YChJ?9TJB@<>=9?@<>DFEHG2TJFYZP>Gs4mQ0TJB@<>A*[SGc=[SGPd\mACLH[SG

;ºgYghcACLH=@?'YZLM?9GC[aE´?9GJIVQH=KhOYZB'YChJ?9TJB@<>=@?@<>DMEHG[]\^YZeH=9ACB@QS?@<>ACL

τa(ω) = C/ωGc=@?Kh9RHAC<>=9<>GkQ~ACESBV='YZ?@<>=dYZ<XB9G

Pd\ <XLHTJFYZPX<X?9TK<XLS<X?@<bYZP>Gτm τ−

]ACESBESLHGB9TcDMEHGJLHhcG°YZ<XeSP>GV[]\mACLH[SGf[SGQ~ACIVQ.YZjGf[SG0.01

NP>GVQSB9AZ¡HP¾GJLPX<XtLHG

5 !"#%$%&'($)*+,&-. /*

−5 −4 −3 −2 −1 0 1 2 3 4 5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

|Zb~Nzpt/τm

h Nn ¤2ª¨¦ 9§¬¤¯xª©%«%¬¨Á²±(¦« ;¨ªµE²´¥¯x¬²±µn¥»£¤2¥2²«©:¢£_ª¤ µnªl¦¬µ ±©%«¬¨ ²´¢xµ2· ¥2°±ª©%«¬¨ 5 ¹ 7<£_¬±¤¸²« º¥¤B¢Ü©:¢¦ 9p¤B¥2°±¢¨¢¦³²´¢b£_¬¯º£_ª ;´¢ ;,²´¢¶µ ª B¬±!¤B¢<¨¬«&¤B¢:3µnª¬±!¤B¢>;¤p«&¦5µnª«&¤+%

ω/ωc = 0.01, 0.1, 0.2, 0.4, 0.8¹

LHAC<XB9Gl¤ijAC<XBPbYV¡HtESB9Glka^F£@£:Gc=@?:QSB9ACQ0ACB@?@<>ACLSLHGJPEsVPbYK[STJB@<XijTcGQSB9GJIV<>UJB9G[SGPbY°ACLHhJ?@<>ACL[SGIKAM[aESPbYZ?@<>ACL ∂f(θ)∂θ

ACB9=9DFEHG PbYB9TcDFEHGJLHhcG[SG Pd\mACLH[SG[SGvQ0ACIVQ.YZjGGc=@?¦YZEStIKGJLF?9TcGtNFGJL6FYZB9[HYZLM?¦hcACLH=@?'YZLF?P>G LHACI4eSB9G

n[SG Q~TJB@<>A*[SGc=

[SGPd\mACLH[SG%;ºghcACLM?9GJLFEHGc=:[HYZLH=vPbY4[aESB9TcG,[SG,PbYIKA*[aESPbYZ?@<>ACLN*PbY?@B'YZLH=9<X?@<>ACLl[SG ∂f(θ)∂θ

sf(θ)

Gc=9?¾B9TOYZPX<>=9TcG<tFQ0ACESBPbY2¤B9TcDMEHGJLHhcG,LHACB@I|YZPX<>=9TcG,[]\mACLH[SGw[SG,Q~ACIVQ.YZjG

ω/ωc = 0.1GJL|PX<XtLHG,tB@<>=¦°ACLHhcTtNOEH=9DME\s

ω/ωc = 0.4GJLfPX<XtLHG

tB@<>=hJPbYZ<XBOHES?9ACESB:[SGPbY4¤B9TcDMEHGJLHhcGω/ωc = 0.6

NHP>G%QSB9AZ¡HP Gc=@?QSB9ACQ~ACB@?@<>ACLSLHGJP,sf(θ)

* Y?@B'YZLH=@<X?@<>ACL[SG ∂f(θ)∂θs

f(θ)[aEHGhsPbY[a<>=@Q0GJB9=@<>ACLl[SGi*<X?9Gc=9=9G:Gc=9?v[SACLHhwACeS?9GJLFEHG2YcijGch,ESLHGwYZEStIKGJLM?'YZ?@<>ACL[SG:PbY2¤B9TcDMEHGJLHhcG,[SGPd\mACLH[SG

[SG%Q0ACIVQ.YZjG;hg ACESBhcACLHhJPXESB9G%=@ESB Pd\^YZL.YZPX*=9Gw[SG,PbYhcACLM?@B@<XeSES?@<>ACLYZEl=9<XtL.YZP~[STJIKA*[aESP>TwQSB9AOijGJL.YZLF?,[SGPbYQ.YZB@?@<>Gwe.YZPX<>=@?@<>DFEHG

[aElh9R.YZIVQYChcACEH=@?@<>DMEHG,;hgNj<XPHdYZES?v=9<XtL.YZP>GJB DMEHGVΩb

hcACLM?@<>GJLF?:hcGJB@?'YZ<XLHGJIKGJLF?[SGPd\ <XLaACB@I|YZ?@<>ACL=@ESBvPd\^YZ?@?9TJLFE.Y®_?@<>ACLf[SGc=¾ACLH[SGc= ;ºgNtIKJIKG:[HYZLH=vP>G:[SACI|YZ<XLHG,[SGB9TcDFEHGJLHhcGc=

Ω 1τ−(ω)

kaktF£oj GJQ~GJLH[HYZLM?ONM[HYZLH=¾Pd\mTcDFE.YZ?@<>ACLkakjyt£oN~PbY|?@B'YZLH=@<X?@<>ACL"[aEWB9TJt<XIKG

τm τ+(ω)YZEgB9TJt<XIKG

τm τ+(ω)Gc=9?w=@ESB@?9ACES?YChchcACIVQ.YZtLHTcGQ.YZB%ESLHG

itYZB@<bYZ?@<>ACL[SG%Pd\^YZIVQSPX<X?@EH[SG4[aE=@<XtL.YZP='YZLH=?@B'YZLH=@ACB@I|YZ?@<>ACLg=@<XtLS<¢¡.hOYZ?@<XijG[SG%Pd\mGJLMijGJP>ACQSQ~GtGIHEH&G u[UcdaIORT.¿L E½a ]XOEQ ¿L'`RLfY E½QÁOR½>T.M½ QSOEQ ¿)a ]RORQÁORQÁ`ba c"`.LfeQ

YkhcACLF?@B@<XeSES?@<>ACLVΩd

YZE=@<XtL.YZP?9AC?'YZPVΩ = VΩb + VΩd

Q.YZB[STJIKA*[aESPbYZ?@<>ACL[aEh'R.YZIVQYChcACEH=@?@<>DMEHGK;hg[a<¢§0EH=@<¢v[a<¢§]UJB9GV[SG

VΩb=9GJESP>GJIKGJLM?Q.YZB2ESLWIESPX?@<XQSPX<>hOYZ?9GJESB6=@Q~GchJ?@B'YZP¦YC[S[a<X?@<>ACLSLHGJP τ−1

s (ω)„τ−1a (ω)+D(ω)Ω2

c20−iΩ

«¤ijAC<XB

P>Gl=9GchcACLH[?9GJB@IKGk=9ACEH=Pd\^YChchcACPbYC[SGk[HYZLH=6P>Gc=TcDFE.YZ?@<>ACLH=kakkF£GJ?kak ¿ £@£o ¥¦LES?@<XPX<>=YZLF?PbYLHAC?'YZ?@<>ACLkanmyj£oNPd\mTcDME.YZ?@<>ACLkak ¿ £ Gc=9?B9TcTchJB@<X?9G=9ACEH=PbY4°ACB@IKGxt

VΩ = VΩb

1 +1

τs(ω)τa(ω) + d

(cg(ω)

c0

)2(Ωτs(ω))2 − iΩτs(ω)

.kaktxj£

¯2YZLH=hcGJ?@?9GTcDME.YZ?@<>ACLkaktxF£oNd ∼ 1

GJ? ( cg(ω)c0

)2≤ 1

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τa(ω) τs(ω)ACEg=9<

τm ∼ 1Ω τs(ω)

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¹ #º¨ªµ ¦§¢5²´¢p¦5²« º¥¤B¢Ü©¦»¤B¥ ;«¯¸¢p¦ 5

YESL²='GJLH=4QSRM*=9<>DFEHGkhJPbYZ<XBltVΩd

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£oNACE=@<¾PbY[a<¢§~EH=@<>ACLL\^YQ.YC=4PX<>GJEQ0GJLH[HYZLF?6Pd\^YChJ?@<>ACL´[aEQ.YCDMEHGJ?4[]\mACLH[SGl;ºg=9ESBP>GKIV<XPX<>GJELHACL"PX<XLHTOYZ<XB9G

τm ∼ 1Ω τs(ω)

£o¯%YZLH=P>Gc=2[SGJEan"hOYC=N0PbYkhcACLF?@B@<XeSES?@<>ACL[SGKPbYlQ.YZB@?@<>GVe.YZPX<>=@?@<>DMEHGV[aEh9R.YZIVQYChcACEH=@?@<>DMEHG4;hg²[SACIV<XLHGt0YfhcACLH[a<X?@<>ACLgLHTchcGc='='YZ<XB9GQ0ACESB%ACeH=9GJB@ijGJBwGonaQ0TJB@<XIKGJLF?'YZP>GJIKGJLF?

VΩdQ~GJES?2YZP>ACB9=%J?@B9G

ACB@I4ESP>TcGhcACIVIKGτs(ω) ≤ τa(ω)

GJ?τs(ω) ≤ 1

Ω

a¥L=@ESQSQ0At='YZLF?:DME\mGJLjTJLHTJB'YZPdNaPd\ <XLHTJFYZPX<X?9T 1τa(ω) 1

τs(ω)

Gc=@?ijTJB@<¢¡.TcG¤Q.YZB9hcG:DMEHG 1

τa(ω) ∼ ωNjYZP>ACB9=¦DMEHG 1

τs(ω) ∼ ω4 £oNC<XPHGc=@?Q0At=9=@<XeSP>G[SG=9TJQ.YZB9GJBtB ChcGºs%[SG ACB@?9Gc=<XLHTJFYZPX<X?9Tc=P>G2=@Q0GchJ?@B9G2[SG2Pd\mTcDFE.YZ?@<>ACLkaktxF£GJL?@B9AC<>=:B9TJt<>ACLH=hOYZB'YChJ?9TJB@<>=@?@<>DFEHGc=

VΩ = VΩb ×

τa(ω)τs(ω)

=@<Ω 1

τa(ω) 1τs(ω) ,

1−iΩτs(ω)

=@< 1τa(ω) Ω 1

τs(ω) ,

1=@< 1

τa(ω) 1τs(ω) Ω .

kaktsj£

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VΩ =

τa(ω)τs(ω) VΩb

=@<τm τa(ω) τs(ω) ,

τmτs(ω)

∫ θ−∞ VΩb(θ

′)dθ′=@<

τa(ω) τm τs(ω) ,

VΩb=@<

τa(ω) τs(ω) τm .

ka ¿ j£

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τa(ω) τm τs(ω)

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¦\ <XLHTJFYZPX<X?9Tτa(ωin) τs(ωin)

Gc=@?=@ESQSQ0At=9TcGíjTJB@<¢¡.TcGds¸PbYµB9TcDFEHGJLHhcG²<XLS<X?@<bYZP>Gωin

N=@<KhcGJPXP>Go_`hJ<lGc=@?=@E_r|='YZIVIKGJLM?QSB9AMh'RHGg[SGPbY¤B9TcDMEHGJLHhcGW[SGhcACESQSESB9G

ωcpmt<bYtstsCzK¤Q.YZB9hcGWDMEHGP>Gg?9GJIVQH=khOYZB'YChJ?9TJB@<>=@?@<>DMEHG"[SG

[a<¢§~EH=9<>ACLτs(ω) ∼ 1

ω4

[a<XIV<XLMEHG:QSPXEH=i*<X?9G:YcijGch:Pd\^YZEStIKGJLF?'YZ?@<>ACLk[SGPbY,¤B9TcDMEHGJLHhcGDFEHGP>G?9GJIVQH=vhOYZB'YChJ?9TJB@<>=@?@<>DMEHG[]\^YZeH=9ACB@QS?@<>ACL

τa(ω) ∼ 1|ω|

£o¾ACB9=9DMEHGPbY"hcACLH[a<X?@<>ACLτa(ωin) τ in

m = nin

(2πωin

) τs(ωin)

Gc=@?KijTJB@<¢¡.TcGYZP>ACB9=ON P>GWQSB9AZ¡HP[aE¸=@<XtL.YZP[STJIKAM[aESP>T"Gc=9?QSB9ACQ~ACB@?@<>ACLSLHGJP5s ∫ θ

−∞ VΩb(θ′)dθ′

Y?@B'YZLH=9<X?@<>ACLijGJB9=kP>GgQSB9AZ¡HPVΩ ∼ VΩb(θ)

Q~GJES?YZP>ACB9=2J?@B9GVGo§]GchJ?@EHTcGK=@<XIVQSP>GJIKGJLM?GJLYZEStIKGJLF?'YZLM?PbYk[aESB9TcGτm

[aE"Q.YCDFEHGJ?[]\mACLH[SG|;ºgGWQ.YC=9='YZjG"[]\ ESL¸LHACI4eSB9G

nin[SGWQ0TJB@<>AM[SGc=;hg»hcACLF?9GJLMEHGc=[HYZLH=P>GWQ.YCDMEHGJ?[]\mACLH[SGsESL¸LHACIeSB9G¡HL.YZP

nfNQ~ACESBP>GcDMEHGJPPd\ <XLHTJFYZPX<X?9T

τfm = nf

(2πωin

) τa(ωin)

hcACIVIKGJLHhcG s´J?@B9G"ijTJB@<¢¡.TcGGc=9?[SACLHhGo§0GchJ?@EHTtj ACIVIKGPd\ <XLHTJFYZPX<X?9T

τ(ω) ≤ min(τa(ω), τs(ω))Gc=@?gB9TOYZPX<>=9TcGtNwYZP>ACB9=W[HYZLH=gP>Gc=W[SGJEan B9TJt<XIKGc=WhcACLH=@<>[STJB9Tc=ON

Pd\ <XLHTJFYZPX<X?9Tτm τ+(ω) ≥ τ−(ω)

Gc=@?lijTJB@<¢¡.TcGt¯%ACLHhtN ='GJP>ACL½Pd\mTcDME.YZ?@<>ACL ka ¿ F£oN [HYZLH=lP>Gc=f[SGJEanµB9TJt<XIKGc=VΩb ' 2

(τ(ω)τm

)∂f∂θ

0YK?@B'YZLH=9<X?@<>ACL[SGτa(ωin) τ in

m τs(ωin)s

τ fm τa(ωin) τs(ωin)

=9G4I|YZLS<¢Gc=9?9GYZP>ACB9=Q.YZB:ESLHG[STJB@<XitYZ?@<>ACL[aEQSB9AZ¡HP<XLS<X?@<bYZPwg

!"#%$%&'($)*+,&-. /*

−5 −4 −3 −2 −1 0 1 2 3 4 5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

|Zb~Nzpt/τm

h NC ¤Zª¨¦ 9§¬¤p¯xª©%«¬¨l²±¦p« ;¨ªµf²´¥¯x¬²±µ ¥.£¤B¥2²«%©:¢.£_ª¤ºµnª<¦¬µ ±_©%«%¬¨P²´¢hµB· ¥2°§±_ª©%«¬¨ 5 ¹ 7¹8 ¢p¦º¨¬¯ ¤B¢p¦²!¢º£¥¤«¬²´¢p¦¸²´¢³¦p«¨±´¦$! ?XB¬¯º£¤«&¦§¢p¦x²ª¨¦³µ ¢ ©:¢¯º£U¦ ²´¢ ¯x¬²±!µnª©%«¬ü¦¬Ü©>²´¢ µ ª µ « ;¨¢³¨¬«&¤B¢ 3lµ ª µ « ;¨¢:;¤p«&¦§µ ª«¤¤B¢p¦£¢B§©%«/.¢¯¸¢Ü© ;

n = 5, 100, 500, 5000¹ω/ωc = 0.9

¹

VΩi ' VΩdi ' 2f(θ) ,

VΩf ' VΩdf ' 2τa(ωin)

τmf

∂f

∂θ.

ka ¿ m£

Y|?@B'YZLH=@ACB@I|YZ?@<>ACL[aEWQSB9AZ¡HP¦[aE"=@<XtL.YZP[STJIKAM[aESP>TV[SAC<X?2J?@B9GACeH=9GJB@ijTcGQ0ACESBncr

N0='YZ?@<>=dYZ<>='YZLM?wPbYlhcACLH[a<X?@<>ACLncr

(2πωin

)∼ τa(ωin)

GJ?VGc=@?KYChchcACIVQ.YZtLHTcG¤ijAC<XBVPd\mTcDME.YZ?@<>ACL ka ¿ m®£@£Q.YZBESLHG[a<XIV<XLFES?@<>ACL·[SGlPd\^YZIVQSPX<X?@EH[SG[aEW=@<XtL.YZPQSB9ACQ~ACB@?@<>ACLSLHGJPXP>Gxs τf

mτa(ωin) 1

0¦\mTJitYZPXE.YZ?@<>ACL"[SG6hcGJ?@?9G4?@B'YZLH=°ACB@I|YZ?@<>ACL[SACLSLHG4[SGc=w<XLa°ACB@I|YZ?@<>ACLH==9ESB|P>G?9GJIVQH=k[]\^YZeH=9ACB@QS?@<>ACL

τa(ω)[SGc=fACLH[SGc=kYChcACEH=@?@<>DMEHGc=l;hg¾GBBvCP>G[aEµQ.YZB'YZIKUJ?@B9G τa(ω)

τm

[HYZLH=fhcGJ?@?9G?@B'YZLH=@ACB@I|YZ?@<>ACL´DFES<YPX<>GJEGJLF?@B9G[SGc=:B9TJt<XIKGc=,[SACIV<XLHTc=Q.YZB:PbYV[STJIKA*[aESPbYZ?@<>ACL[]\ ESLh'R.YZIVQgYChcACEH=@?@<>DMEHG;hg[a<¢§0EH=@<¢o£%Gc=9?4=@<XIV<XPbYZ<XB9GW[aEQ0AC<XLF?[SGKi*EHGKQSRM*=9<>DFEHG£4YZEBBvCP>Gl[aEQ.YZB'YZIKUJ?@B9G τR

τL

[HYZLH=Pd\mGonShJ<X?'YZ?@<>ACLÁCQS?@<>DFEHG[]\ <XIVQSESP>=9<>ACLH=YChcACEH=9?@<>DFEHGc=6Q.YZBjTJLHTJB'YZ?@<>ACL±[SGfQSPbYC=@I|YGJP>GchJ?@B9ACLa_d?@B9ACE´[HYZLH=4P>Gc==9GJIV<¢_`hcACLH[aEHhJ?9GJESB9=p A EH='stx®z` hJ<dN

τLGc=@?%PbYl[aESB9TcGV[SG6Pd\ <XIVQSESP>=@<>ACLPbYC=9GJB2GJ?

τRGc=@?%P>G6?9GJIVQH=[SG6B9GchcACIeS<XL.YZ<>='ACL[aEWQSPbYC=@I|YkTJP>GchJ?@B9ACLa_d?@B9ACE

jTJLHTJB9TtqMESBPbY,¡HtESB9Gkaryj£oNtGc=@?QSB9Tc=9GJLM?9TcG:PbYw?@B'YZLH=°ACB@I|YZ?@<>ACLf[aEK=9<XtL.YZPSQ.YZB'YZIKTJ?@B@<>DFEHGQSB9Tc[a<X?9GQ.YZB¦PbY%=9ACPXES?@<>ACL|[SG

Pd\mTcDME.YZ?@<>ACL"kaktxF£oNM[HYZLH=vP>GwhOYC=vAilPbYhcACLM?@B@<XeSES?@<>ACLYZEl=9<XtL.YZP~[STJIKA*[aESP>TQSB9AiM<>GJLM?QSB@<XLHhJ<XQ.YZP>GJIKGJLF?[aEkh9R.YZIVQ;hg·[a<¢§0EH=@<¢@8 YfACLHhJ?@<>ACL"[SG6IKAM[aESPbYZ?@<>ACLGc=9?wESLHG6FYZEH=9=9<>GJLSLHGV[SG6[aESB9TcGhOYZB'YChJ?9TJB@<>=@?@<>DMEHG

τm = n(

2πω

) 0Gc=i*<X?9Gc=9=9Gc=|Q.YZB@?@<>hJESPbYZ<XB9Gc=[SGc=k=@<XtL.YZEan·=9ACLF?kLHACB@I|YZPX<>=9TcGc=vYB9TcDFEHGJLHhcGg[SGPd\mACLH[SGW[SGQ~ACIVQ.YZjGWGc=@?lh9RHAC<>=@<>GQSB9A*h9RHG4[SG2PbYB9TcDMEHGJLHhcG[SGhcACESQSESB9GtN

ω = 0.9ωcHGc=?9GJIVQH=,hOYZB'YChJ?9TJB@<>=@?@<>DFEHGc=%[SG[a<¢§0EH=@<>ACLgGJ?,[]\^YZeH=9ACB@QS?@<>ACL

='ACLF?:?9GJP>=:DME\ <XP>==YZ?@<>=ACLM?:Pd\ <XLHTJFYZPX<X?9Tτa(ωin) τm(ωin) τs(ωin)

Q0ACESBPbY6B9TcDFEHGJLHhcG2<XLS<X?@<bYZP>Gωin

GJ?:Q~ACESBP>GwLHACI4eSB9G2<XLS<X?@<bYZP[SGwQ~TJB@<>A*[SGc=:;hg[HYZLH=P>GwQ.YCDMEHGJ?:[]\mACLH[SG

ninaGc=hcACLH=@?'YZLF?9Gc=

C1GJ?

C2[SG

τa(ω) = C1ω

GJ?τs(ω) = C2

ω4

=9ACLM?[SACLHh[STJ?9GJB@IV<XLHTcGc=,tB ChcG5s6Pd\ <XLHTJFYZPX<X?9T<XLS<X?@<bYZP>Gt.ACB9=9DMEHG2P>G2LHACI4eSB9Gn[SG2Q0TJB@<>AM[SGc=;ºg´Gc=@?

YZEStIKGJLM?9TtN.PbYK[aESB9TcG[aEQ.YCDMEHGJ?,[]\mACLH[SG6YZEStIKGJLF?9G6GJ?:¡HLS<X?:Q.YZBw[SGJijGJLS<XB,=@ESQ0TJB@<>GJESB9G4YZE?9GJIVQH=w[]\^YZeH=9ACB@QS?@<>ACLτa(ωin)

j GchJ<~IKUJLHGsPd\ <XLHTJFYZPX<X?9Tw¡HL.YZP>Gτm(ωf ) τa(ωin) τs(ωin)

NaDFES<0hcACB@B9Gc=@Q0ACLH[ sPbY4hcACLH[a<X?@<>ACLQ~ACESBACeH='GJB@ijGJB ESLlQSB9AZ¡HP[STJIKA*[aESP>T,QSB9ACQ0ACB@?@<>ACLSLHGJP s ∂f(θ)

∂θ

*Y4?@B'YZLH=@<X?@<>ACL[SGf(θ)

s ∂f(θ)∂θ

Gc=@?YZP>ACB9= eS<>GJLACeS?9GJLFEHG

¹ #º¨ªµ ¦§¢5²´¢p¦5²« º¥¤B¢Ü©¦»¤B¥ ;«¯¸¢p¦ N

−5 −4 −3 −2 −1 0 1 2 3 4 5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

|b~Nzpt/τm

h f ¤Zª¨¦ 9§¬¤p¯xª©%«¬¨P²±4¦« ;¨ªµf²!¥¯x¬²±µn¥b£¤B¥2²«%©:¢ £ª¤ µ ª<¦§¬µ ±©%«¬¨ ²!¢ µB· ¥2°§±_ª©%«¬¨ 5 ¹ 7 ¬©:¢¨±¢5ª .¢2²« º¥¤2¢Ü©:¢p¦ 9¤2¥2°§±¢¨¢p¦ ;,²± ;¤«)¦¶µnª«¤5ª±l¨¬«&¤+%

ω/ωc = 0.9, 0.65, 0.3, 0.1¹

[HYZLH=hcGJ?@?9GVACQ0TJB'YZ?@<>ACL ACESBn = 5

P>GV=@<XtL.YZP¦Gc=@?2QSB9ACQ0ACB@?@<>ACLSLHGJP.sf(θ)

YZP>ACB9=DFE\ <XP¦=9GB'YZQSQSB9A*h9RHG|[SG ∂f(θ)∂θQ~ACESB

n = 5000

G,HEHCB hYLN]RM½*E½:a ] a ]RORQUMPORQÁ`ba c"`.LfeQSÀRLN¿:½M E½\TRQUM → O.½>T.M½ QUM

ACESB|ACeH=9GJB@ijGJBKPbYW?@B'YZLH=9<X?@<>ACL·[aE±B9TJt<XIKG[SACIV<XLHTQ.YZBKP>Gk?@B'YZLH=9Q~ACB@?|[a<¢§0EH=@<¢2[SGkPd\mTJLHGJB@t<>G[SGc=K;hgijGJB9=P>GVB9TJt<XIKGK[SACIV<XLHT|Q.YZBP>GV?@B'YZLH=@Q0ACB@?e.YZPX<>=@?@<>DMEHGK[SGc=4;ºgN]<XP¾Gc=9?2Q~At=9=9<XeSP>GK[SGKhcACIVIKGJLHhcGJBVYcijGchKPbYhcACLH[a<X?@<>ACL<XLS<X?@<bYZP>G

τa(ωin) τ inm = nin

(2πωin

) τs(ωin)

WYZ<>=2hcGJ?@?9G6AC<>=@_`hJ<dN8<XPGc=@?%ES?@<XP>G[SGV[a<XIV<XLFEHGJBPbYKB9TcDMEHGJLHhcG[SGPd\mACLH[SGK[SGVQ0ACIVQ.YZjG

ω='YZLH=h'R.YZLSjGJBP>GVLHACI4eSB9G

n[SGVQ0TJB@<>AM[SGc=;hgºhcACLF?9GJLMEHGc=4[HYZLH=P>GQ.YCDFEHGJ?[]\mACLH[SG

n = nin = const

£o0¯2YZLH=hcGKhOYC=ONτa(ω)

GJ?τm(ω)

ijACLM?4YZEStIKGJLF?9GJB4QSB9ACQ~ACB@?@<>ACLSLHGJPXP>GJIKGJLM? s 1ω

NYZP>ACB9=DMEHGτs(ω)

itY:YZEStIKGJLF?9GJBQSPXEH=B'YZQS<>[SGJIKGJLM?: ∼ 1ω4

£oN®hcG¦DMES<FitY<XLHTJiM<X?'YZeSP>GJIKGJLM?IKGJLHGJBYZEB9TJt<XIKGτa(ωf ), τs(ωf )

τfm = nin

(2πωf

) Q~ACESB,ESLHGB9TcDMEHGJLHhcG¡HL.YZP>Gω = ωf

q*GJP>ACLgPd\mTcDME.YZ?@<>ACL±ka ¿ M£oNHPbY|?@B'YZLH=ACB@I|YZ?@<>ACL"[aEQSB9AZ¡HP[STJIKAM[aESP>T[SG

VΩ =∫ θ−∞ VΩb(θ

′)dθ′ijGJB9=

VΩ ∼ VΩb(θ)[SGJiMB'YZ<X?¾J?@B9GACeH=9GJB@ijTcGtj PS°YZES?¦='G=9ACESijGJLS<XB¾DME\^YcijGch:PbY

hcACLH[a<X?@<>ACLτa(ωin) τ in

m = nin τs(ωin)N.Pd\ <XLHTJFYZPX<X?9T

τ inm τ+(ωin) τ−(ωin)

Gc=@?:ijTJB@<¢¡.TcGtN~GJ?P>GQSB9AZ¡HP<XLS<X?@<bYZPGc=@?[STchJB@<X?Q.YZB:PbYQSB9GJIV<>UJB9GGonaQSB9Gc=9=@<>ACL[SGPd\mTcDFE.YZ?@<>ACL´ka ¿ m®£oSACB9=9DMEHG2PbY6B9TcDFEHGJLHhcG[SG2Q~ACIVQ.YZjGGc=@?[a<XIV<XLFEHTcGtN~<XP Gc=9?,=@ESQSQ0At=9TDMEHGQ0ACESB,PbYV¤B9TcDMEHGJLHhcG2¡HL.YZP>G

ωfPbYK[a<>=@Q0GJB9=@<>ACLgGc=@?,=@<°YZ<XeSP>G4DFEHG4ht\mGc=@?wPd\ <XLHTJFYZPX<X?9T

τ+(ωf ) τ fm τ−(ωf )

QSPXES?BvC?:DFEHGwPd\ <XLHTJFYZPX<X?9Tτ+(ωf ) τ−(ωf ) τ f

mDMES<]Gc=@?ijTJB@<¢¡.TcGtSYZB:hcACLH='TcDFEHGJLM?ON

=9GJP>ACLPd\mTcDME.YZ?@<>ACLkakjyt£GJ?Pd\mTcDME.YZ?@<>ACLka ¿ j£oNSP>G%QSB9AZ¡HP¡HL.YZPwgGc=9?[STchJB@<X?:Q.YZB

V fΩ ' V f

Ωb 'τ(ωf )

τfm

∂f

∂θ.

ka ¿ kj£Gc= TcDFE.YZ?@<>ACLH=2ka ¿ j£¾GJ?wka ¿ kF£IKACLF?@B9GJLF?DMEHG:PbY?@B'YZLH=@<X?@<>ACLk[aE|B9TJt<XIKGw[SACIV<XLHT,Q.YZBvP>Gc=vACLH[SGc=;hg"[a<¢§~EH=9<XijGc=ijGJB9=P>G·B9TJt<XIKG·[SACIV<XLHT½Q.YZBP>Gc=ACLH[SGc=;hg e.YZPX<>=@?@<>DFEHGc=N4Y¸PX<>GJE¶Q~ACESB

ωcrNDMES<KQ~GJES?J?@B9G·Gc=@?@<XIKTcG½[SG

τ crm ≡ nin

2πωcr

∼ τs(ωcr).j GJ?@?9G?@B'YZLH=9<X?@<>ACL·=9GkI|YZLS<¢°Gc=@?9GQ.YZBKPbYW[STJB@<XitYZ?@<>ACL½[aE²=@<XtL.YZP,[STJIKAM[aESP>TGJ?|Q.YZB

Pd\^YZEStIKGJLF?'YZ?@<>ACLg[SGw='ACLYZIVQSPX<X?@EH[SG%QSB9ACQ0ACB@?@<>ACLSLHGJPXP>G5s τ(ωf)

τfm

1S¾\mACeH=9GJB@iCYZ?@<>ACL[SG2hcGJ?@?9Gw?@B'YZLH=@<X?@<>ACL[SACLSLHG

[SGc=<XLa°ACB@I|YZ?@<>ACLH==@ESBP>G2?9GJIVQH=hOYZB'YChJ?9TJB@<>=@?@<>DFEHGτs(ω)

[SGc=ACLH[SGc=:;hg

!"#%$%&'($)*+,&-. /*

qMESB PbY:¡HtESB9G2ka^xF£oNPbY?@B'YZLH=°ACB@I|YZ?@<>ACLV[aE=9<XtL.YZP*[STJIKA*[aESP>T[aEHG sPbY:?@B'YZLH=9<X?@<>ACL[aEB9TJt<XIKG[SACIV<XLHT Q.YZB PbY[a<¢§0EH=@<>ACL|[SGc=¾ACLH[SGc=v;hggijGJB9=¦P>GB9TJt<XIKG,[SACIV<XLHTQ.YZB¾PbY%QSB9ACQ.YZFYZ?@<>ACLfe.YZPX<>=@?@<>DMEHG[SGc=¦ACLH[SGc= ;ºgWGc=@?¾QSB9Tc=9GJLF?9TcGt YVhcACLH[a<X?@<>ACL<XLS<X?@<bYZP>G

τa(ωin) = C1/ωin τ inm = nin

(2πωin

) τs(ωin) = C2/ω

4in

GJ?:PbY¤B9TcDMEHGJLHhcG<XLS<X?@<bYZP>G[SGPd\mACLH[SGf[SGVQ0ACIVQ.YZjG

ωin = 0.9ωcN0Q~GJB@IKGJ?@?9GJLM?6[SG|[STJ?9GJB@IV<XLHGJBP>Gc=hcACLH=9?'YZLF?9Gc=

C1GJ?

C2]YkB9TcDFEHGJLHhcG

[SG4Pd\mACLH[SG[SGQ0ACIVQ.YZjGGc=@?%GJLH=@ES<X?9G[a<XIV<XLFEHTcGxs|Q.YZB@?@<XB%[SGωin = 0.9ωc

EH='DFE\sωf = 0.1ωc

GJLgFYZB9[HYZLF?2P>GLHACI4eSB9G

n[SG6Q0TJB@<>AM[SGc=%hcACLF?9GJLMEHGc=[HYZLH=wP>G6Q.YCDFEHGJ?2[]\mACLH[SGV;hg·hcACLH=@?'YZLM?O j GchJ<IKUJLHGxsKPbYlhcACLH[a<X?@<>ACLW¡HL.YZP>G

τa(ωf ), τs(ωf ) τ fm = nin

(2πωf

) vY?@B'YZLH=9<X?@<>ACLµ[aEµQSB9AZ¡HPf(θ)

YZE½QSB9AZ¡HP ∂f(θ)∂θ

Gc=@?fACeS?9GJLMEHGgGJLM?@B9GPbYB9TcDMEHGJLHhcG2[SG2Q~ACIVQ.YZjG

ω = 0.9ωcGJLPX<XtLHGtB@<>=9GGJ?PbY6B9TcDFEHGJLHhcG

ω = 0.1ωcGJLPX<XtLHGLHAC<XB9Gt

j GJQ~GJLH[HYZLM?ONSPbY=@<[email protected]?@<>ACL[HYZLH=PbYCDMEHGJPXP>G2Pd\mTJ?'YZ?¡HL.YZPAiPd\ <XLHTJFYZPX<X?9Tτ+(ωf ) τ−(ωf ) τ f

mGc=@?ijTJB@<¢¡.TcGtN

Q0GJES?GJLQSB@<XLHhJ<XQ0GYZEH=9=9<6YcijAC<XBgPX<>GJE%¯2YZLH=hcGhOYC=ONPbY·[STJB@<XiCYZ?@<>ACL [aE QSB9AZ¡HPwg¼DMES<<XLM?9GJB@iM<>GJLM?"[HYZLH=PbY?@B'YZLH=9<X?@<>ACLGJLF?@B9G[a<¢§0EH=@<>ACLGJ?QSB9ACQ.YZFYZ?@<>ACLe.YZPX<>=@?@<>DMEHG|¤QSB9Tc[a<X?9G[HYZLH=Pd\mTcDFE.YZ?@<>ACLka ¿ M£@£ Q~GJES?:J?@B9GI|YC=9DFEHTcGQ.YZB,Pd\ <XLF?9TJtB'YZ?@<>ACL[aEgQSB9AZ¡HPwg²[aEHGYZEanGo§]GJ?9=,[SG6[a<>=@Q0GJB9=@<>ACL[SG4i*<X?9Gc=9=9Gf¤QSB9Tc[a<X?9=%[HYZLH=,Pd\mTcDFE.YZ?@<>ACL²kakjyj£@£o P:Gc=@?K[SACLHhe0GOYZEHhcACESQ²QSPXEH=f[a< r|hJ<XP>G[]\mGon*?@B'YZ<XB9G[SGc=V<XLa°ACB@I|YZ?@<>ACLH=f=@ESBVP>Gc=KQ.YZB'YZIKUJ?@B9Gc=f[aE²IV<XPX<>GJE·P>ACB9=9DFEHG[a<¢§]TJB9GJLF?9=Go§]GJ?9=lQSRFa=@<>DMEHGc=k<XLa³HEHGJLHhcGJLF?=@<XI4ESPX?'YZLHTJIKGJLF?P>G°ACLHhJ?@<>ACLSLHGJIKGJLF?[SGgPd\^YZLF?9GJLSLHGWQ.YZB'YZIKTJ?@B@<>DMEHGtACESB¾Pd\^YZL.YZPXa=9G:[SGhcGc=v=@<[email protected]?@<>ACLH=ONt<XP8Gc=@?LHTchcGc=9='YZ<XB9G,[SG[a<>=@Q~At='GJB¾[SGIKA*[SUJP>Gc=¦QSPXEH=QSB9TchJ<>=¾Q~ACESB¦PbY[STJQ~GJLH[HYZLHhcGGJLkB9TcDMEHGJLHhcG[SG%Pd\^YZeH=9ACB@QS?@<>ACL[SGc=ACLH[SGc=,YChcACEH=@?@<>DMEHGc=ONH[SG%PbYV[a<¢§0EH=@<>ACLGJ?[SG%PbYViM<X?9Gc=9='G2[SG%QSB9ACQ.YZFYZ?@<>ACL

Gc=fGo§]GJ?9=fQSB9Tc[a<X?9=l[SG[STJB@<XiCYZ?@<>ACL¸ACEµ[]\ <XLF?9TJtB'YZ?@<>ACL[aE·QSB9AZ¡HP%[STJIKA*[aESP>TP>ACB9='DFEHGP>Gc=fhOYZB'YChJ?9TJB@<>=@?@<>DMEHGc=[SGPd\mACLH[SG[SGQ0ACIVQ.YZjG

ωNτm£iCYZB@<>GJLM?ON¦Q0GJESijGJLF?lYZEH='=@<:J?@B9GI|YC=9DMEHTc=KQ.YZBKP>Gc=fGo§0GJ?9=|[SG[a<¢§~B'YChJ?@<>ACLµDFES<

YZQSQ.YZB'YZ<>='=9GJLF?[HYZLH=ESLHGhcACLa¡HtESB'YZ?@<>ACLjTcACIKTJ?@B@<>DMEHG?@B@<>[a<XIKGJLH=9<>ACLSLHGJPXP>GtGc=,hcACLH=9?'YZLF?9Gc=

CNC1

GJ?C2N.DMES<YZQSQ.YZB'YZ<>='=9GJLF?2[HYZLH=,P>Gc=,GonaQSB9Gc=9=@<>ACLH=w[SGc=Q.YZB'YZIKUJ?@B9Gc=%hOYZB'YChJ?9TJB@<>=@?@<>DMEHGc=

[]\^YZeH='ACB@QS?@<>ACLgACEg[SG[a<¢§0EH=@<>ACLNHQ0GJESijGJLF?wQSB9GJLH[aB9G4[SGc=,itYZP>GJESB9=,[a<¢§]TJB9GJLF?9Gc=%=9GJP>ACLPbYKhcACLa¡HtESB'YZ?@<>ACL"[aEIV<XPX<>GJEYZB"GonaGJIVQSP>GtN%=@<4PbY²QSB9Gc='=@<>ACL =@?'YZ?@<>DFEHGÝZQSQSPX<>DMEHTcG=9ESBPd\^YC=9='GJIeSPbYZjGtB'YZLFESPbYZ<XB9G±Gc=@?gitYZB@<>TcGtNwhcG=9ACLM?ghcGc=hcACLH=9?'YZLF?9Gc=¾DMES<.=9ACLM?¾IKA*[a<¢¡.TcGc=¦Q0ACESB¦QSB9GJLH[aB9GGJLfhcACIVQS?9G:PbY%itYZB@<bYZ?@<>ACLK<XLH[aES<X?9G[]\^YZeH='ACB@QS?@<>ACLlACE|[SG[a<¢§~EH=@<>ACL¯\^YZES?@B9Gc=%dYChJ?9GJESB9=GonaQ~TJB@<XIKGJLM?'YZEan"Q~GJESijGJLM?<XLF?9GJB@ijGJLS<XB6[HYZLH=hcGc=2hcACLH=@?'YZLM?9Gc=hcACIVIKGVPd\mTJ?'YZ?[SGV=9ESB°YChcGV[SGc=eS<XPXP>Gc=:Q.YZBGonaGJIVQSP>Gt

¹ ¡¥¦p±µ ©ª©¦<¢ )p£¥¤«&¯¸¢¨©ª± ) U

#%$ ?IF8356+9+F 3 1?-07=@3),+9:3G,H,HJ ½M`ba M½ E½W QVI`b[Y´½cdQU]GELN¿

¹,L[a<>=@Q~At=9<X?@<¢GonaQ~TJB@<XIKGJLM?'YZP []\ ES?@<XPX<>='YZ?@<>ACLg=9ACESQSP>GtNHDMES<]Q~GJB@IKGJ?,[SG%IKGJLHGJBQSPXEH=@<>GJESB9=? MQ0Gc=[]\mGonaQ0TJB@<>GJLHhcGc=Y±TJ?9Th'RHAC<>=@<t:QSPXEH=@<>GJESB9=? MQ0Gc=[SGI|YZ?9TJB@<bYZEantB'YZLFESPbYZ<XB9Gc="=9ACLM?[a<>=@Q0ACLS<XeSP>Gc=°YC='=9GJIeSPbYZjGc=W[SG"eS<XPXP>Gc=g[SGijGJB@B9GK[SG

2mmGJ?

0.15mm[SGK[a<bYZIKUJ?@B9G£oN[a<¢§]TJB9GJLF?9=³HES<>[SGc=='YZ?@ESB'YZLM?9=P>Gc=2eS<XPXP>Gc=N[a<¢§0TJB9GJLM?9Gc=4hcACLM?@B'YZ<XLF?9Gc=

=@?'YZ?@<>DMEHGc=ONv[a<¢§]TJB9GJLF?9Gc=[a<>=@?'YZLHhcGc=[SGgQSB9ACQ.YZFYZ?@<>ACLN[a<¢§0TJB9GJLM?9=?`*Q~Gc=k[]\mTJIKGJ?@?9GJESB9=GJ?[SGgB9TchcGJQS?9GJESB9=O¬ ACEH=hcGc=KQ.YZB'YZIKUJ?@B9Gc=fiCYZB@<bYZeSP>Gc=l[SGPd\mGonaQ~TJB@<>GJLHhcGACLM?fDME\ <XP±Y"ESL²tB'YZLH[·LHACIeSB9Gg[SGB9TJt<XIKGc= sTJ?@EH[a<>GJBO¾Gc=B9Tc=@ESPX?'YZ?9=VQSB9Tc=9GJLM?9Tc=|<>hJ<Q0ACB@?9GJLF?l=@ESBVPd\mTJ?@EH[SG[]\ ESLHGhcACLa¡HtESB'YZ?@<>ACLµQ.YZB@?@<>hJESPX<>UJB9G#tI|YZ?9TJB@<bYZE½tB'YZLFESPbYZ<XB9G=9GchLHACLa_`hcACLH=9ACPX<>[STtN0hcACLH=@?@<X?@EHT[SGeS<XPXP>Gc=%[SG4ijGJB@B9G4[SG6[a<bYZIKUJ?@B9G

0.15mmN]YOijGch6[SGc=whcACLF?@B'YZ<XLM?9Gc==@?'YZ?@<>DMEHGc=2YZPXPbYZLF?

[SG7.2 Pa

s72 kPa

°ACB9hcGc=¾=9?'YZ?@<>DFEHGc=¾YZQSQSPX<>DMEHTcGc=¾GJLM?@B9G1GJ?

10 kN£GJ?Q0ACESB¾[SGc=¾[a<>=@?'YZLHhcGc=v[SGQSB9ACQ.YZFYZ?@<>ACL

YZPXPbYZLF?[SG4s

30 cm.¯,<¢§]TJB9GJLF?9Gc=Q~At='=@<XeS<XPX<X?9Tc=:AZ§]GJB@?9Gc=:[HYZLH=P>G2h9RHAC<¢n[aE?@B'YZLH=9[aEHhJ?9GJESB[SG%B9TchcGJQS?@<>ACLACLF?TJ?9T

?9Gc=@?9TcGc= tYChchcTJP>TJB9ACIKUJ?@B9GtN?@B'YZLH='[aEHhJ?9GJESBESPX?@B'YC=9ACLHACB9GQS<>G>OAZ_`TJP>GchJ?@B@<>DMEHGK<>[SGJLF?@<>DMEHG¸slPd\mTJIKGJ?@?9GJESBON0¡HPXI¦w¯ gNiM<XeSB9ACIKUJ?@B9GPbYC=9GJBO¦¦\mTJIV<>=9=9<>ACL½TJ?'[email protected]|[SGc=|?@B'YZLH=9[aEHhJ?9GJESB9=KQS<>G>OAZ_`TJP>GchJ?@B@<>DMEHGc=lP>ACLSt<X?@EH[a<XL.YZEan·[SG?`*Q~G2YZL.YZIKGJ?@B@<>hc=w ¿ À k¤[email protected][SGhcGJLF?@B9TcG4=@ESB

100 kHzNS[SGB'YcjACL ∼ 2 cm

£oYhcACLSL.YZ<>=9=YZLHhcG[SG%PbYB9TJQ0ACLH=9G2GJLk¤B9TcDMEHGJLHhcG2[aE?@B'YZLH=9[aEHhJ?9GJESBw[SG%B9TchcGJQS?@<>ACLGc=@?LHTchcGc=9='YZ<XB9GQ~ACESBYOijAC<XB

YChchcUc=YZEº=9<XtL.YZP2YChcACEH=9?@<>DFEHGgQSB9Tc=9GJLM?[HYZLH=kP>GIV<XPX<>GJEGJ?kQ0ACESBfQ0ACESijAC<XBhcACIVQ.YZB9GJBhcGg[SGJB@LS<>GJBYZEºIKA*[SUJP>G?@RHTcACB@<>DFEHGg[STJijGJP>ACQSQ0T[HYZLH=khcGh'R.YZQS<X?@B9GtGc=f¡HPXIK=f%¯ g¶[a<>=@Q0ACLS<XeSP>Gc=kYZE½PbYZe0ACB'YZ?9AC<XB9GACLM?fESLHGB9TJQ0ACLH=9GDFE.YC=9<¢_dQSPbYZ?9G|=@ESB4ESLHGKPbYZB@jGKFYZIVIKGl[SGVB9TcDMEHGJLHhcGc=I|YZ<>=4=@ESQSQ~ACB@?9GJLM?4I|YZPvPbYhcACLM?@B'YZ<XLF?9G=@?'YZ?@<>DMEHGK=@ESB4P>GJESB9=TJP>GchJ?@B9AM[SGc=j ACESB ESLHGES?@<XPX<>='YZ?@<>ACL|¡8YZeSP>GtNFhcG? MQ0G:[SGB9TchcGJQS?9GJESBvLHGQ0GJES?¾=O\ ES?@<XPX<>=9GJB DFE\mGJL|=9ESB°YChcG:[aE|I|YZ?9TJB@<bYZEtB'YZLFESPbYZ<XB9GtNP)sAiP>Gc=6hcACLF?@B'YZ<XLM?9Gc=K=@?'YZ?@<>DMEHGc=6='ACLF?6P>Gc=QSPXEH=4dYZ<XeSP>Gc=ONP>ACB9=9DMEHGfP>Gc=4=9GJESP>Gc=hcACLM?@B'YZ<XLF?9Gc=V[HYZLH=6P>GIV<XPX<>GJE=9ACLM?[aEHGc=>s6PbYQ0Gc='YZLF?9GJESBO

¦\^YChchcTJP>TJB9ACIKUJ?@B9GYESLHG|B9TJQ0ACLH=9GlDFE.YC=9<¢_dQSPbYZ?9GfGJL¤B9TcDMEHGJLHhcGfI|YZ<>=YOijGch|ESLHGfB9GJPbYZ?@<XijGJIKGJLF?dYZ<XeSP>Gfe.YZLH[SGQ.YC=9='YZLM?9GK[SG2DFEHGJP>DMEHGc=

Hzs

20 kHz£GJ?ESLHG2=9GJLH=@<XeS<XPX<X?9T%PX<XIV<X?9TcGtH P0Q~GJES?J?@B9G%ES?@<XPX<>=9T2GJL=@ESBdYChcG2hcACIVIKG%P>G

¡HPXI¼%¯hgÝZ<XLH=@<DME\ <XIVIKGJB@jT[HYZLH=P>G%IV<XPX<>GJEUqaY6B9TJQ0ACLH=9GwQ~GJES?_`J?@B9GhcACLH=9<>[STJB9TcG2hcACIVIKG%¡8YZeSP>G%?'YZLF?DMEHGwPbYP>ACLStEHGJESB¦[]\mACLH[SGYChcACEH=@?@<>DMEHGGc=@?tB'YZLH[SG:[SGJitYZLF?¦=9Gc=[a<XIKGJLH=@<>ACLH=NthcGDFES<aPX<XIV<X?9G:=9ACLKYZQSQSPX<>hOYZ?@<>ACL!j GJQ0GJLH[HYZLF?ONhcG%B9TchcGJQS?9GJESB,Gc=@??@B9Uc=:ES?@<XP>G2Q~ACESB,hOYZPX<XeSB9GJB[SGc=B9TchcGJQS?9GJESB9=,QSPXEH=:=9GJLH=@<XeSP>Gc=:GJ?,DFES<=9ACLM?:ES?@<XPX<>=9Tc=:[HYZLH=PbY=@ES<X?9Gt

G|?@B'YZLH=9[aEHhJ?9GJESB4QS<>G>OAZ_`TJP>GchJ?@B@<>DMEHGlL\^YQ.YC=4[SGKB9TJQ0ACLH=9GKQSPbYZ?9GlGJLB9TcDMEHGJLHhcGtGl[SACI|YZ<XLHGf[]\ ES?@<XPX<>=YZ?@<>ACLGJLB9TchcGJQS?@<>ACLµ¤e.YC='=9Gc=2B9TcDFEHGJLHhcGc=

0 → 20 kHz£=9GV=@<X?@EHGf=@ESB2P>GKe0ACB9["<XLa°TJB@<>GJESB4[SGVPbYke.YZLH[SG|Q.YC=9='YZLM?9G|[aE

?@B'YZLH=9[aEHhJ?9GJESBOqaY:B9TJQ~ACLH='G¦GJL2°ACLHhJ?@<>ACL4[SGPbY B9TcDMEHGJLHhcGtN[HYZLH=hcG¦B9TJt<XIKGtNGc=@?IKACLHAC?9ACLHG EH='DFE\^YZEan2?@B9Uc=e.YC='=9Gc=¤B9TcDMEHGJLHhcGc=%¤IKAC<XLH= [SG

500 Hz£oM¾\ <XLF?9TJB9J?[SG,hcG?@B'YZLH=9[aEHhJ?9GJESBGc=@?v=Y2tB'YZLH[SG%=9GJLH=@<XeS<XPX<X?9TwGJ?¾PbYQ~At='=@<XeS<XPX<X?9T[SG

Pd\ <XIVIKGJB@jGJB2?9AC?'YZP>GJIKGJLF?6[HYZLH=%P>G4IV<XPX<>GJE`P=@ESQSQ0ACB@?9G?@B9Uc=%eS<>GJL"P>Gc=%QSB9Gc=9=@<>ACLH=2=@?'YZ?@<>DMEHGc=%<XIVQ~ACB@?'YZLM?9Gc=OuwACEH=YcijACLH= ES?@<XPX<>=9TQSB@<XLHhJ<XQ.YZP>GJIKGJLF?:hcG,?@B'YZLH=9[aEHhJ?9GJESBON*?9ACES? GJLfijTJB@<¢¡8YZLF?PbY2°[email protected]<X?'YZ?@<XijG%[SGc=v=9<XtL.YZEanfYcijGch,P>Gc=YZES?@B9Gc=hOYZQS?9GJESB9=O8¥LQ.YZB@?@<>hJESPX<>GJBON.QSPXEH=@<>GJESB9=wGon*Q0TJB@<>GJLHhcGc=DME.YZPX<X?'YZ?@<XijGc=,ACLM?IKACLM?@B9T4DMEHG2PbY6°ACB@IKG2?9GJIVQ0ACB9GJPXP>G[SG=@<XtL.YZEanéSB9Go=KYChchcTJP>TJB9ACIKTJ?@B@<>DMEHGc=|QSB9Tc='GJLF?9GJLF?fESLHG[STJB@<XijTcGYC[S[a<X?@<>ACLSLHGJPXP>GQ.YZBVB'YZQSQ0ACB@?¸sgPbY°ACB@IKG[SGc==@<XtL.YZEan<>=9=@EH=w[SGhcG4?@B'YZLH=9[aEHhJ?9GJESBO0 P Gc=@?w[SACLHh6YC=9=9<XIV<XP>T<sKESLghOYZQS?9GJESB%[SGi*<X?9Gc=9=9Gt8 ACESB2[SGc=wACLH[SGc=wQSPbYZLHGc=ON[HYZLH=lESL½IV<XPX<>GJEº='YZLH=k[a<>=@Q0GJB9=@<>ACLN¾PbYQSB9Gc=9=@<>ACL¸YChcACEH=@?@<>DMEHG

pQ0GJES?l=O\mGonaQSB@<XIKGJBkGJL·°ACLHhJ?@<>ACL¸[SGPbYiM<X?9Gc='=9G

Q.YZB@?@<>hJESPbYZ<XB9GwYChcACEH=9?@<>DFEHGtNp = ρc0 v = ρc0 ∂U/∂t

N*YZ<XLH=@<.DME\mGJLK°ACLHhJ?@<>ACLf[SG:PbY[SToACB@I|YZ?@<>ACLYChcACEH=@?@<>DMEHG[HYZLH=P>GIV<XPX<>GJE

p = −ρc20εN]Ai

ρGJ?

c0=9ACLF?2B9Gc=9Q~GchJ?@<XijGJIKGJLM?PbY[SGJLH=@<X?9TKYZEB9GJQ~At=GJ?2PbYli*<X?9Gc=9=9GV[SGQSB9ACQ.YZFYZ?@<>ACL

YChcACEH=@?@<>DMEHG[aEIV<XPX<>GJEqMESB PbY%¡HtESB9G ka^sSNjP>Gw=9h9RHTJI|Y4[aEk[a<>=@Q~At=9<X?@<¢]GonaQ0TJB@<XIKGJLF?'YZP][SGIKGc=9ESB9Gc= Gc=9?¾QSB9Tc=9GJLM?9Tt*Gc=v=@<XtL.YZEanfjTJLHTJB9Tc=

=9ACLF?%[SG4?`*Q~G6=@<XLFEH=%;hg±IKA*[aESP>T4GJL"YZIVQSPX<X?@EH[SGQ.YZB,ESLHG°ACLHhJ?@<>ACL A YZEH='=@<>GJLSLHGl¤ijAC<XB2[STJeSES?%[aEgh'R.YZQS<X?@B9Gxkj£o`P>=¾=9ACLF?¾GJLMijAOjTc=bswESLfYZIVQSPX<¢¡.hOYZ?9GJESB[SG:QSES<>=9=YZLHhcGYcijGchESLKFYZ<XL|[SG Àt¿ [ShJPbYC='=9G:2£oNtQSES<>=vYZEK?@B'YZLH='[aEHhJ?9GJESBQS<>G>OAZ_`TJP>GchJ?@B@<>[email protected][SGvhcGJLF?@B9T¾=9ESB

100 kHz[HYZLH=Pd\^YZ<XBONcQ.YZBPd\ <XLM?9GJB@IKTc[a<bYZ<XB9G[]\ ESLHG¦e0AC<X?@<>GJB[]\^YC[HYZQS?'YZ?@<>ACL

[]\ <XIVQ~Tc[HYZLHhcGfQ0ACESBESLHGKIKGJ<XPXP>GJESB9GlGr|hOYChJ<X?9Tt ,QSB9Uc=4B9TchcGJQS?@<>ACLQ.YZBP>G|?@B'YZLH=9[aEHhJ?9GJESBON P>Gc=4=@<XtL.YZEanwg=9ACLM?¡HPX?@B9Tc= QSES<>=YZIVQSPX<¢¡.Tc=:Q.YZBvESLkQSB9To_ YZIVQSPX<¢¡.hOYZ?9GJESBdYZ<XeSP>GeSB@ES<X?[SGwFYZ<XLlI|Y®n*<XI4ESI 4[S:£oj Gc==@<XtL.YZEanl=9ACLM?

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¹,LHGGonaQ~TJB@<>GJLHhcG<XLF?9TJB9Gc=9='YZLM?9G´hcACLH=@<>=@?9G s²IKAM[a<¢¡.GJBgP>GB9TJt<XIKG[SG?@B'YZLH=9Q~ACB@?g[SGPd\mACLH[SG[SGQ0ACIVQ.YZjGh9R.YZLSjGJIKGJLM?W[SGPd\^YZeH=9ACB@QS?@<>ACLNw[SGPbY²[a<¢§0EH=@<>ACLGJ?[SGPbY²iM<X?9Gc=9='G[SGQSB9ACQ.YZFYZ?@<>ACL8£GJLdYZ<>='YZLF?iCYZB@<>GJBPbY¤B9TcDMEHGJLHhcG[aEl=@<XtL.YZP0TJIV<>=ONF='YZLH= h9R.YZLSjGJB PbY%°ACLHhJ?@<>ACLk[SG:IKA*[aESPbYZ?@<>ACLkGJLkYZIVQSPX<X?@EH[SGta¯%YZLH= hcGJ?@?9GGonaQ0TJB@<>GJLHhcGtNPbYlB9TcDFEHGJLHhcGf[SGVQ~ACIVQ.YZjG|<XLS<X?@<bYZP>G|Gc=@?4[SG

fp = 60 kHzQSES<>=hcGJ?@?9GVB9TcDMEHGJLHhcG|Gc=@?4YZEStIKGJLM?9TcG|Q.YZB4Q.YC=[SG

20 kHzEH=9DME\s

fp = 300 kHz,j ACIVIKGVP>Gc==@<XtL.YZEan"GonaQ~TJB@<XIKGJLM?'YZEan=9ACLF?YlQSB@<>ACB@<¦B9GJEGJL[SGJRHACB9=[SGPbY

B9TJt<>ACL[]\ <XLF?9GJB'YChJ?@<>ACLLHACLPX<XLHTOYZ<XB9GtNaGJ?[HYZLH=P>G%h9R.YZIVQP>AC<XLF?'YZ<XLNaP>G%=@<XtL.YZP][STJIKA*[aESP>T2wgGc=@?[a<¢§0B'YChJ?9TtN*hcG%DMES<<XIVQSPX<>DFEHGlESLHGl[STJB@<XiCYZ?@<>ACL²=9ESQSQSP>TJIKGJLF?'YZ<XB9G[SGc==@<XtL.YZEanQ.YZBB'YZQSQ0ACB@?KYZEhOYC=ESLS<>[a<XIKGJLH=@<>ACLSLHGJP:[aEÍKA*[SUJP>GYZL.YZPXM?@<>DMEHG[STJijGJP>ACQSQ0T2QSB9TchcTc[SGJIVIKGJLF?p A EH='s ¿ N.u,Aiaxjyz`

¦\mACLH[SGw[SG:Q0ACIVQ.YZjG,Gc=9?¾<XLS<X?@<bYZP>GJIKGJLF?IKA*[aESP>TcGQ.YZBvESLHG:°ACLHhJ?@<>ACL A YZEH=9=@<>GJLSLHGt_j Gh9RHAC<¢nlGc=@?¾IKAC?@<XijT,Q.YZBPbY|=9<XIVQSPX<>hJ<X?9T6[SG4Pd\mACQ0TJB'YZ?@<>ACLW[SG4?@B'YZLH=°ACB@IKTcG[SG5gHACESB@<>GJBYZQSQSPX<>DMEHTcG sKESLHGACLHhJ?@<>ACL A YZEH='=@<>GJLSLHGtN~YZ<XLH=9<DMEHGQ.YZBPbY6°[email protected]?@<>DMEHG2B9GJPbYZ?@<XijGJIKGJLF?%=@<XIVQSP>G[SG2='Gc=:[STJB@<XijTcGc==@EHhchcGc='=@<XijGc=O

qMESBPbY¡HtESB9G4kanmHN=9ACLM?4?@B'YChcTc=4Q~ACESB6DFE.YZ?@B9G|B9TcDFEHGJLHhcGc=6[SGKQ0ACIVQ.YZjGf[a<¢§]TJB9GJLF?9Gc=fp = 60, 160, 220,GJ?

300 kHz£,I|YZ<>=6YcijGchVESLHGKIKAM[aESPbYZ?@<>ACL<>[SGJLF?@<>DMEHGtNP>Gc==@<XtL.YZEan"TJP>GchJ?@B@<>DMEHGc=4[SGQ0ACIVQ.YZjGKGJ?P>Gc==9<XtL.YZEan

[STJIKAM[aESP>Tc=%YC=9=9A*hJ<>Tc=ONSB9GJ?'YZB9[STc=%[aE?9GJIVQH=,[SGQSB9ACQ.YZFYZ?@<>ACLW[HYZLH=,P>GIV<XPX<>GJEtB'YZLFESPbYZ<XB9Gt0¹,LHG[STJB@<XijT4QSB9GJIV<>UJB9G[SG A YZEH=9=@<>GJLSLHGGc=@?QSB'YZ?@<>DFEHGJIKGJLM?,ACeH=9GJB@ijTcGQ~ACESB:P>G2=9<XtL.YZP[STJIKAM[aESP>T?@B'YChcT[HYZLH=PbYQ.YZB@?@<>G|°Yj£ [SGPbY4¡HtESB9GkanmaNahcACB@B9Gc=@Q~ACLH[HYZLM?ºs4ESLHG,¤B9TcDMEHGJLHhcG%[SG,Q0ACIVQ.YZjG

fp = 60 kHzaACESB

fp = 160 kHzNa[HYZLH=PbY4Q.YZB@?@<>GK¤e8£

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¬ GJIVQH=,[SG%ijACPτ0 ¤*; >®£ mtNk À aN ¿ x

10−4 =£ aNryCx 10−3 =£x ¤*; >®£ mtN ¿tÀ aN ¿ y

10−4 =£ aNrytyCs10−3 =£

m ¤a; >®£ mtN = À aN ¿tÀ 10−4 =£ aNrytyC

10−3 =£mk ¤a; >®£ mtN^ À aN ¿ =

10−4 =£ aNrytyC10−3 =£

m= ¤a; >®£ mtN^x À aN ¿ =10−4 =£ aNryty À

10−3 =£mO ¤a; >®£ kaN À aN ¿ =

10−4 =£ aNrytyCx10−3 =£

mOx ¤a; >®£ mtN^s À aN ¿t¿ 10−4 =£ aNryty2=

10−3 =£k ¤a; >®£ kaNk aN ¿t¿

10−4 =£ aNryty2=10−3 =£

kk ¤a; >®£ kaNry aN ¿t¿ 10−4 =£ aNrytyC

10−3 =£k = ¤a; >®£ kaN À aN ¿t¿

10−4 =£ aNrytyk10−3 =£

kt ¤a; >®£ mtN^s À aN ¿t¿ 10−4 =£ aNryCtx

10−3 =£ktx ¤a; >®£ mtN^s À aN ¿t¿

10−4 =£ aNryty 10−3 =£¿ ¤a; >®£ mtN^s À aN ¿t¿

10−4 =£ aNryCtx10−3 =£

f%U ª¤«%ª©%«%¬ü²´¢p¦ £_ª¤Zª¯ ©%¤B¢p¦ ²´¢¶µ2· ª±´¦©:¢¯¸¢Ü© ¢¨ 9§¬¨§©%«¬ü²´¢¶µ ª 9¤2¥2°§±¢¨¢x²´¢b£_¬¯>£ª1;´¢

[SGPbY%¡HtESB9G kanmaNjP>Gw=@<XtL.YZP8[STJIKA*[aESP>T,B9GJEfGc=@?vQSB9ACQ~ACB@?@<>ACLSLHGJPs2PbY[STJB@<XijTcG,='GchcACLH[SG,[]\ ESLHG A YZEH=9=@<>GJLSLHGtM¹LHG[STJB@<XitYZ?@<>ACLk[aEl=@<XtL.YZP0[STJIKA*[aESP>T,=O\mGc=9? [SACLHh,QSB9AM[aES<X?9GwGJLF?@B9G

fp = 60 kHzGJ?

fp = 160 kHz_qM<~PbY2B9TcDMEHGJLHhcG

[SG:Q~ACIVQ.YZjG6°YC=9=@<XIV<XP>TcG:<>hJ<s2PbYwB9TcDFEHGJLHhcGQ~ACB@?9GJEH='G:ACEKB9TcDFEHGJLHhcGhcGJLF?@B'YZP>Gw[aE|Q.YCDMEHGJ?¾[]\mACLH[SG,[SG:Q~ACIVQ.YZjG£Gc=9? GJLHhcACB9GYZEStIKGJLM?9TcGtNaESLHGw[STJB@<XijTcG%?@B9AC<>=9<>UJIKG2[SG A YZEH=9=@<>GJLSLHG2YZQSQ.YZB'Y>?Q~ACESB

fp = 220 kHz¤Q.YZB@?@<>GKh£v[SG

PbY%¡HtESB9GºkanmF£oj¥¦La¡HLN*YZES?9ACESB [SG:PbYwB9TcDFEHGJLHhcGfp = 300 kHz

NjESLHG[STJB@<XijTcG='GchcACLH[SGGc=@? ACeH=9GJB@ijT¤Q.YZB@?@<>G[.£[SG%PbY¡HtESB9G¶kanmF£oS,<XLH=@<dNS[SGJEan[STJB@<XiCYZ?@<>ACLH=,=@ES<XiM<>Gc=[]\ ESLHG2<XLF?9TJtB'YZ?@<>ACLW[aE=@<XtL.YZP[STJIKA*[aESP>T=9ACLM?ACeH=9GJB@ijTcGc=GJLYZEStIKGJLM?'YZLF?wPbY6B9TcDFEHGJLHhcG2[SG2Q0ACIVQ.YZjG[SG

fp = 60 kHzs

fp = 300 kHz

:¡HLk[SG%hcACLa¡HB@IKGJB:hcGJ?@?9GwACeH='GJB@iCYZ?@<>ACLDME.YZPX<X?'YZ?@<XijGtNaESLf?@B'YZ<X?9GJIKGJLM?:LFESIKTJB@<>DMEHG2=@<XIVQSP>G2Y6TJ?9Tw<XIVQSP>TJIKGJLF?9Tt¾\mACB9[aB9Gg[SGW[STJB@<XitYZ?@<>ACLº?9GJIVQ~ACB9GJPXP>GWQ.YZB@?@<>GJPXP>G

pYTJ?9TgES?@<XPX<>=9TgQ0ACESBlACeS?9GJLS<XBESLµQSB9AZ¡HP2?9GJIVQ0ACB9GJPwIKA*[STJPX<>=9T

[STJB@<XijTcGQ.YZB@?@<>GJPXP>G±[]\ ESLHG A YZEH=9=9<>GJLSLHG£P>GQSPXEH=gQSB9A*h9RHGQ~At='=@<XeSP>G´[SGc=QSB9AZ¡HP>=W?9GJIVQ0ACB9GJP>="GonaQ~TJB@<XIKGJLM?'YZEan0ACESBhcGdYZ<XB9GtN]PbYk?@B'YZLH=@ACB@IKTcGK[SG¸gHACESB@<>GJB4[SGVPd\mGJLMijGJP>ACQSQ~Gf[SGVIKA*[aESPbYZ?@<>ACLQ0GJES?J?@B9G|TchJB@<X?9GK=9ACEH=PbYlACB@IKGS(f) = TF [s(t)] = (−j2πf)pe−

f2τ2m

4 e−2πjfτ0Ai

τ0Gc=@?kP>GW?9GJIVQH=[SGWijACP[aE=@<XtL.YZPGJ?

τmP>GW?9GJIVQH=

hOYZB'YChJ?9TJB@<>=9?@<>DFEHG,[SGPd\mGJLFijGJP>ACQSQ0G[SGIKA*[aESPbYZ?@<>ACLg[aESB9TcG,hOYZB'YChJ?9TJB@<>=@?@<>DMEHG[aE|=@<XtL.YZP8[STJIKAM[aESP>T£oMG:Q.YZB'YZIKUJ?@B9GhOYZB'YChJ?9TJB@<>=9?@<>DFEHG

τmNES?@<XPX<>=9Tf[HYZLH=6P>GfhOYZP>hJESP[SG|IV<XLS<XIV<>='YZ?@<>ACL±Gc=@?GJLdYZ<X?ESLQ.YZB'YZIKUJ?@B9Gf¡SnaT|Q.YZB4PbYACB@IKG

[aE=@<XtL.YZP[SG%Q0ACIVQ.YZjGTJIV<>=Oa PGc=@?[SACLHhY6QSB@<>ACB@<hcACLSLFENSitYZB@<bYZeSP>GGonaQ~TJB@<XIKGJLM?'YZP>GJIKGJLF?ON8GJ?:=9GJP>ACLP>G%IKAM[SUJP>G[STJijGJP>ACQSQ0T4QSB9TchcTc[SGJIVIKGJLF?[HYZLH=%hcG4h'R.YZQS<X?@B9GtN~<XPGc=@? sKPbYVAC<>=whOYZB'YChJ?9TJB@<>=@?@<>DMEHG[aEg=@<XtL.YZP[SG4Q~ACIVQ.YZjG6GJ?w[aE=9<XtL.YZP~[STJIKA*[aESP>TtUj GJQ~GJLH[HYZLM?ONS[]\^YZES?@B9Gc=QSRHTJLHACIKUJLHGc=ONaLHACLkQSB@<>=GJLkhcACIVQS?9G%[HYZLH=P>G,IKAM[SUJP>G%[STJijGJP>ACQSQ0T2=9ACLF?=9EH=9hcGJQS?@<XeSP>Gc=:[SG=9G%I|YZLS<¢°Gc=@?9GJBONHGJ?:ht\mGc=9?:Q~ACESB:hcGJ?@?9GB'YZ<>=9ACLDMEHG2hcG2Q.YZB'YZIKUJ?@B9GGc=@?QSB@<>=hcACIVIKGY® EH=9?'YZeSP>Gt

Gc=B9Tc=9ESPX?'YZ?9=g[aE?@B'YZ<X?9GJIKGJLF?"LMESIKTJB@<>DFEHG´=9ACLM?QSB9Tc='GJLF?9Tc=W[HYZLH=gP>G?'YZeSP>GOYZE kanmQ0ACESBSmk±itYZP>GJESB9=g[SGB9TcDMEHGJLHhcGc=6[SGfQ0ACIVQ.YZjGlGJLM?@B9G

fp = 60 kHzGJ?

fp = 300 kHz A P>ACe.YZP>GJIKGJLF?ONESLHGf°YZ<XeSP>Gl[STchJB9AC<>=9='YZLHhcG

[SGc=Q.YZB'YZIKUJ?@B9Gc=[SGK?9GJIVQH=τ0

GJ?τm

Gc=@?ACeH=9GJB@ijTcGtGc=4iCYZB@<bYZ?@<>ACLH=6[SGKPd\mACB9[aB9Gf[SG|[STJB@<XitYZ?@<>ACLphcACLa¡HB@IKGJLM?

P>Gc=ACeH=9GJB@iCYZ?@<>ACLH=wDME.YZPX<X?'YZ?@<XijGc=B9TOYZPX<>=9TcGc=,=@ESB:PbY6¡HtESB9G<kanmSHACESBfp = 60 kHz

NHPd\mACB9[aB9G[SG[STJB@<XitYZ?@<>ACLpGc=9?

QSB9A*h9RHGw[SG¶mZmtk À £oNMQSES<>=¾<XP]YZEStIKGJLF?9G2YcijGch,PbY¤B9TcDMEHGJLHhcG[SG,Q~ACIVQ.YZjG<t!sfp = 160 kHz

Np ' 2.05

[STJB@<XijTcG='GchcACLH[SG£ EH='DFE\s

fp = 220 kHzAi

p ' 2.7¤QSB9AMh'RHG[]\ ESLHG4[STJB@<XijTcG?@B9AC<>=@<>UJIKG£oH¥¦La¡HLNH<XP[a<XIV<XLMEHGQ0ACESB:=9G

=9?'YZeS<XPX<>=9GJBYZES?9ACESB,[SG2Q0ACESB

fp = 300 kHz[STJB@<XijTcG=9GchcACLH[SG£o

qMESB6PbY¡HtESB9G kanmmjNP>Gc==@<XtL.YZEan[STJIKA*[aESP>Tc=V<>=9=9EH=6[SGl?@B9AC<>=4B9TcDFEHGJLHhcGc=V[SGlQ0ACIVQ.YZjG[a<¢§0TJB9GJLM?9Gc=°YcijGchPbYfIKJIKGGJLFijGJP>ACQSQ0GK[SG6IKA*[aESPbYZ?@<>ACL8£w=9ACLF?2=@ESQ0GJB@Q~At='Tc=%Y®¡HL[SGhcACIVQ.YZB9GJB%dYChJ<XP>GJIKGJLF?P>GJESB9=%?9GJIVQH=2[SG6ijACP>=OGc=,QSB9AZ¡HP>=2[SG6hcGc=%=@<XtL.YZEang='ACLF?%QSB9AMh'RHGc=2[SGPbYl[STJB@<XijTcG6?@B9AC<>=@<>UJIKG[]\ ESLHG A YZEH=9=9<>GJLSLHG4GJ?%LHG4iCYZB@<>GJLM?2Q.YC=w[SG

¹ ¡¥¦p±µ ©ª©¦<¢ )p£¥¤«&¯¸¢¨©ª± )

0 2 4 6

x 10−4

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

|Zb~,n

zE|Z~~|I|,&$ |ZzZ|

h f%U U« ;¨ª± )5²´¥¯ ¬*²±!µ ¥p¦,£_¬±!¤©%¤2¬«&¦ 9¤B¥2°±¢¨¢p¦²´¢,£_¬¯º£_ª1;´¢h²« º¥¤B¢Ü©:¢p¦ ;fp = 170, 200

¢§©230 kHz

¹8 ¢p¦l©:¢¯º£U¦P²´¢ .¬µº²´¢p¦ ¦« ;¨ª± )S²«¯³«&Ü±¢¨©µ ¬¤¦§°±¢µnª 9p¤B¥2°±¢¨¢#²´¢³£_¬¯º£_ª1;´¢ª± ;¯x¢Ü©:¢¹ 8 ª¯x¬²±µnª©%«%¬¨²± ¦« ;¨ªµh²!¢x£_¬¯º£_ª1;´¢ ¢p¦©³©%¤2ª¥B¢4£_¬±¤(²´¬¨¨¢¤±!¨f¢¤2¥C9¥¤B¢¨¢*¹

P0 ' 300 kPa% ¢©

d ' 12 cm5ª±©%¤B¢

B¬¨12 ;±!¤2ª©%«¬¨#¢ )p£¥¤p«¯¸¢Ü©ªµ ¢ °±¢<¢µµn¢£¤B¥p¦¢Ü©:¥B¢5¦±!¤¶µ ª 2G;±¤2¢ ¹.7¹

°YcACL<XIVQ0ACB@?'YZLF?9Gt¦Gk=@<XtL.YZP[STJIKA*[aESP>TYC=9=9A*hJ<>T4sPbY¤B9TcDMEHGJLHhcGk[SGfQ0ACIVQ.YZjGfp = 230 kHz

GonaRS<Xe~GlESL¤B9ACLM?IKACLF?'YZLM?VDFES<¾YZB@B@<XijGfGJLQSB9GJIV<>GJBON=9ES<XiM<¾[aE=@<XtL.YZP¾[STJIKA*[aESP>TK<>=9=@E[aEQ.YCDFEHGJ?6[]\mACLH[SGskPbYk¤B9TcDMEHGJLHhcGfp = 200 kHz

QSES<>=[aE¸=@<XtL.YZPYC='=9AMhJ<>Tusfp = 170 kHz

>j GWhcACIVQ0ACB@?9GJIKGJLF?IKGJ?eS<>GJLGJLµitYZP>GJESBPbY[STchJB9AC<>=9='YZLHhcGv[SGc=Q.YZB'YZIKUJ?@B9Gc= ?9GJIVQ~ACB9GJP>=

τ0GJ?

τmACeH=9GJB@ijTcG¾[HYZLH=P>G¦?'YZeSP>GOYZE<kanmwhcGB9Tc=@ESPX?'YZ?TJ?'YZLF?hcGJQ~GJLH[HYZLM?

ACeS?9GJLFE[HYZLH=ESLHGhcACLa¡HtESB'YZ?@<>ACLGonaQ0TJB@<XIKGJLF?'YZP>G[a<¢§]TJB9GJLF?9G£oj GKhcACIVQ0ACB@?9GJIKGJLF?[SGK[STJB@<XitYZ?@<>ACLH=GJ?<XLM?9TJtB'YZ?@<>ACL[aE=9<XtL.YZP¦?9GJIVQ~ACB9GJP [STJIKAM[aESP>T|L\^YkQ.YC=TJ?9TfACeH=9GJB@ijT

QSB9TchcTc[SGJIVIKGJLF?O~`PGc=9?:hcGJB@?'YZ<XLHGJIKGJLF?2YC=9=9A*hJ<>TYZEan[a<XijGJB9=IKTchOYZLS<>=@IKGc=:QSRM*=9<>DFEHGc=QSB9Tc='GJLF?9=w[HYZLH=P>Gc=:IV<XPX<>GJEantB'YZLFESPbYZ<XB9Gc=ONSDMES<~<XLa³HEHGJLHhcGJLF?P>G,ACLHhJ?@<>ACLSLHGJIKGJLM?:[SGwPd\^YZLF?9GJLSLHGtS¹,LHG,[SGc=9hJB@<XQS?@<>ACLDME.YZPX<X?'YZ?@<XijG%[SG%hcGc= Go§]GJ?9=GJ?[SGP>GJESBw<XLa³HEHGJLHhcG=@ESBwPd\^YZLF?9GJLSLHGQ.YZB'YZIKTJ?@B@<>DMEHG6Gc=9?,B9TOYZPX<>=9TcG6[HYZLH=wPbY|='GchJ?@<>ACLg=@ES<XitYZLF?9GGJLgiMEHG6[]\ <XLF?9GJB@QSB9TJ?9GJBP>Gc=B9Tc=@ESPX?'YZ?9=:GonaQ0TJB@<XIKGJLF?'YZEanACeS?9GJLFEH=

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P>GJESB9=VhcACLH='TcDFEHGJLH=9Gc=|=@ESBVP>GlQSB9AZ¡HP[aE²=@<XtL.YZP:[STJIKA*[aESP>TGc=@?VI|YZ<XLF?9GJL.YZLM?|LHTchcGc=9='YZ<XB9GY®¡HL½[]\ <XLF?9GJB@QSB9TJ?9GJBONvYZEIKAC<XLH=2DME.YZPX<X?'YZ?@<XijGJIKGJLF?ONP>Gc=%?@B'YZLH=@<X?@<>ACLH=2ACeH=9GJB@ijTcGc=2GonaQ~TJB@<XIKGJLM?'YZP>GJIKGJLF?Oj GJ?@?9GV=9MLM?@RHUc=9GGc=@?%B9TOYZPX<>=9TcG6<>hJ<GJ?B9Tc=9ESIKTcG[HYZLH=P>G%?'YZeSP>GOYZE kaka

Yl[a<>=@Q~GJB9=9<>ACLW[SG6i*<X?9Gc=9=9GkDFES<~@ACEHG6ESLgBBvCP>G6P>ACB9=9DMEHGτ− τm

£oN~tB ChcGxsfPbYl[a<¢§0TJB9GJLHhcGVGJLM?@B9GPbYfiM<X?9Gc=9='G[SGftB9ACESQ0Gf[aEQ.YCDFEHGJ?V[]\mACLH[SGl[SGlQ~ACIVQ.YZjG

cg(ω)GJ?

c0NPbYi*<X?9Gc=9=9G|[SGfQSR.YC=9Gk[aE´=@<XtL.YZP [STJIKA*[aESP>TtNQ~GJES?

GJLM?@B'Y>LHGJBESLHG2<XLM?9TJtB'YZ?@<>ACL[aEQSB9AZ¡HP?9GJIVQ~ACB9GJP[STJIKA*[aESP>TtYk[a<¢§~EH=9<>ACL[SGc=2ACLH[SGc=[SGQ~ACIVQ.YZjGtN][]\^YZQSB9Uc=P>G6IKAM[SUJP>GV[STJijGJP>ACQSQ0TVQSB9TchcTc[SGJIVIKGJLF?ON Gc=@?=@EH=9hcGJQS?@<XeSP>GN

P>ACB9='DFEHGτm τs τa

Q.YC=9=9G»sτs τm τa

NS[SGwACESB@LS<XB:ESLHG<XLF?9TJtB'YZ?@<>ACL[aE=@<XtL.YZP[STJIKA*[aESP>TtY%[a<¢§~B'YChJ?@<>ACLf[aE°YZ<>=9hcGOYZEV[]\mACLH[SGc=v[SGQ~ACIVQ.YZjG¤P>ACB9=9DMEHG

(Ω/ω)à `dNZB9TJt<XIKG:[SGGJB@*?'YcH£oNtGJLF?@B'Y>LHG

ESL<XLM?9TJtB'YZ?@<>ACLg[aE=9<XtL.YZP[STJIKAM[aESP>TGJLhcACIVQ.YZB'YZ<>=9ACLYOijGchP>G%B9TJt<XIKG[SG%Gc=9?9GJB@ijGJPX?O¦\mACeH=9GJB@itYZ?@<>ACL[aEk=9<XtL.YZP0[STJIKA*[aESP>T%[HYZLH==9ACLh9R.YZIVQQSB9A*h9RHG%Q0GJES?GJLF?@B'Y>LHGJBESLHGw<XLF?9TJtB'YZ?@<>ACL[aEkQSB9AZ¡HPdN

GJLlhcACIVQ.YZB'YZ<>=9ACLYcijGch,ESLHGACeH=9GJB@itYZ?@<>ACLlPbYZB@jGJIKGJLF?GJLl[SGJRHACB9= [SG,PbY2B9TJt<>ACLk[]\ <XLF?9GJB'YChJ?@<>ACLLHACL|PX<XLHTOYZ<XB9Gtjw\mGc=@?ESLGo§]GJ?:QSESB9GJIKGJLM?PX<>TYZEhOYZB'YChJ?9UJB9G?@B@<>[a<XIKGJLH=@<>ACLSLHGJP [aEdYZ<>=9hcGOYZE[]\mACLH[SG[STJIKA*[aESP>TcGt

g <XL.YZP>GJIKGJLM?ON PbYhcACLH[a<X?@<>ACLµPX<XIV<X?9GQ0ACESB|P>G=@<XtL.YZPw[STJIKA*[aESP>TtNYZE·LS<XijGOYZEº[SGPbY=9ESB°YChcG[SGPd\mTJIKGJ?@?9GJESB[]\mACLH[SGc=[SGQ0ACIVQ.YZjGtNGc=@?hcACLH=@<>[STJB9TcGhcACIVIKGB@<Xt<>[SG[HYZLH=P>G"IKAM[SUJP>G?@RHTcACB@<>DFEHG[STJijGJP>ACQSQ~Tt, ACESB@?'YZLM?ONhcGJ?@?9G±hcACLH[a<X?@<>ACL¶L\mGc=@?"Q.YC="Q.YZBdYZ<X?9GJIKGJLF?B9TOYZPX<>='TcG²[HYZLH="P>Gc="GonaQ~TJB@<>GJLHhcGc="IKGJLHTcGc=¥L Go§]GJ?ON%P>G´B'YZQSQ~ACB@?[]\ <XIVQ0Tc[HYZLHhcGYChcACEH=@?@<>DMEHGGJLF?@B9GkP>GlIV<XPX<>GJE±tB'YZLFESPbYZ<XB9GGJ?6PbYW=9ESB°YChcGk[aE´?@B'YZLH=9[aEHhJ?9GJESBKPXES<¢_dIKJIKGkL\mGc=@?Q.YC=LHTchcGc='='YZ<XB9GJIKGJLF?Q0GJ?@<X?[HYZLH=P>Gc=hcACLa¡HtESB'YZ?@<>ACLH=¦Gon*Q0TJB@<XIKGJLF?'YZP>Gc=NjGJ??9ACES?Q.YZB@?@<>hJESPX<>UJB9GJIKGJLM? s,e.YC=9=9GvB9TcDFEHGJLHhcGt¹,[email protected]°YZ<X?9GJIKGJLM?PX<XeSB9GKGJLM?@B'Y>LHGJB'YZ<X?ESLHGK[STJB@<XitYZ?@<>ACL[aE=@<XtL.YZP¦[STJIKA*[aESP>Ttj GJ?@?9GVhcACLH[a<X?@<>[email protected]@?@<>GJPXP>GJIKGJLF?:B9TOYZPX<>='TcGP>ACB9=9DMEHG:PbYP>ACLStEHGJESB[SG:Pd\^YZLM?9GJLSLHG2=9G:B'YChchcACESB9hJ<X?ONaGJ?vDMEHG,P>Gc=vGo§]GJ?9=vhJESI4ESPbYZ?@<¢=[SGQSB9ACQ.YZFYZ?@<>ACLLHGw=9ACLM? QSPXEH=Gr|hOYChcGc=ONa[SGJijGJL.YZLM?9=:hcACIVQ.YZB'YZeSP>Gc=>sPd\mGo§0GJ?[SG,PbYB9To³.Gona<>ACL=@ESBvPbY6=@ESBdYChcG ¨ DFE.YC=@<¢_PX<XeSB9Goª%Q0ACESB:P>Gc=:ACLH[SGc=:[SG2e.YC='=9Gw¤B9TcDMEHGJLHhcG=9G%QSB9ACQ.YZjGOYZLM?,[HYZLH=P>G2=9GJLH=:ACQSQ0At=9TYZEanACLH[SGc=:[SG2Q0ACIVQ.YZjGt

`P8°YZES?:hcGJQ0GJLH[HYZLF?B9Gc=9?9GJBQSB@EH[SGJLF?[HYZLH=P>Gc=TJijGJLM?@EHGJPXP>Gc=:I|YZLS<¢°Gc=@?'YZ?@<>ACLH=:[SG%hcGc=Go§0GJ?9=QSRM*=9<>DFEHGc=LHAC?'YZI_IKGJLM?P>ACB9=9DMEHGQSPXEH=@<>GJESB9=[]\mGJLF?@B9G2GJEanlACLM?PX<>GJE=@<XIESPX?'YZLHTJIKGJLM?Oj GJQ~GJLH[HYZLM?ONS<XPYZB@B@<XijG%DFEHG%hcGc=Go§]GJ?9=vQSES<>='=9GJLF?J?@B9Gw[a<>=@?@<XLStEHTc=='YZLH=YZIeS<XtES<X?9T%tB ChcG%YZEanfhcACLH[a<X?@<>ACLH=GonaQ~TJB@<XIKGJLM?'YZP>Gc=Q~ACESB P>Gc=9DMEHGJPXP>Gc=¾<XP>= <XLF?9GJB@i*<>GJLSLHGJLF?:ACELHACL0YZBGonaGJIVQSP>GtN0P>G6B9TJt<XIKGV[SG[a<¢§0EH=@<>ACL"[SGc=2ACLH[SGc=2[SGQ~ACIVQ.YZjGGc=@?%?9AC?'YZP>GJIKGJLF?<XLHhcACIVQ.YZ?@<XeSP>G|YcijGchP>Gc=Go§]GJ?9=:[SG2[a<>=@Q0GJB9=@<>ACLYC='=9AMhJ<>Tc=s[SGc=,Go§0GJ?9=hJESIESPbYZ?@<¢°=[]\mACLH[SGc=:QSB9ACQ.YZFYZ?@<XijGc=

¯%YZLH=4P>Gc=Gon*Q0TJB@<>GJLHhcGc=6QSB9Tc=9GJLF?9TcGc=QSB9TchcTc[SGJIVIKGJLM?ONP>GfhcACIVQ0ACB@?9GJIKGJLF?K[SGfPd\^YZLF?9GJLSLHG[HYZLH=4P>GfB9TJt<XIKGl[SGGc=9?9GJB@ijGJPX?ON,='YZLH=<XLa³HEHGJLHhcG´[SGc=Go§]GJ?9=QSRM*=9<>DFEHGc=g=@ESea_dIKGJLF?@<>ACLSLHTc=Nw='G?@B'YC[aES<X?WhcACIVIKG[HYZLH=Pd\mGOYZENQ.YZBESLHG[STJB@<XijTcG=9GchcACLH[SG[aE±QSB9AZ¡HP[]\ <XLF?9GJLH=9<X?9T[SGkQ0ACIVQ.YZjGQ~ACESBVP>G=9<XtL.YZP:[STJIKAM[aESP>TACeH='GJB@ijT=@ESBVPd\^Y®naG[SGQSB9ACQ.YZFYZ?@<>ACLjw\mGc=@?2P>GVhOYC=[HYZLH=2P>Gc=GonaQ0TJB@<>GJLHhcGc=2QSB9Tc=9GJLF?9TcGc=Q~ACESB

fp ' 160 kHzAi

p ' 2 j GJPbY=@<XtLS<¢¡.G

DME\sQSPXEH=Ve.YC=9=9GfB9TcDFEHGJLHhcG[SGkQ0ACIVQ.YZjGtNP>GkQSB9AZ¡HP:Gc=@?<XLF?9TJtB9TtN¾QSES<>=9DME\ <XPGc=9?VQSB9ACQ~ACB@?@<>ACLSLHGJPºsPbYW[STJB@<XijTcGQSB9GJIV<>UJB9G|[]\ ESLHG A YZEH=9=@<>GJLSLHGt¦PXEH=@<>GJESB9=Go§]GJ?9=2QSRFa=@<>DMEHGc==9ACLF?[SACLHhVhOYZLH[a<>[HYZ?9=Q~ACESBhcGJ?@?9GK<XLF?9TJtB'YZ?@<>ACL[aE=9<XtL.YZPd]¬ACES?4[]\^YZe0ACB9[]N]PbY[a<¢§0EH=@<>ACL[SGc=ACLH[SGc=4[SGVQ0ACIVQ.YZjGtNDFES<¾YZQSQ.YZB'Y>?4P>ACB9=9DMEHGPbYkP>ACLStEHGJESB4[]\mACLH[SGK[SGQ0ACIVQ.YZjGK[SGJi*<>GJLF?hcACIVQ.YZB'YZeSP>G4skESLHGK[a<XIKGJLH=@<>ACLhOYZB'YChJ?9TJB@<>=@?@<>DFEHGf[]\ <XLSRHACIKACjTJLHTJ<X?9Tc=[HYZLH=2P>GVIV<XPX<>GJEN]L\mGc=@?Q.YC=|P>Ge0ACL·hOYZLH[a<>[HYZ?Ov Y"I|YZLS<¢°Gc=@?'YZ?@<>ACL·Q0At=9=@<XeSP>G[SGPbY"[a<¢§0EH=@<>ACL½[SGc=fACLH[SGc=f[SGQ0ACIVQ.YZjG=9ESBKP>G=@<XtL.YZP[STJIKA*[aESP>T,YTJ?9TYZL.YZPX*='TcG[HYZLH=¦PbY=9GchJ?@<>ACL4ka =H ¿ M`PHGc=@?¾QSB9Tc[a<X?vDMEHGPbY2[a<¢§0EH=@<>ACLf[SGc=vACLH[SGc=¾[SGQ0ACIVQ.YZjG:IKUJLHGsPd\ <XLF?9TJtB'YZ?@<>ACLg[aEkQSB9AZ¡HP0[STJIKA*[aESP>T%P>ACB9=9DMEHGwPbYB9TcDMEHGJLHhcGw[SGwQ0ACIVQ.YZjG2Gc=@?:YZEStIKGJLF?9TcG|¤PbY4P>ACLStEHGJESB:[]\mACLH[SG[SGwQ0ACIVQ.YZjG2=9GwB'YZQSQSB9AMh'RHG2[SGwPbY6?'YZ<XPXP>Gw[SGc=<XLSRHACIKACjTJLHTJ<X?9Tc=w[aEkIV<XPX<>GJE8£oUjw\mGc=@?<>hJ<]ESLHG%[STJB@<XiCYZ?@<>ACL[aE=@<XtL.YZPDMES<Gc=@?2ACeH=9GJB@ijTcGKYcijGchPd\^YZEStIKGJLM?'YZ?@<>ACL[SG6PbYfB9TcDFEHGJLHhcG[SGQ0ACIVQ.YZjGtN]hcACLF?@B'YZ<XB9GJIKGJLM?³sfPd\mGo§]GJ?2Q~At=9=9<XeSP>G6[SGPbY[a<¢§0EH=@<>ACL[SGc=ACLH[SGc=:[SG2Q0ACIVQ.YZjG=@ESBP>G=@<XtL.YZP[STJIKA*[aESP>Tt

G|=9GchcACLH[hOYZLH[a<>[HYZ?4Q0ACESB@B'YZ<X?4J?@B9GfP>GVQ.YC=9=YZjG|[aEB9TJt<XIKGl[SG|GJB@*?'Yc#sfp = 60 kHz

YZEB9TJt<XIKGl[SGGc=9?9GJB@ijGJPX?s

fp = 160 kHzj GJQ0GJLH[HYZLF?ONCYcijGchvP>Gc=Gc=9?@<XI|YZ?@<>ACLH= [SG¾PbYP>ACLStEHGJESB [SGv[a<¢§~B'YChJ?@<>ACL s

fp = 60 kHzGJ? [SG¦PbYP>ACLStEHGJESB[]\^YZeH=9ACB@QS?@<>ACL[SG¦Pd\mACLH[SG¾[SG¾Q~ACIVQ.YZjGtN`d = πfp/c(ωp)a

2 ' 25 cmGJ?

20 cm . à . 50 cmN

¹ #º¨ªµ ¦§¢<¢©w²«&¦±!¦¦«%¬¨

¥§]GJ?:QSRFa=@<>DMEHG j ACLH[a<X?@<>ACLH= ¥§]GJ?=@ESBP>G2=@<XtL.YZP gSB9TcDFEHGJLHhcG4[SG%Q~ACIVQ.YZjGhcACLH=@<>[STJB9T [SG%I|YZLS<¢Gc=9?'YZ?@<>ACL [STJIKAM[aESP>Tl=@ESBPd\^Y®nSG£

¯w<>=@Q0GJB9=@<>ACL[SG2i*<X?9Gc=9=9Gτ− τm

<XLM?9TJtB'YZ?@<>ACLfp¨ e.YC=9=9Gc=@ª

¯w<¢§0EH=@<>ACL[SGc=τs τm τa

<XLM?9TJtB'YZ?@<>ACLfp¨ R.YZES?9Gc=ª

ACLH[SGc=[SG%Q0ACIVQ.YZjG«TJt<XIKG[SG2GJB@*?'Yc

(Ω/ω)à `d<XLM?9TJtB'YZ?@<>ACL

fp¨ e.YC=9=9Gc=@ª

jvR.YZIVQQSB9AMh'RHG2Q~ACESBà ∼ d

<XLM?9TJtB'YZ?@<>ACLfp¨ e.YC=9=9Gc=@ª

P>Gw°YZ<>=9hcGOYZE gj ACLH[a<X?@<>ACLPX<XIV<X?9G

λΩ ≥ à, `[STJB@<XiCYZ?@<>ACL

fp¨ R.YZES?9Gc=ª

PX<XeSB9GGJLz = 0

f)f ' º¢§©¦£® ¦p«°±¢¦ ¦±´¦§¢:£©%« µ ¢p¦Á² · «¨.±¢¨p¢¤uµn¢ 9§¬¨§©%«¬¨¨¢¯x¢Ü©¸²´¢µ2· ªÜ©:¢¨¨¢£ª¤Zª¯x¥§©%¤p«°±¢¹8 ¢p¦¢ º¢§©¦l¦p±¤lµ ¢ ¦« ;¨ªµ>²´¥¯x¬²±!µ ¥P¦¬¨©5²¬Ü¨¥p¦l¤B¢µ ª©%«/.¢¯x¢Ü© ª± ¤B¥ ;«¯x¢(²!¢ 7 ¢p¦©:¢¤+.¢µ © µnª¦p¦p«°±¢ 5¦ª¨¦²«&¦£¢¤p¦p«¬¨% ²« b±!¦p«¬¨'¢©R£_¬±¤¸±¨¢ B¬¨²«%©%«%¬¨Sµ «&¯³«%©:¢)¤« ;«²´¢ ¢¨

z = 07¹8 ¢¸¦p« ;¨ªµ.²!¢»£¤2¢¦¦p«¬¨"¢p¦©hªµ ¬¤p¦

£¤2¬p£_¬¤©%«%¬¨¨¢µG3¸µnª4²!¥¤p« .¥B¢¶¦¢2B¬¨²´¢³²´¢¶µ ª 9§¬¨§©%«¬¨(²´¢¶¯ ¬*²±!µnª©%«¬¨¹

PbY|hcACLH[a<X?@<>ACLWLHTchcGc=9=YZ<XB9G(Ω/ω)à `d

L\mGc=@?wQ.YC=wB9GJIVQSPX<>GV[HYZLH=,P>Gc=%GonaQ~TJB@<>GJLHhcGc=8¦\^YZLM?9GJLSLHGVQ.YZB'YZIKTJ?@B@<>DMEHGACQ~UJB9G2[SACLHh?9ACECACESB9=,[HYZLH=P>G2B9TJt<XIKG[SG%Gc=@?9GJB@ijGJPX?w[HYZLH=P>Gc=:GonaQ~TJB@<>GJLHhcGc=QSB9Tc=9GJLF?9TcGc=

G=@<XtL.YZP[STJIKA*[aESP>TtN]P>ACB9=9DMEHG4PbY|B9TcDFEHGJLHhcG[SG6Q0ACIVQ.YZjGVGc=@?w°YZ<XeSP>G¤?`*QS<>DMEHGJIKGJLF?4YZES?9ACESB[SG60 kHz

£Q~GJES?4LHG|Q.YC=J?@B9GlhOYZQS?9Tf[HYZLH==9ACL´h9R.YZIVQ´P>AC<XLM?'YZ<XLNht\mGc=9?³s[a<XB9G|Q0GJES?6LHG|Q.YC=6J?@B9Gf[a<¢§0B'YChJ?9Tt¥L´Go§0GJ?ON PbYP>ACLStEHGJESB2[]\^YZ?@?9TJLFE.YZ?@<>ACLN]QSB@<XLHhJ<XQ.YZP>GJIKGJLM?YC=9=9A*hJ<>TcG<s|Pd\^YZeH=9ACB@QS?@<>ACL"[HYZLH=%hcG6hOYC=NHL\mGc=@?%Q.YC=,?@B9Uc=,<XLa°TJB@<>GJESB9G sPbY[a<>=9?'YZLHhcGw[]\mACeH='GJB@iCYZ?@<>ACL

d ' 20 cm£oj GJQ~GJLH[HYZLM?ONSPbY?@B'YZLH=9<X?@<>ACLACeH=9GJB@ijTcGwijGJB9=vPbY6[STJB@<XijTcG%=9GchcACLH[SG%[]\ ESLHG

A YZEH=9=@<>GJLSLHGtN0hOYZB'YChJ?9TJB@<>=@?@<>DMEHGK[]\ ESLHGVACeH=9GJB@itYZ?@<>ACL[HYZLH=%P>GVh9R.YZIVQ"P>AC<XLM?'YZ<XL[aE=@<XtL.YZP[STJIKA*[aESP>T¤B9TJt<XIKGV[SGGc=@?9GJB@ijGJPX?£oN.Gc=9?:QSB9ACtB9Gc=9=@<XijGEH=9DME\s

fp = 160 kHzH¯%Gc=Gc=@?@<XI|YZ?@<>ACLH=,Gon*Q0TJB@<XIKGJLF?'YZP>Gc=%Go§]GchJ?@EHTcGc=:Q0ACESB[SG

°YZ<XeSP>Gc=[a<>=@?'YZLHhcGc=:[SG2QSB9ACQ.YZFYZ?@<>ACL´¤?`*QS<>DFEHGJIKGJLM?3 − 5 cm

£ ACLF?[STJIKACLM?@B9TDMEHG%PbYV[a<>=@?'YZLHhcGhOYZB'YChJ?9TJB@<>=@?@<>DMEHG[]\^YZ?@?9TJLFE.YZ?@<>ACLQ~ACESBESLHG6B9TcDMEHGJLHhcGK[SG

100 kHzGc=9?[SGK=9GJESP>GJIKGJLF?4DFEHGJP>DMEHGc=hcGJLF?@<XIKUJ?@B9Gc=kGJLFi*<XB9ACL

5 cmN

I|Y®n*<XI4ESI10 cm

=9GJP>ACL´PbYghcACLa¡HtESB'YZ?@<>ACL8£o¦`PGc=@?[SACLHhk[a< r|hJ<XP>G[SG4 EH=@?@<¢¡.GJBVDMEHGlPd\mACeH=9GJB@iCYZ?@<>ACL²[aE±=@<XtL.YZP[STJIKAM[aESP>Tk=9G|dYZ<X?6[HYZLH==9ACL´h9R.YZIVQ±QSB9A*h9RHG shcGc=B9TcDMEHGJLHhcGc=6[SGlQ0ACIVQ.YZjG"s

fp = 100 kHzPd\mACB9[aB9Gk[SG

[STJB@<XiCYZ?@<>ACLp ' 1.45

£oNDMES<w=9ACLM?kYZ?@?9TJLMEHTcGc=keS<>GJLYcitYZLF?kPbY[a<>=@?'YZLHhcGW[]\mACeH=9GJB@itYZ?@<>ACLGg=@<XtL.YZPw[STJIKA*[aESP>T[a<>=@Q0At=9GtNS[HYZLH=:hcGhOYC=ONS[]\ ESLHG[a<>=@?'YZLHhcG[SG2QSB9ACQ.YZFYZ?@<>ACLPX<XeSB9G=@E_r|='YZLM?9G2Q~ACESB:J?@B9G[a<¢§~B'YChJ?9Tt

G[SGJB@LS<>GJBlhOYZLH[a<>[HYZ?f=@EH='hcGJQS?@<XeSP>G[]\mGon*QSPX<>DMEHGJBKPd\ <XLM?9TJtB'YZ?@<>ACLºDMES<<XLF?9GJB@i*<>GJLF?lQ0ACESBKP>Gc=VQSPXEH=VdYZ<XeSP>Gc=B9To_DFEHGJLHhcGc=[SG2Q~ACIVQ.YZjGGc=@?PbYV[a<>=@Q0GJB9=@<>ACL[SG2i*<X?9Gc=9=9GtS¥LES?@<XPX<>='YZLF?:P>GIKAM[SUJP>G[SG2[a<>=9Q~GJB9=@<>ACL[SG2iM<X?9Gc='=9G%[]\ ESLHGh9R.Y>LHGw[SGeS<XPXP>Gc=vESLS<>[a<XIKGJLH=@<>ACLSLHGJPXP>GK¤QSB9Tc=9GJLM?9T%[HYZLH=¾P>Gwh9R.YZQS<X?@B9G³m£oNj<XP0Gc=@?vQ~At=9=9<XeSP>G[]\mGc=@?@<XIKGJB P>Gc=¾itYZP>GJESB9= [SGc=Q.YZB'YZIKUJ?@B9Gc=¾[aEKIKA*[SUJP>G[STJijGJP>ACQSQ0TQSB9TchcTc[SGJIVIKGJLM?[HYZLH=¾hcG:h9R.YZQS<X?@B9G

τ−(ω)YOijGch

cg(ω)Nc0GJ?

τt¥¦L|QSB9GJL.YZLF?

ESLHGP>ACLStEHGJESB[]\^YZeH=9ACB@QS?@<>ACLàGc=9?@<XIKTcG[SGPd\mACB9[aB9G,[SG

30 cmNtP>G?9GJIVQH= hOYZB'YChJ?9TJB@<>=@?@<>DMEHG[]\^YZ?@?9TJLFE.YZ?@<>ACL¤QSB@<XLa_

hJ<XQ.YZP>GJIKGJLF?:Q.YZB:YZeH=9ACB@QS?@<>ACL s4hcGc= B9TcDMEHGJLHhcGc=£¾iCYZES?:YZP>ACB9=τ ' à/cg(ω) ' 0.3/300 = 10−3 s

N*hcG%DMES<]Gc=@?[SGPd\mACB9[aB9G[SG2tB'YZLH[SGJESBw[aE?9GJIVQH=,[SG2IKAM[aESPbYZ?@<>ACL

τmj GJQ0GJLH[HYZLF?ON8ESLYZES?@B9G°YChJ?9GJESB,YZQSQ.YZB'Y>?%[HYZLH=:P>G?9GJIVQH=

hOYZB'YChJ?9TJB@<>=@?@<>DMEHGτ−(ω)

N]ht\mGc=@?2PbYk[a<¢§]TJB9GJLHhcGK[SGi*<X?9Gc=9=9Gc=c0 − cg(ω)

DMES<¦Gc=@?2?@B9Uc=wdYZ<XeSP>GQ~ACESB2P>Gc=2e.YC='=9Gc=w¤B9To_DFEHGJLHhcGc=¾[SGQ~ACIVQ.YZjG<XIVQSPX<>DMEHTcGc=¦<>hJ<dNCPbY,B9TcDFEHGJLHhcG[SGhcACESQSESB9G[aEVIV<XPX<>GJE|TJ?'YZLF?

fc = c0/(2πR) ' 500 kHz

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!"#%$%&'($)*+,&-. /*

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c0£

YZES?9ACB@<>='GtN8YcijGchP>Gc=Gc=@?@<XI|YZ?@<>ACLH=%[SGτ−

hcACB@B9Gc=9Q~ACLH[HYZLM?9Gc=ON~Pd\ <XLF?9TJtB'YZ?@<>ACLgQ.YZB@?@<>GJPXP>GJIKGJLF?%B9TOYZPX<>=9TcG6Gon*Q0TJB@<XIKGJLF?'Y®_P>GJIKGJLM?v[aEVQSB9AZ¡HP~[STJIKAM[aESP>TtjGc=¾hcACLH[a<X?@<>ACLH= []\mACeS?9GJLF?@<>ACLk[SGhcGJ?@?9G:?@B'YZLH=@<X?@<>ACL|=9ESB¦P>Gc=Q.YZB'YZIKUJ?@B9Gc= [aEKIKAM[SUJP>G='ACLF?TJLHACLHhcTcGc=w[HYZLH=:PbYV='GchJ?@<>ACLka =Hnmjj ACIVIKG4hcG2QSB9AMhcGc='=@EH=:Gc=@?hJESI4ESPbYZ?@<¢N.P>Gc=ACLH[SGc=,[SG2Q~ACIVQ.YZjG4[SAC<XijGJLF?='GQSB9ACQ.YZjGJB2=@ESBwESLHG[a<>=@?'YZLHhcG[]\^YChchJESIESPbYZ?@<>ACLIV<XLS<XI4ESIQ~ACESB%DMEHGPd\mGo§]GJ?2[SG4PbYf[a<>=@Q0GJB9=@<>ACL"=9G4I|YZLS<¢Gc=@?9GtqM<8P>Gc=ACLH[SGc= [SGQ~ACIVQ.YZjG%=9ACLM?YZ?@?9TJLMEHTcGc=:YcitYZLF?hcGJ?@?9Gw[a<>=@?'YZLHhcG,[]\^YChchJESI4ESPbYZ?@<>ACLNSPd\ <XLF?9TJtB'YZ?@<>ACLYC='=9AMhJ<>TcGhsPbY[a<>=9Q~GJB9=@<>ACLL\^YQ.YC=:PX<>GJEN.IKJIKG=@<PbYK[a<¢§0TJB9GJLHhcG4[SG2iM<X?9Gc='=9Gc=[SG2QSB9ACQ.YZFYZ?@<>ACLWGJLF?@B9G4=9ACESB9hcGc=:LHACLPX<XLHTOYZ<XB9Gc=,GJ?=9<XtL.YZP[STJIKA*[aESP>TGc=@?w[SG6QSPXEH=%GJLgQSPXEH=,<XIVQ0ACB@?'YZLF?9GVGJLYZEStIKGJLM?'YZLF?PbYK¤B9TcDMEHGJLHhcG[SGQ0ACIVQ.YZjGt0¦\ <XLa³HEHGJLHhcG[SG|PbY[a<>=@Q0GJB9=@<>ACL[SG|i*<X?9Gc=9=9Gg¤<XLF?9TJtB'YZ?@<>ACL±[aEQSB9AZ¡HP [STJIKAM[aESP>T£4[a<>[email protected]'YZ<X?[SACLHhfP>ACB9=9DFEHG|Pd\^YZ?@?9TJLFE.YZ?@<>ACL²;hgYZEStIKGJLM?9GtN*<dmGtMP>ACB9='DFEHG,PbY2B9TcDMEHGJLHhcG,[SG,Q~ACIVQ.YZjG%Gc=@?YZEStIKGJLF?9TcGtS,<XLH=@<dNFPd\ <XLF?9TJtB'YZ?@<>ACL[aEk=@<XtL.YZP0[STJIKAM[aESP>T[a<>=9Q.YZB'YZ<X?P>ACB9=9DMEHGPbY6B9TcDMEHGJLHhcG[SGQ0ACIVQ.YZjG6Gc=@?,YZEStIKGJLF?9TcG[SG

60 kHzs

160 kHzN.hcGDMES<B9GJi*<>GJLF?sVESLHG

[STJB@<XitYZ?@<>ACL¹LHGYZES?@B9GACeH=9GJB@iCYZ?@<>ACL"DMES<Gc=9?wGJLYChchcACB9["YcijGchhcGJ?@?9G6<XLF?9GJB@QSB9TJ?'YZ?@<>ACLGc=9?,PbYl[a<XIV<XLFES?@<>ACL[aEWQ.YZB'YZIKUJ?@B9G

[]\^Y®EH=@?9GJIKGJLM?τm

P>ACB9=9DMEHGKPbY¤B9TcDMEHGJLHhcGl[SGKQ0ACIVQ.YZjGfGc=@?YZEStIKGJLF?9TcGkQ0ACESBP>Gc=6QSPXEH=°YZ<XeSP>Gc=4iCYZP>GJESB9=[SGfp¤ijAC<XB,P>G?'YZeSP>GOYZEukanm®£oj GJ? ¨ TJPbYZB@t<>=9=9GJIKGJLM?ª6[aE=9<XtL.YZP[STJIKA*[aESP>TQ0GJES?QSB9AijGJLS<XBw[]\ ESLgQSRHTJLHACIKUJLHG6LHACLQSB@<>=

GJLhcACIVQS?9G[HYZLH=:P>G%IKAM[SUJP>G2[STJijGJP>ACQSQ0T2DFES<Gc=9? PbY[a<>=9Q~GJB9=@<>ACL[SG%i*<X?9Gc=9=9G%[SG%tB9ACESQ0GtS¯2YZLH=hcG2hOYC=ONaP>G%Q.YCDFEHGJ?[]\mACLH[SGg[SGgQ~ACIVQ.YZjGgQ0GJES?lJ?@B9GIKA*[STJPX<>=9TQ.YZBkESLµ?9GJIVQH=khOYZB'YChJ?9TJB@<>=@?@<>DFEHG"[SGIKA*[aESPbYZ?@<>ACLºDMES<2YZEStIKGJLF?9GYOijGch|PbY[a<>=@?'YZLHhcGl[SG|QSB9ACQ.YZFYZ?@<>ACLNQ0ACESB4?9GJLS<XBhcACIVQS?9Gf[SGfPd\mTJ?'YZP>[email protected]?@<bYZP[SGlhcGKQ.YCDMEHGJ?YZEhcACESB9=6[SG=YQSB9ACQ.YZFYZ?@<>ACL shOYZEH='Gf[SG|PbY[a<>=@Q0GJB9=@<>ACL´[SG|i*<X?9Gc=9=9Gt ¦\mTJPbYZB@t<>=9=9GJIKGJLM?K[aEQ.YCDMEHGJ?6[]\mACLH[SGk;hg¸P>ACB9=[SG|='YQSB9ACQ.YZFYZ?@<>ACL¸<XIVQSPX<>DMEHGESLºTJPbYZB@t<>='=9GJIKGJLF?[aEº=@<XtL.YZP%[STJIKAM[aESP>TgjTJLHTJB9Tt>j GJ?@?9GWGo§0GJ?k[a<>[email protected]'Y>?kP>ACB9=9DMEHGPbY[a<>=9?'YZLHhcGw[]\^YChchJESI4ESPbYZ?@<>ACLLHTchcGc='='YZ<XB9GwQ~ACESBP>GwQSB9A*hcGc=9=@EH=[SG%[a<>=@Q0GJB9=@<>ACLkL\mGc=@?QSPXEH=:YZ?@?9GJ<XLF?9G2Q.YZBP>Gc=ACLH[SGc=:[SGQ0ACIVQ.YZjGtN.YZ?@?9TJLMEHTcGc=,=@ESBESLHG[a<>=@?'YZLHhcG2<XLa°TJB@<>GJESB9Gt

YltB'YZLH[SGK[a< r|hJESPX?9TKGc=@?2[SGLHGQ.YC=[a<>=9Q~At=9GJBON][HYZLH=PbYlPX<X?@?9TJB'YZ?@ESB9GtN[SGV[SACLSLHTcGc=GonaQ~TJB@<XIKGJLM?'YZP>Gc=¡8YZeSP>Gc==9ESB¦PbY2[a<>=@Q0GJB9=@<>ACL|[SGiM<X?9Gc=9='G:[HYZLH=¦P>Gc=¾IV<XPX<>GJEan|hcACLH=@<>[STJB9Tc=v=9ESB¦ESL|tB'YZLH[SGQSPbYZjG[SGB9TcDMEHGJLHhcGc=,¤?`*QS<>DMEHGJIKGJLF?[SGWDMEHGJP>DFEHGc=

kHzs

150 kHzYZE¸IKAC<XLH=o£o Gc=[SGJEanµhOYZLH[a<>[HYZ?9=Q0At=9=@<XeSP>Gc=ON PbY[a<>=@Q0GJB9=@<>ACL¸[SGgiM<X?9Gc=9='GgGJ?

Pd\mACeH='GJB@iCYZ?@<>ACL [aE =@<XtL.YZPV[STJIKA*[aESP>T±[HYZLH="=9ACL h9R.YZIVQ QSB9A*h9RHGtN%Q0GJB@IKGJ?@?9GJLF?[]\mACeS?9GJLS<XB[SGc=W<XLa°ACB@I|YZ?@<>ACLH=<XIVQ0ACB@?'YZLF?9Gc==@ESB P>Gvh'R.YZIVQ6[SGvR.YZES?9Gc= ¤B9TcDMEHGJLHhcGc= tB ChcG s:PbY[STJB@<XitYZ?@<>ACL6[aE6QSB9AZ¡HPa[STJIKAM[aESP>T ACeH=9GJB@ijTcGvP>ACB9=9DFEHGPbYB9TcDFEHGJLHhcG2[SG%Q0ACIVQ.YZjG%Q.YC=9=9G%[SG

60 kHzs

160 kHzUj GchJ<]Q~ACESB@B'Y=9G,dYZ<[email protected]<XLS<XIV<>='YZ?@<>ACL

[SGc=vGJB@B9GJESB9=vGJLM?@B9GP>GIKAM[SUJP>G,[STJijGJP>ACQSQ~T6°YcijGch,PbY2QSB@<>=9G,GJL|hcACIVQS?9Gw[SG:PbY%jTcACIKTJ?@B@<>Gw?@B@<>[a<XIKGJLH=@<>ACLSLHGJPXP>G£vGJ?¦P>Gc=QSB9AZ¡HP>=¾Gon*Q0TJB@<XIKGJLF?'YZEan]FGc=¦Q.YZB'YZIKUJ?@B9Gc=v[]\^Y®EH=@?9GJIKGJLF? =9ACLF?¾YZP>ACB9=whcACLSL.YZ<>=9='YZLM?¦P>G?9GJIVQH=¦hOYZB'YChJ?9TJB@<>=@?@<>DMEHG[SGIKA*[aESPbYZ?@<>ACL

τmNCPbY%iM<X?9Gc='=9G[SGQSR.YC=9G>s%e.YC='=9G ¤B9TcDMEHGJLHhcG

c0GJ?P>G:[a<bYZIKUJ?@B9G,[aEK?@B'YZLH=9[aEHhJ?9GJESBv[SGQ0ACIVQ.YZjG£oNjP>G

?9GJIVQH=hOYZB'YChJ?9TJB@<>=@?@<>DFEHG[]\^YZ?@?9TJLFE.YZ?@<>ACLτ(ω)

GJ?PbY6i*<X?9Gc=9=9G2[SG%tB9ACESQ0G[aEQ.YCDMEHGJ?[]\mACLH[SG[SG%Q0ACIVQ.YZjGcg(ω)

¥LF?@B9G4P>Gc=B9TcDFEHGJLHhcGc=w[SGQ0ACIVQ.YZjG

fp = 160 kHzGJ?

fp = 220 kHzNSESLHG6[STJB@<XiCYZ?@<>ACLWYC[S[a<X?@<>ACLSLHGJPXP>G[aE

QSB9AZ¡HP[STJIKA*[aESP>T6Gc=9?wACeH=9GJB@ijTcGk¤ijAC<XB%PbYV¡HtESB9G kanmfGJ?,Pd\mACB9[aB9G[SG6[STJB@<XiCYZ?@<>ACL"[aEgQSB9AZ¡HP[STJIKAM[aESP>Tp[HYZLH=wP>G

¹ #º¨ªµ ¦§¢<¢©w²«&¦±!¦¦«%¬¨

?'YZeSP>GOYZEkanm®£oG%=9GJESP0Go§0GJ?Q0GJB@IKGJ?@?'YZLF?:[]\mGonaQSPX<>DFEHGJBhcGJ?@?9G%[STJB@<XitYZ?@<>ACLYC[S[a<X?@<>ACLSLHGJPXP>G2Gc=@?ONahcACLaACB@IKTJIKGJLM?YZEanlTJP>TJIKGJLF?9=

hcACPXP>GchJ?9Tc=2[HYZLH=%P>G6?'YZeSP>GOYZE#kakSN~PbYfhcACLH[a<X?@<>ACL"PX<XIV<X?9GVGJLx = 0

NNs|PbYl=@ESBdYChcG[aEW?@B'YZLH=9[aEHhJ?9GJESB[SG6Q~ACIVQ.YZjGt¯2YZLH=P>G2IKA*[SUJP>G%QSB9Tc=9GJLF?9T4[HYZLH=hcG2h9R.YZQS<X?@B9GtN~hcGJ?@?9GhcACLH[a<X?@<>ACLPX<XIV<X?9GGc=@?QSB@<>=9G ¨ B@<Xt<>[SGoª|hcACLH[a<X?@<>ACLPX<XIV<X?9G[SGu,GJESI|YZLSLN~<dmGt~P>G6[STJQSPbYChcGJIKGJLF?[STJIKA*[aESP>TGJL

x = 0Gc=9?,LMESP £oNj GJQ0GJLH[HYZLF?ON]GJL"hcACIVQ.YZB'YZLM?2P>Gc=,<XIVQ0Tc[HYZLHhcGc=

YChcACEH=@?@<>DMEHGc=[aE6IV<XPX<>GJEK[SG QSB9ACQ.YZFYZ?@<>ACLZ¤P>GIV<XPX<>GJE6tB'YZLMESPbYZ<XB9G£GJ?[SG PbYw=@ESB°YChcG[aE?@B'YZLH=9[aEHhJ?9GJESB¾TJIKGJ?@?9GJESB

Z0Na<XPGc=@?:TJi*<>[SGJLF?,DFEHG2hcGJ?@?9GhcACLH[a<X?@<>ACLPX<XIV<X?9GYC[SACQS?9TcG[HYZLH=:P>G%IKAM[SUJP>G2L\mGc=9?:Q.YC=Q.YZBdYZ<X?9GJIKGJLF?B9TOYZPX<>='TcG[aE

Q~AC<XLM?K[SGi*EHGlGonaQ0TJB@<XIKGJLF?'YZPdN¾=9ESB@?9ACES?fYZEane.YC='=9Gc=6¤B9TcDMEHGJLHhcGc=K[aE²=@<XtL.YZP,[STJIKAM[aESP>ThcACLH=9<>[STJB9TcGc=V<>hJ<d¦ACESB=@<XIVQSP>GgGc=@?@<XI|YZ?@<>ACLN

Z = ρc0 ' 5 105 kg.m−2.s−1 GJ?Z0 ' 2 106 kg.m−2.s−1 ¤PbYZIKGc=[]\^YC[HYZQS?'YZ?@<>ACL

[]\ <XIVQ~Tc[HYZLHhcG%[aEl?@B'YZLH=9[aEHhJ?9GJESBQS<>G>OAZ_`TJP>GchJ?@B@<>DMEHG£oSACB9=9DMEHGP>Gc=ACLH[SGc=[SGQ0ACIVQ.YZjGwQ~TJLHUJ?@B9GJLM?:=@E_r|='YZIVIKGJLM?QSB9AZACLH[STJIKGJLM?6[HYZLH=P>GVIV<XPX<>GJEtB'YZLMESPbYZ<XB9GtNPd\mGo§]GJ?[SGVPbYhcACLH[a<X?@<>ACLPX<XIV<X?9GKGJL

x = 0Q~GJES?LHGKQ.YC=J?@B9GV?@B9Uc=

<XIVQ~ACB@?'YZLM?=@ESB%P>GV=@<XtL.YZP[STJIKA*[aESP>T¸slhOYZEH=9GV[SGc=Go§]GJ?9=%LHACL"PX<XLHTOYZ<XB9Gc==@*LHh9RSB9ACLHGc=hJESI4ESPbYZ?@<¢=4DFES<=9ACLM?2IV<>=GJLkGJEYZEhcACESB9=[SGKPbY|QSB9ACQ.YZFYZ?@<>ACLijGJB9=2P>Gc=

xQ0At=@<X?@<¢°=O0¥LGo§]GJ?ON0P>GV=@<XtL.YZP¦[STJIKAM[aESP>T|Gc=@?YChchJESIESP>T|[HYZLH=

PbY[a<XB9GchJ?@<>ACL´[SGKQSB9ACQ.YZFYZ?@<>ACL±[SGc=6=9ACESB9hcGc=LHACLPX<XLHTOYZ<XB9Gc=xQ0At=@<X?@<¢°=£oNGJ?PbYB9To³.Gon*<>ACL´=@ESB4PbY=@ESBdYChcG|[SG|PbY

Q.YZB@?@<>G[aEµ=@<XtL.YZPw[STJIKA*[aESP>TDMES<,=\mGc=@?|QSB9ACQ.YZjTcGgGJL·QSB9GJIV<>GJBlPX<>GJE·ijGJB9=fP>Gc=xLHTJFYZ?@<¢=lGJ?fDMES<,Gc=9?fGonShJ<X?9TcG

='YZLH==@*LHh9RSB9ACLS<>=@IKG4[SGJiM<>GJLM?:LHTJtPX<XjGOYZeSP>Gt.ACB9=9DFEHG%P>Gc=ACLH[SGc=[SG%Q0ACIVQ.YZjG2LHG%Q~TJLHUJ?@B9GJLM?Q.YC=QSB9AZ°ACLH[STJIKGJLF?[HYZLH=P>GkIV<XPX<>GJE²tB'YZLFESPbYZ<XB9GtN¾P>Gc=VGo§0GJ?9=|hJESIESPbYZ?@<¢°=KLHGkQ0GJESijGJLF?VQSPXEH=VQSB9GJLH[aB9GQSPbYChcGtN¦Pd\mGo§0GJ?K[SGkPbYB9To³.Gona<>ACL=@ESBPbY=9ESB°YChcGfTJIKGJ?@?@B@<>hcGtN GJ?6[SACLHh|[SGVPbYhcACLH[a<X?@<>ACLPX<XIV<X?9GfGJL

x = 0[SGJiM<>GJLM?4<XIVQ0ACB@?'YZLF?OES?@B9GJIKGJLM?6[a<X?ON

P>ACB9=9DMEHG2PbYB9TcDFEHGJLHhcG4[SGQ0ACIVQ.YZjG4YZEStIKGJLM?9GtN8PbYKP>ACLStEHGJESBwhOYZB'YChJ?9TJB@<>=9?@<>DFEHG6[]\^YZ?@?9TJLFE.YZ?@<>ACL²¤Q.YZBwYZeH='ACB@QS?@<>ACLGJ?:[a<¢§~EH=9<>ACL8£[SGc=ACLH[SGc=,[SG2Q~ACIVQ.YZjGtN.[a<XIV<XLFEHGtN.GJ?Pd\ <XLa³HEHGJLHhcG[SG2PbYVhcACLH[a<X?@<>ACLPX<XIV<X?9GGJL

x = 0Gc=@?:[SG2QSPXEH=

GJL|QSPXEH=v<XIVQ~ACB@?'YZLM?9GtS¹LHG,hcACLH[a<X?@<>ACLkPX<XIV<X?9G ¨ PX<XeSB9Goª,GJLx = 0

hcACLH[a<X?@<>ACLk[SG,¯w<XB@<>h9RSP>GJ?ONahcACLF?@B'YZ<XLM?9Gc=:[SG:Pd\mACLH[SG[STJIKAM[aESP>TcGLFESPXP>Gc=£GJLF?@B'Y>LHG4ESLHG[STJB@<XiCYZ?@<>ACLg[aE=@<XtL.YZP[STJIKA*[aESP>T=9ESB:Pd\^Y®naG[SG2QSB9ACQ.YZFYZ?@<>ACLp A EH='s ¿ z`HACESBfp = 220 kHz

N~Pd\mACB9[aB9G6[SG6[STJB@<XitYZ?@<>ACL·¤ijAC<XBwP>G4?'YZeSP>GOYZE kanm£,Gc=@?w[SG2.7

N0hcG6DMES<hcACB@B9Gc=@Q~ACLH[gQSB9Gc=9DMEHG³s|ESLHG[STJB@<XiCYZ?@<>ACLg=@ESQSQSP>TJIKGJLF?'YZ<XB9G6[aEQSB9AZ¡HP[STJIKAM[aESP>TQ.YZBB'YZQSQ0ACB@?ºs

fp = 160 kHzQ0ACESB:PbYCDMEHGJPXP>G

p ' 2.¥LM?@B9G

hcGc=¾[SGJEanB9TcDFEHGJLHhcGc=v[SG:Q0ACIVQ.YZjGtNjPbY%B9TJt<>ACLl[SGc=¾=9ACESB9hcGc=vLHACLKPX<XLHTOYZ<XB9Gc= Gc=@?¾GJLHhcACB9G:B'YChchcACESB9hJ<>GtN*hcGDMES<H<XLSRS<Xe0GP>Gc=vGo§]GJ?9=hJESIESPbYZ?@<¢°=[SGjTJLHTJB'YZ?@<>ACL[aEk=@<XtL.YZP0[STJIKAM[aESP>TtaY4B9TJt<>ACLk=9ACESB9hcG%[SGJiM<>GJLM? DME.YC=@<¢_`=@ESBdYChJ<>DFEHGtNaGJ?=9GPX<XIV<X?9GYZEijAC<>=@<XL.YZjG4[aE?@B'YZLH=9[aEHhJ?9GJESB,TJIKGJ?@?9GJESBO

¥La¡HLNGJLM?@B9Gfp = 220 kHz

GJ?fp = 300 kHz

NESLHG<XLM?9TJtB'YZ?@<>ACL Gc=@?ACeH=9GJB@ijTcGtNACE¸[aEºIKAC<XLH=ONESLB9GJ?9ACESBsESLHG%[STJB@<XijTcG2=9GchcACLH[SG2[SG A YZEH='=@<>GJLSLHGwQ~ACESBP>GwQSB9AZ¡HP[STJIKAM[aESP>Tpm¬ ACE ktYCzà

300 kHzN*PbY4P>ACLStEHGJESB

[]\mACLH[SG%[SG,Q0ACIVQ.YZjG2Gc=@?[SGwPd\mACB9[aB9Gw[SG1 mm

NMht\mGc=@?s6[a<XB9G%GJLFi*<XB9ACL[a<¢nK°AC<>= =9ESQ~TJB@<>GJESB9GYZElB'YcjACL[SGc=eS<XPXP>Gc=OGc=K<XLSRHACIKACjTJLHTJ<X?9Tc=[aE·IV<XPX<>GJEºYC=9='AMhJ<>TcGc=4sWPd\^YZB@B'YZLSjGJIKGJLF?[STc=9ACB9[SACLSLHTg[SGc=|eS<XPXP>Gc=f=9ACLF?l[SACLHh[SGPd\mACB9[aB9G[SG|tB'YZLH[SGJESBK[SGlPbYP>ACLStEHGJESBK[]\mACLH[SGtGlQSRHTJLHACIKUJLHG[SGk[a<¢§~EH=9<>ACL´[SGc=ACLH[SGc=V[SGlQ0ACIVQ.YZjGtN[ST 2sIV<>=VGJLTJiM<>[SGJLHhcG4Gon*Q0TJB@<XIKGJLF?'YZP>GJIKGJLM?Kpmt<bYtstsCz`N8Gc=@?[SACLHh6=@EH=9hcGJQS?@<XeSP>G6[]\mJ?@B9GB9Gc=@Q0ACLH='YZeSP>G6[SGPd\ <XLM?9TJtB'YZ?@<>ACLWACeH='GJB@ijTcGtNhcACLaACB@IKTJIKGJLM?,YZEankQSB9Tc[a<>hJ?@<>ACLH=?@RHTcACB@<>DFEHGc=[SG2hcG2h9R.YZQS<X?@B9G|=9GchJ?@<>ACLPka =H ¿ £oqM<]P>G%Q.YZB'YZIKUJ?@B9G

τmGc=@?QSB@<>=[SG

Pd\mACB9[aB9G,[SG0.5 10−4 s

hcACIVIKG%[HYZLH= P>Gc=vGonaQ0TJB@<>GJLHhcGc=vQSB9Tc='GJLF?9TcGc=ON*P>G,Q.YZB'YZIKUJ?@B9G2hOYZB'YChJ?9TJB@<>=@?@<>DFEHG%[]\^YZeH=9ACB@QS?@<>ACLτa[SAC<X?:PXES< J?@B9G2?@B9Uc==@ESQ0TJB@<>GJESBQ0ACESB:Q0GJB@IKGJ?@?@B9G2Pd\mACeH=9GJB@itYZ?@<>ACL[SG2Pd\ <XLM?9TJtB'YZ?@<>ACL"[aEQSB9AZ¡HP[STJIKA*[aESP>TtN.hcGDMES<

<XIVQSPX<>DFEHGτa & 0.5 10−3 s

j GchJ<,hcACB@B9Gc=@Q0ACLH[dsESLHGP>ACLStEHGJESBkhOYZB'YChJ?9TJB@<>=@?@<>DMEHG[]\^YZeH=9ACB@QS?@<>ACLN [SGPd\mACB9[aB9G[SG

10 cmYcijGch

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ω1+ω2N2ω1

N2ω2

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ω1GJ?

ω2=@< |ω1 −ω2| ω1, ω2

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(1 + m cos(ωmt)) cos(ωt)N0Ai

mGc=9?Pd\ <XLH[a<>hcG4[SG2IKA*[aESPbYZ?@<>ACLNωm

Gc=@?PbY6¤B9TcDMEHGJLHhcG[SGIKAM[aESPbYZ?@<>ACLωGc=@?:PbY6B9TcDFEHGJLHhcGQ~ACB@?9GJEH='GtHG=9Q~GchJ?@B9G

[]\ ESL?9GJP=@<XtL.YZPhcACIVQ0ACB@?9G?@B9AC<>=:hcACIVQ0At='YZLF?9Gc=NωNω − ωm

GJ?ω + ωm

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ω1GJ?

ω2M¯2YZLH= hcG

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GJ?ω2N8GJ?,PbY

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GJ?ω2

ω1 > ω2

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ω1 + ω2N

2ω1N2ω2

N0GJ?

ω1 − ω2NDMES<2Gc=@?fESLHG"ACLH[SGW[SGge.YC=9=9G

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ω1GJ?

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dYcACLLHACLPX<XLHTOYZ<XB9G2ijGJB9=ESL=@<XtL.YZP s6PbY6B9TcDMEHGJLHhcGΩ = ω1 − ω2

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U(t) = A cos(ω0t)e− t2

2τ2m ,

¿ nm£U(ω) = Aτm

√2πe−(ω−ω0)2

τ2m2 ,

¿ kj£Ai

τmGc=@?P>G?9GJIVQH=¦hOYZB'YChJ?9TJB@<>=@?@<>DMEHG:[SGIKA*[aESPbYZ?@<>ACLf[aEVQ.YCDMEHGJ?¦[]\mACLH[SG<XLS<X?@<bYZP>GJIKGJLF?TJIV<>=¦GJ?

ω0PbY,B9TcDFEHGJLHhcG

hcGJLM?@B'YZP>Gf[SGfhcGVQ.YCDMEHGJ?O¯2YZLH=4hcGfhOYC=ON]P>G|IKTJPbYZLSjG|[SGVB9TcDMEHGJLHhcGc=6YkPX<>[email protected]|Q.YZ<XB9Gf[SGVB9TcDFEHGJLHhcG[aEk=@Q0GchJ?@B9G A YZEH=9=@<>GJLaYZBhcACLH='TcDFEHGJLM?ONaPbY4='ACPXES?@<>ACL

UgaussΩ (n, t, ω0, τm)

AinGc=@?PbYhcA*ACB9[SACLSLHTcG2[]\[email protected]£

Q0ACESBPbY[STJIKA*[aESPbYZ?@<>ACL[aEQ.YCDFEHGJ?4[]\mACLH[SG A YZEH=9=9<>GJL;ºgºGc=@?4GJLB9GJPbYZ?@<>ACLYOijGchKPbY=9ACPXES?@<>ACLUΩ(n, t, ω0,Ω)[STchJB@<XitYZLF?Pd\mGonahJ<X?'YZ?@<>ACL [SGWPbYB9TcDMEHGJLHhcG"[a<¢§0TJB9GJLHhcG

Ω = ω1 − ω2N[HYZLH=P>G"hOYC=[]\ ESL IKTJPbYZLSjG[SG"[SGJEan

B9TcDMEHGJLHhcGc=OS¦\ <XLF?9TJtB'YZ?@<>ACLW=@ES<XitYZLF?9G[SAC<X?,J?@B9G2B9TOYZPX<>=9TcGxt

UgaussΩ (n, t, ω0, τm) =

∫ +∞

−∞UΩ(n, t, ω0,Ω)e−Ω2τ2

mdΩ , ¿ ¿ £

Aiω0 = ω1

GJ?ω0 − Ω = ω2

GJ?4Aie−Ω2τ2

mGc=@?4QSB9ACQ~ACB@?@<>ACLSLHGJP:YZE=@Q~GchJ?@B9Gf[SGKPd\mGJLMijGJP>ACQSQ~Gk[SGfPd\ <XLF?9GJLH=@<X?9T

[SGlQ0ACIVQ.YZjGt¯%YZLH=PbYg=9GchJ?@<>ACL ¨ ,L.YZPX*='Goª ¿ ¿ [SGkhcGlh'R.YZQS<X?@B9GtNP>Gk[STJQSPbYChcGJIKGJLF?|[STJIKA*[aESP>TUΩ(n, t, ω0,Ω)

!¹ ®_¥B¬¤p«¢

aa − |δ0|

a − |δ0|

g|Izb " 3

|,zbn

|p z§:p *|:p|α

p|Izb " . "!

h N%U ®_¥¯xª ²±³£¤2¬µ ¯¸¢³B¬¨¦«%²!¥¤2¥*¹

=9GJB'YGon*QSB@<XIKTWtB ChcGsESLHGg°ACLHhJ?@<>ACL[SGW?@B'YZLH=°GJB@?G(ω0,Ω)

YZBhcACLH=9TcDMEHGJLF?ONESL¸h9R.YZLSjGJIKGJLM?[HYZLH=PbY[STJQ~GJLH[HYZLHhcGGJL

Ω[SG

G(ω0,Ω)itY='Gf?@B'YC[aES<XB9GkQ.YZBESLHGlIKA*[a<¢¡.hOYZ?@<>ACL·[SGlPbY°ACB@IKGl?9GJIVQ~ACB9GJPXP>G[aE´QSB9AZ¡HP

[STJIKAM[aESP>TUgauss

Ω (n, t, ω0, τm)

B,H&G,HG (`.`RYaVE½:cdL E½:a ] \,TRLfORYL E½:\,TEQ OEQ ¿ [U\,TEL E½a ]XORT cda T QUcdQU]GEGQSB9ACeSP>UJIKG,hcACLH=@<>[STJB9TQ~ACB@?9G,=@ESB¦PbYQSB9ACQ.YZFYZ?@<>ACLk[]\mACLH[SGc= TJPbYC=@?@<>DMEHGc=¾[HYZLH=vESLHGh'R.Y>LHG,=9GJIV<¢_d<XLa¡HLS<>G<mo_`¯

[SG=@QSRHUJB9Gc=2<>[SGJLF?@<>DMEHGc=ONhcACIVIKG<XP¦Gc=9?w<XPXPXEH=9?@B9TV=@ESB%PbYf¡HtESB9G ¿ nmj0¯%YZLH=PbYf=9ES<X?9GtN]=9GJESP>=%P>Gc=%QSRHTJLHACIKUJLHGc=DMES<ACLF?VPX<>GJE'sg[SGc=VTch9RHGJPXP>Gc=|[SGk?9GJIVQH=Ve~GOYZEHhcACESQ±QSPXEH=P>ACLStEHGc=|DFEHGlP>Gk?9GJIVQH=V[SGkijACP:[]\ ESLHGACLH[SGYChcACEH=@?@<>DMEHG[HYZLH=P>GVI|YZ?9TJB@<bYZEhcACLH=@?@<[email protected]?6P>Gc=eS<XPXP>Gc=6=@ESB2ESL[a<bYZIKUJ?@B9G|[SGKeS<XPXP>Ght\mGc=@?<s[a<XB9Gs[SGc=Tch'RHGJPXP>Gc=6[SGK?9GJIVQH=<XLaTJB@<>GJESB9Gc=»s

10−6sQ0ACESBw[SGc=weS<XPXP>Gc=2[SG4ijGJB@B9G6[SG[a<bYZIKUJ?@B9G

a = 2mm£oN0=9ACLM?whcACLH=9<>[STJB9Tc=O0Gc=%QSRHTJLHACIKUJLHGc=

TJ?@EH[a<>Tc=¾Q0GJESijGJLF?[SACLHhJ?@B9G,hcACLH=@<>[STJB9Tc= hcACIVIKG,DME.YC=@<¢_`=@?'YZ?@<>DMEHGc=¦Q0ACESB¾PbY[SToACB@I|YZ?@<>ACL s2Pd\mTch'RHGJPXP>G,[]\ ESLHGeS<XPXP>GtNI|YZ<>=¾hcGchJ<8L\ <XIVQSPX<>DFEHGQ.YC=¾DMEHG:PbY%P>ACLStEHGJESB []\mACLH[SG,[]\ ESLHG,ACLH[SGTJPbYC=@?@<>DMEHG:[HYZLH=vPbY2h9R.Y>LHGw=9AC<X?¦tB'YZLH[SG,[SGJiCYZLM?P>Gf[a<bYZIKUJ?@B9Gk[SGc=6eS<XPXP>Gc=

a¤PbYiM<X?9Gc=9='G|[SGlQSB9ACQ.YZFYZ?@<>ACL²YChcACEH=@?@<>DFEHGk[HYZLH=6PbYgh9R.Y>LHGkGc=@?6?@B9Uc=6<XLaTJB@<>GJESB9G sPbY

iM<X?9Gc='=9G:[SG:QSB9ACQ.YZFYZ?@<>ACLYChcACEH=@?@<>DMEHG[HYZLH=vP>G:I|YZ?9TJB@<bYZEkhcACLH=@?@<[email protected]? P>Gc=veS<XPXP>Gc=£oF,<XLH=@<dNtP>G,hOYZB'YChJ?9UJB9G,[a<>=9hJB9GJ? [SGPbYKh'R.Y>LHG4Gc=@?,?9ACECACESB9=%hcACLH=@<>[STJB9TtNj ACIVIKG4P>Gc=,[STo°ACB@I|YZ?@<>ACLH=%TJPbYC=@?@<>DFEHGc=w=9ACLM?whcACLHhcGJLM?@B9TcGc=YZEijAC<>=@<XL.YZjG6[aEPX<>GJEK[SGhcACLF?'YChJ? GJLF?@B9G:[SGJEanVeS<XPXP>Gc=ONC<XPHGc=@?Q0At=9=@<XeSP>G[SGIKA*[STJPX<>=9GJB¦PbY%h'R.Y>LHG:tB'YZLMESPbYZ<XB9G¶mo_`¯ºhcACIVIKGESLHG:h'R.Y>LHG[SGI|YC=9=9Gc=%Q0ACLHhJ?@EHGJPXP>Gc=

m = πa3ρ/6Ai

ρGc=9?2PbYk[SGJLH=@<X?9TK[aE"I|YZ?9TJB@<bYZEhcACLH=@?@<[email protected]?P>Gc=%eS<XPXP>Gc=£YZ?@?'YCh9RHTcGc=

GJLF?@B9GlGJPXP>Gc=6Q.YZB4[SGc=4B9Gc=9=9ACB@?9=4<>[SGJLF?@<>DMEHGc=LHACLPX<XLHTOYZ<XB9Gc=VYcFYZLM?ESLHGKB9GJPbYZ?@<>ACLACB9hcGK_%[STJQSPbYChcGJIKGJLM?V[SGf;wGJB@? >pmYZL.xtSNSjACR.x À zRt

F0 + Fd ∝ (|δ0| + δd)3/2 ,

¿ =M£Ai

F0Gc=@?PbYw°ACB9hcG=@?'YZ?@<>DFEHG,YZQSQSPX<>DFEHTcG:=9ESB¦PbY2h9R.Y>LHGtN

FdGc=@?ESLHGQ0GJ?@<X?9GACB9hcG[aML.YZIV<>DMEHG:=@ESQ0GJB@Q~At='TcGtN

δ0Gc=@?

PbYV[STJi*<bYZ?@<>ACLg=@?'YZ?@<>DMEHG[SGPbYV[a<>=@?'YZLHhcG4<XLF?9GJB_deS<XPXP>Gc=sQ.YZB@?@<XB%[SGahOYZB'YChJ?9TJB@<>=@?@<>DFEHG6[SG2PbY|h9R.Y>LHG4LHACLh9R.YZB@jTcG

YcijGchF0 = 0

£GJ?δd

|δd| |δ0|£Gc=@?:PbY6Q0GJB@?@[email protected]?@<>ACLgYC=9='AMhJ<>TcG¶s6Pd\mACLH[SG6YChcACEH=@?@<>DMEHGtH¥LhcACLH=@<>[STJB'YZLM?,P>G2hOYC=

F0 FdNaPbYVB9GJPbYZ?@<>ACL ¿ =*£vQ0GJES?:J?@B9G[STJijGJP>ACQSQ~TcG4GJL=9TJB@<>Gc=»t

F0 + Fd ∝ |δ0|3/2 +3

2|δ0|1/2δd +

3

8|δ0|−1/2δ2

d + . . . . ¿ À £

GQSB9GJIV<>GJB¾?9GJB@IKG:LHACLKPX<XLHTOYZ<XB9G:[SG:hcG[STJijGJP>ACQSQ0GJIKGJLF?hcACLF?@<>GJLM?δ2d

NtDFES<8Gc=@?¦DFE.YC[aB'YZ?@<>DMEHGt!j G:[STJijGJP>ACQSQ0GJIKGJLF?[SGPbY2B9GJPbYZ?@<>ACLf°ACB9hcG_[STo°ACB@I|YZ?@<>ACLlGJLM?@B9G[SGJEanKeS<XPXP>Gc=vQ~GJES? YZEH=9=@<.J?@B9G,ACeS?9GJLFE s2Q.YZB@?@<XBv[SGPbY%B9GJPbYZ?@<>ACLkTJLHGJB@t<>G

4! P#& P+ /x : N$&u

Q0AC?9GJLF?@<>GJPXP>G_¾[STJQSPbYChcGJIKGJLF?wQ~ACESBPbY6?9AC?'YZPX<X?9T4[SG%PbYVh9R.Y>LHG4ACE[aEB9Tc=9GOYZE#mo_`¯TcDMES<XiCYZP>GJLM?4¤ijAC<XB¡HtESB9G ¿ nm£ht

Ep = Ep0 +α

2!

∑

n

[U(n) − U(n + 1)]2 ¿ ^j£

+β

3!

∑

n

[U(n) − U(n + 1)]3 + . . .

¯2YZLH= hcGJ?@?9G TcDFE.YZ?@<>ACLNU(n)

B9GJQSB9Tc='GJLF?9G P>G¾[STJQSPbYChcGJIKGJLM?¦[SG¦PbYI|YC=9=9G,ACE4eS<XPXP>G£]LMESIKTJB9AnNαGc=@?PbY,hcACLH=@?'YZLM?9G

TJPbYC=9?@<>DFEHG6PX<XLHTOYZ<XB9GV[]\ ESLhcACLF?'YChJ?GJ?βGc=@?%Q.YZB'YZIKUJ?@B9GV[SGLHACLa_dPX<XLHTOYZB@<X?9TKDME.YC[aB'YZ?@<>DMEHGV[aEgIKJIKGVhcACLM?'YChJ?OG

?9GJB@IKGkhJESeS<>DMEHG[SGlPd\mTcDFE.YZ?@<>ACL ¿ ^j£4hcACB@B9Gc=@Q0ACLH[ sPbYgLHACLa_dPX<XLHTOYZB@<X?9TDME.YC[aB'YZ?@<>DFEHG[SGlPd\mTcDME.YZ?@<>ACL ¿ À £o¯wGhcGJ?@?9GB9GJPbYZ?@<>ACLWLHACLPX<XLHTOYZ<XB9GtN~<XP Gc=@?,Q~At='=@<XeSP>G[SG6[STc[aES<XB9GPd\mTcDME.YZ?@<>ACLW[aEIKACESijGJIKGJLM?wLHACLgPX<XLHTOYZ<XB9GYZEg=9GchcACLH[ACB9[aB9GGJLhcAMACB9[SACLSLHTcGc=,[a<>=9hJB9UJ?9Gc=:Q0ACESB:h9R.YCDMEHGI|YC=9='G2[SG%PbYVh9R.Y>LHGt

m∂2U(n)

∂t2= F (n) = − ∂Ep

∂U(n)= α [U(n + 1) − 2U(n) + U(n − 1)]

¿ ryt£

− β

2[U(n + 1) − 2U(n) + U(n − 1)] [U(n + 1) − U(n − 1)] .

Gc==9ACPXES?@<>ACLH=[SGVhcGJ?@?9GKTcDME.YZ?@<>ACL[aEIKACESijGJIKGJLM?LHACL"PX<XLHTOYZ<XB9GVQ0ACESB2P>GVQSRHTJLHACIKUJLHGK[]\^YZES?9AZ_`[STJIKA*[aESPbYZ?@<>ACL='ACLF?:ACeS?9GJLFEHGc=%YcijGchPbYIKTJ?@RHAM[SG[SGc=YZQSQSB9Acna<XI|YZ?@<>ACLH=,=9EHhchcGc=9=@<XijGc=pmuwAOiSxjyz`BIH&G,H B D(QU¿L E½a ]XOEQSOR½M`bQY´M½a ]X`ba TRYu¿QUM a ]ROEQMlLN^_a TRM&E½:\,TRQUMlOELN].M¿:L'^RL]EQ

¥LWQSB9GJIV<>GJBwPX<>GJEN~P>Gc=%QSB9ACQSB@<>TJ?9Tc=%PX<XLHTOYZ<XB9Gc=[]\ ESLWB9Tc=9GOYZEWB9TJtESPX<>GJB mo_`¯ =9ACLF?%B'YZQSQ0GJP>TcGc=ON0GJ?%GJLgQ.YZB@?@<>hJESPX<>GJBONPbYB9GJPbYZ?@<>ACL±[SGf[a<>=@Q0GJB9=@<>ACL,j Gc=4QSB9ACQSB@<>TJ?9Tc=6PX<XLHTOYZ<[email protected]?6PbYQSB9ACQ.YZFYZ?@<>ACL±[]\mACLH[SGc=TJPbYC=@?@<>DFEHGc=[HYZLH=P>Gc=B9Tc=9GOYZEanESLS<¢_`[a<XIKGJLH=@<>ACLSLHGJP>=[SGKI|YC='=9Gc=GJ?6[SGVB9Gc=9=9ACB@?9=4<>[SGJLF?@<>DMEHGc=4ACLM?6TJ?9TKe~GOYZEHhcACESQ´TJ?@EH[a<>TcGc=fpm B@< À Zz` Y|B9GJPbYZ?@<>ACL[SG6[a<>=@Q0GJB9=@<>ACL"Gc=@?2ACeS?9GJLFEHGV[SG6Pd\mTcDME.YZ?@<>ACLW[aEWIKACESijGJIKGJLM?PX<XLHTOYZ<XB9GV[SG6PbYfh'R.Y>LHG ∀n

¤PbYfQ.YZB@?@<>GPX<XLHTOYZ<XB9G[SG2Pd\mTcDFE.YZ?@<>ACL ¿ ryj£@£ht

m∂2U(n)

∂t2− α[U(n + 1) − 2U(n) + U(n − 1)] = 0 .

¿ ^xj£Gc=:IKA*[SGc=QSB9ACQSB9Gc=w[SGPbYKh9R.Y>LHG4<XLa¡HLS<>G4=9ACLM?,ACeS?9GJLMEH=,=9ACEH=PbY°ACB@IKG

U(n) = A(ω)eiωt−ikan N.Ai ωGc=@?

PbY|QSESP>='YZ?@<>ACLNA(ω)

Pd\^YZIVQSPX<X?@EH[SGV=@Q0GchJ?@B'YZP>GtN8GJ?kP>G6LHACIeSB9G[]\mACLH[SGhcACIVQSP>GonSGt~¥¦LW=9ESeH=@?@<[email protected]?%PbYf[SGJB@LS<>UJB9G

°ACB@IKG[SGU(n)

[HYZLH=%Pd\mTcDME.YZ?@<>ACL· ¿ ^xM£oN8PbYfB9GJPbYZ?@<>ACL[SGV[a<>=@Q0GJB9=@<>ACLWeS<>GJL"hcACLSLFEHGtN]B9GJLHhcACLM?@B9TcG|YZE"h9R.YZQS<X?@B9GlmtNGc=9?:ACeS?9GJLFEHG¸t

ω2 =4α

msin2

(ka

2

).

¿ ^sj£¥¦L<XLF?@B9A*[aES<>='YZLF?WP>Gc=LHAC?'YZ?@<>ACLH=

ωc = 2√

αm

Q0ACESBPbY±QSESP>='YZ?@<>ACL [SGhcACESQSESB9GtN,ht\mGc=@?_s®_`[a<XB9GPbY´B9TcDFEHGJLHhcGI|Y®na<XIESI [SGc=ACLH[SGc=QSB9ACQ.YZFYZ?@<XijGc=ONwGJ?

kc = πa

Q~ACESBP>GLHACIeSB9G[]\mACLH[SGº¤B9TcGJP £kI|Y®na<XIESI [SGc=ACLH[SGc=QSB9ACQ.YZFYZ?@<XijGc=NS<XPGc=@?Q0At=9=@<XeSP>G2[SG2B9TcTchJB@<XB9GPbYB9GJPbYZ?@<>ACL ¿ ^sF£ =9ACEH=:PbY4ACB@IKG=9ES<XiCYZLM?9Gxt

sin2

(π

2

k

kc

)=

(ω

ωc

)2

. ¿ nmj£

¯wGc=4=9ACPXES?@<>ACLH=QSESB9GJIKGJLM?B9TcGJPXP>Gc=Q0ACESBP>GKLHACI4eSB9G|[]\mACLH[SGkGon*<>=9?9GJLF?4='GJESP>GJIKGJLF?=@< |ω| ≤ ωc

¯2YZLH=hcGhOYC=NaPbY6B9GJPbYZ?@<>ACL[SG2[a<>=@Q0GJB9=@<>ACLGc=@?»t

π

2

k

kc= ± arcsin

(ω

ωc

),

¿ nmm£

!¹ ®_¥B¬¤p«¢

AiP>G4=@<XtLHG ¨+ªhcACB@B9Gc=@Q~ACLH[WYZEanACLH[SGc=w=9GQSB9ACQ.YZjGOYZLM?wijGJB9=,PbYK[aB9AC<X?9Gk[a<XB9GchJ?@<>ACLgQ~At=@<X?@<XijG£GJ?P>G4=@<XtLHG ¨ − ªhcACB@B9Gc=@Q0ACLH[²YZEanACLH[SGc=K=9GlQSB9ACQ.YZjGOYZLM?|ijGJB9=VPbYFYZEHh9RHG[a<XB9GchJ?@<>ACL²LHTJFYZ?@<XijG£o¦¥L±LHGhcACLH=9<>[STJB'YZLF?|DMEHGlP>Gc=

ACLH[SGc=:=9G%QSB9ACQ.YZjGOYZLM?,ijGJB9=PbYK[aB9AC<X?9GtNSPbY6B9GJPbYZ?@<>ACL[SG2[a<>=@Q0GJB9=@<>ACLGc=9?:ACeS?9GJLFEHG¸t

k =2

πkc arcsin

(ω

ωc

) Q0ACESB − ωc ≤ ω ≤ ωc . ¿ nmkj£

qM< |ω| > ωcNaPd\mTcDME.YZ?@<>ACL ¿ nmF£L\^YQ.YC=[SG='ACPXES?@<>ACLQSESB9GJIKGJLF?wB9TcGJPXP>GtH PGc=@?:LHTchcGc='='YZ<XB9G[SGhcACLH=@<>[STJB9GJB,P>G

LHACIeSB9G4[]\mACLH[SGk=9ACEH=PbY4°ACB@IKG

k = k′ + ik′′ Ai k′ GJ? k′′ ='ACLF?:B9TcGJP>=O.P>ACB9=ONs6Pd\^YZ<>[SG[SG2Pd\mTcDFE.YZ?@<>ACL ¿ nmF£oN<XPGc=@?Q0At=9=@<XeSP>G2[]\mTchJB@<XB9GESLHG2GonaQSB9Gc=9=@<>ACLiCYZPbYZeSP>G2Q0ACESBP>G2LHACI4eSB9G[]\mACLH[SGTJitYZLHGc=9hcGJLF?ON.=9ACEH=PbY6°ACB@IKG t

k = kc=@<XtL

(ω) − i2

πkcYZB9hcAt=9R ( |ω|

ωc

) Q~ACESB |ω| > ωc . ¿ nm ¿ £

`P.°YZES?LHAC?9GJBDFEHGwPd\mTcDME.YZ?@<>ACL ¿ nm ¿ £¦[STchJB@<X?PbY4[a<>=@Q0GJB9=@<>ACLl[SGc= IKA*[SGc=TJiCYZLHGc='hcGJLF?9=DMES<~=9ACLM?YZ?@?9TJLMEHTc=:[HYZLH= PbY[a<XB9GchJ?@<>ACL|Q0At=@<X?@<XijGtNjht\mGc=@?_s®_`[a<XB9G,P>ACB9=9DMEHG

nYZEStIKGJLF?9Gt*Y[SGc='hJB@<XQS?@<>ACLfhcACIVQSP>UJ?9G[SGPbY%B9GJPbYZ?@<>ACLf[SG[a<>=@Q~GJB9=9<>ACL

[SG%PbYVh9R.Y>LHGGc=@?:[SACLSLHTcGQ.YZB:PbYhcACI4eS<XL.YZ<>=9ACL[SGPd\mTcDFE.YZ?@<>ACL ¿ nmkM£ GJ?[SG%Pd\mTcDFE.YZ?@<>ACL´ ¿ nm ¿ £oB,H&G,H ¼,a ¿:T E½:a ] `ba TEYu¿)a ]RORQ LNT,Ea§OE[UcdaIORT.¿[Q

¹,LHGYZLF?9GJLSLHGQ.YZB'YZIKTJ?@B@<>DMEHGTJIKGJ?W[SGc=g=@<XtL.YZEan [SGe.YC=9='G"B9TcDFEHGJLHhcGµ=@<XtL.YZEan [STJIKA*[aESP>Tc=£ktB ChcGSsPd\^YZES?9AZ_`[STJIKAM[aESPbYZ?@<>ACLºLHACL²PX<XLHTOYZ<XB9G¤B9GchJ?@<¢¡.hOYZ?@<>ACL8£f[]\mACLH[SGc=K<XLF?9GJLH='Gc=f[SGR.YZES?9Gc=VB9TcDFEHGJLHhcGc=ON¦IKAM[aESP>TcGc=lGJLYZIVQSPX<X?@EH[SGACLH[SGc=fQSB@<XI|YZ<XB9Gc=£oN GJ?|B'YOjACLSLHTcGc=lGJL½QSB9GJIV<>GJBfPX<>GJEµ[HYZLH=|P>GIV<XPX<>GJE¾YZBlhcACLH='TcDFEHGJLM?ONv[HYZLH=fPbYB9TJt<>ACL·[]\ <XLF?9GJB'YChJ?@<>ACLNv<XPGona<>=@?9GkESLHGtB'YZLH[SG[a<¢§]TJB9GJLHhcG[]\^YZIVQSPX<X?@EH[SGgGJLF?@B9GP>Gc=fACLH[SGc=VQSB@<XI|YZ<XB9Gc=lGJ?VPd\mACLH[SG[STJIKAM[aESP>TcGt¥¦L±ES?@<XPX<>='YZLF?KPbYgIKTJ?@RHA*[SG[SGc=KYZQSQSB9Acna<XI|YZ?@<>ACLH=|=9EHhchcGc=9=@<XijGc=ONht\mGc=@?_s®_`[a<XB9GGJL²hcACLH=9<>[STJB'YZLF?|DMEHG[HYZLH=PbY%B9TJt<>ACLl[]\ <XLF?9GJB'YChJ?@<>ACLNMP>Gc=¦ACLH[SGc=¦QSB@<XI|YZ<XB9Gc= ACLF?¦ESLHG,YZIVQSPX<X?@EH[SG:e0GOYZEHhcACESQ|QSPXEH=tB'YZLH[SG,DFEHGP>Gc=¾ACLH[SGc=[STJIKAM[aESP>TcGc=LHACLPX<XLHTOYZ<XB9GJIKGJLM?ONPd\mTcDME.YZ?@<>ACLYZEQSB9GJIV<>GJB6ACB9[aB9G|[]\^YZQSQSB9AOn*<XI|YZ?@<>ACLQ~ACESBP>Gc=ACLH[SGc=QSB@<XI|YZ<XB9Gc=;ºg½Gc=@?2TcDFES<XitYZP>GJLF?9Gs|Pd\mTcDME.YZ?@<>ACL· ¿ ^xF£oN]YcijGchPbYfLHAC?'YZ?@<>ACL

U ≡ UωQ~ACESB2[STc=@<XtLHGJB2P>G[STJQSPbYChcGJIKGJLM?6YC=9=9A*hJ<>T

YZEanÁCLH[SGc=|QSB@<XI|YZ<XB9Gc=K<XLM?9GJLH=9Gc=O¾¡HL²[SG[STchJB@<XB9GPd\mGonahJ<X?'YZ?@<>ACL·[aE·IV<XPX<>GJE·Q.YZB|[SGJEan´R.YZES?9Gc=VB9TcDFEHGJLHhcGc=NPbYhcACLH[a<X?@<>ACLPX<XIV<X?9Gf=@ES<XitYZLF?9GtN Q~ACESBPbYI|YC=9=9G|LFESIKTJB9A

0=@<X?@EHTcGsPbYPX<XIV<X?9Gl[SGKPbYh'R.Y>LHG|<XLa¡HLS<>GtNGc=@?ES?@<XPX<>='TcG

¤ijAC<XB¡HtESB9G ¿ nm£oNUω(n = 0, t) = <e[Aω1(0)e

iω1t + Aω2(0)eiω2t] .

¿ nm=M£¯2YZLH=,hcGJ?@?9G6TcDFE.YZ?@<>ACLN

Aω1(0)GJ?

Aω2(0)=9ACLM?,P>Gc=%YZIVQSPX<X?@EH[SGc=2hcACIVQSP>GonaGc=%[SGc=%ACLH[SGc=2YZEan¤B9TcDMEHGJLHhcGc=

ω1GJ?

ω2sKPbYVPX<XIV<X?9G

n = 08 Y|=9ACPXES?@<>ACLW[SGPd\mTcDME.YZ?@<>ACL± ¿ ^xF£:='YZ?@<>=dYZ<>='YZLF?%hcGJ?@?9GhcACLH[a<X?@<>ACLWPX<XIV<X?9G6GJ?PbYfhcACLH[a<X?@<>ACL

[SG¶q*ACIVIKGJBGJP>[ s6Pd\ <XLa¡HLS<[HYZLH=:PbY[a<XB9GchJ?@<>ACLQ~At=9<X?@<XijGGc=@?»tUω(n, t) = <e[Aω1(0)e

iω1t−ik(ω1)an + Aω2(0)eiω2t−ik(ω2)an] ,

¿ nm À £Ai

k(ω)ACe~TJ<X?ºs6PbY6B9GJPbYZ?@<>ACLg[SG2[a<>=@Q0GJB9=@<>ACL[STchJB@<X?9G2Q.YZB:Pd\mTcDME.YZ?@<>ACL ¿ nmkF£ GJ?:Pd\mTcDME.YZ?@<>ACL ¿ nm ¿ £o

hOYZEH=9G4[aEQSRHTJLHACIKUJLHG[SG4e.YZ?@?9GJIKGJLF?GJLF?@B9Gω1

GJ?ω2N.P>G4=9<XtL.YZP ?9AC?'YZPGc=@?,TcDMES<XitYZP>GJLF?»sVESLW=@<XtL.YZP;ºg

IKAM[aESP>TGJLYZIVQSPX<X?@EH[SGtG¦[STJQSPbYChcGJIKGJLM? [STJIKA*[aESP>T¾wg

UΩGc=@?ACeS?9GJLME<sQ.YZB@?@<XB [SGPd\mTcDFE.YZ?@<>ACLYZE=9GchcACLH[ACB9[aB9G¦[]\^YZQSQSB9AOn*<XI|YZ?@<>ACL t

m∂2UΩ(n)

∂t2− α [UΩ(n + 1) − 2UΩ(n) + UΩ(n − 1)] =

− β

2[Uω(n + 1) − 2Uω(n) + Uω(n − 1)] [Uω(n + 1) − Uω(n − 1)] ,

¿ nmOj£Ai¸PbY±=9ACPXES?@<>ACL¸Q0ACESB

UωGc=9?=@ESeH=@?@<X?@EHTcG"[HYZLH=P>GWIKGJIeSB9G[SG"[aB9AC<X?9G[SGWPd\mTcDFE.YZ?@<>ACL ¿ ryj£oN:YZP>ACB9=DMEHGWPbY

hcACLF?@B@<XeSES?@<>ACL[SGUΩ

YZEan?9GJB@IKGc=%LHACL"PX<XLHTOYZ<XB9Gc=2Gc=@?%LHTJtPX<XjTcGt]¥LWLHG4B9GJ?9GJL.YZLM?=9GJESP>GJIKGJLM?DFEHG4P>Gc=%?9GJB@IKGc=»s

4! P#& P+ /x : N$&u

PbYB9TcDFEHGJLHhcG%[a<¢§]TJB9GJLHhcGΩ = ω1 −ω2

N*<XP]Gc=@? Q~At=9=9<XeSP>Gw[SG,B9TcTchJB@<XB9G%P>GwIKGJIeSB9G2[SG%[aB9AC<X?9G2[SGwPd\mTcDFE.YZ?@<>ACL ¿ nmOF£hcACIVIKG¸t

2β=m

Aω1(0)A

∗ω2

(0)eiΩt−i[k1−k∗

2 ]an

[cos(

k1 + k∗2

2a) − cos(

k1 − k∗2

2a)

]sin(

k1 − k∗2

2a)

,

¿ nmyt£Ai

k2 = k(ω2)Nk1 = k(ω1)

GJ?k∗

2

Gc=@?P>GKhcACIVQSP>GonSG|hcACLCEStEHT|[SGk2Y='ACPXES?@<>ACLjTJLHTJB'YZP>G|[SGKPd\mTcDFE.YZ?@<>ACL

¿ nmOj£Gc=@?wPbY|='ACIVIKG6[SG4PbYf=9ACPXES?@<>ACLWjTJLHTJB'YZP>G6[SG4Pd\mTcDME.YZ?@<>ACLgRHACIKACjUJLHG6Q0ACESB,P>Gc=%ACLH[SGc=%[SG4¤B9TcDMEHGJLHhcGΩ=9G

QSB9ACQ.YZjGOYZLM?,[HYZLH=PbY[a<XB9GchJ?@<>ACLQ0At=@<X?@<XijG2GJ?[]\ ESLHG=9ACPXES?@<>ACLQ.YZB@?@<>hJESPX<>UJB9G[SG%Pd\mTcDME.YZ?@<>ACL<XLSRHACIKACjUJLHG¸t

UΩ(n, t, ω1, ω2) = =m[UΩ(n, ω1, ω2)eiΩt] ,

UΩ(n, ω1, ω2) = C1e−ik(Ω)an + C2e

−i∆k an ,

C2 =βAω1(0)A

∗ω2

(0)[cos(

k1+k∗

22 a) − cos(

k1−k∗

22 a)

]sin(

k1−k∗

22 a)

2α[sin2(∆k

2 a) − Ω2

ω2c

] . ¿ nmOxj£

hJ<dN∆k = k∗

2 − k1NC1

GJ?C2

=9ACLM?w[SGJEanhcACLH=@?'YZLM?9Gc=O~¥L=@ESQSQ0At='YZLM?wDME\ <XPY|YZeH='GJLHhcG4[SGIKACESijGJIKGJLF?2wgsVPbYVPX<XIV<X?9G

n = 0N8PbYVB9GJPbYZ?@<>ACL

C1 = −C2Gc=@?ijTJB@<¢¡.TcGt~Y|=9ACPXES?@<>ACLQ0ACESBP>G4[STJQSPbYChcGJIKGJLF?2[STJIKA*[aESP>TQSB9GJLH[

YZP>ACB9=:PbY4ACB@IKG%¡HL.YZP>G¸t

UΩ(n, t, ω1, ω2) = ¿ nmOsj£

=m

βAω1 (0)A∗

ω2(0)

»cos(

k1+k∗22

a)−cos(k1−k∗2

2a)

–sin(

k1−k∗22

a)

2α

»Ω2

ω2c−sin2(∆k

2a)

–[1 − ei[k(Ω)−∆k]an

]ei(Ω)t−ik(Ω)an

.

!¹ #º¨ªµ ¦§¢

$ )<96F3¯2YZLH=|hcGJ?@?9G=9GchJ?@<>ACLNP>GB9Tc=9ESPX?'YZ?K?@RHTcACB@<>DFEHG ¿ nmOsF£Gc=@?lYZL.YZPX*=9T[HYZLH=KQSPXEH=@<>GJESB9=fhOYC=KPX<XIV<X?9Gc=ON¾Q~ACESBf[SGc=

ACLH[SGc=[SG2Q0ACIVQ.YZjGQSB9ACQ.YZFYZ?@<XijGc=%GJ?TJitYZLHGc=9hcGJLM?9Gc=O.¥LQSB9GJIV<>GJBPX<>GJEN.PbY6ACB9hcGLHACLPX<XLHTOYZ<XB9GB9Gc=9Q~ACLH='YZeSP>G4[aEQSB9AMhcGc='=@EH=[SGf[STJIKAM[aESPbYZ?@<>ACL´Gc=@?4=9<XIVQSPX<¢¡.TcGtN hcGfDFES<¦Q0GJB@IKGJ?[SG|hcACIVQSB9GJLH[aB9Gl=9Gc=hcACIVQ0ACB@?9GJIKGJLF?9=V[SG|e.YC=9Gt¥L[SGJRHACB9=%[SGPbYVB9TJt<>ACLW[]\mGon*<>=@?9GJLHhcG6[SGhcGJ?@?9G°ACB9hcGtNHPd\^YZL.YZPXa=9G4[aEg[STJQSPbYChcGJIKGJLF?[STJIKA*[aESP>T<>='=@E[]\mACLH[SGc=%[SGQ~ACIVQ.YZjGc=6QSB9ACQ.YZFYZ?@<XijGc=VGc=@?GJLH=@ES<X?9GfIKGJLHTcGtYg=9ACEH=_`=9GchJ?@<>ACL ¿ ¿ kakghcACIVQ~ACB@?9GlPd\^YZL.YZPXa=9GfDME.YZPX<X?'YZ?@<XijGk[aEhcACIVQ~ACB@?9GJIKGJLM?|[aE²=@<XtL.YZP:[STJIKA*[aESP>TPsPd\ <XLF?9TJB@<>GJESBl[SGkPbYgB9TJt<>ACL²[SGc=V=9ACESB9hcGc=VLHACL±PX<XLHTOYZ<XB9Gc=KQSB9ACQ.YZFYZ?@<XijGc= ACESB:¡HLS<XBONSPd\^YZL.YZPX*='G[aE[STJQSPbYChcGJIKGJLF?w[STJIKAM[aESP>T<>=9=@E[]\mACLH[SGc=[SGQ~ACIVQ.YZjGc=,GJiCYZLHGc=9hcGJLM?9Gc=Gc=@?B9TOYZPX<>='TcGtB,H B,HJ u]RLf¿ MQSOEQ ¿:L WZaY´^_QS]Ea ] ¿½:]E[ULf½:YQ

¥LYChchcACB9[gYcijGchPd\mTcDFE.YZ?@<>ACL´ ¿ ryj£oNSPd\mGonaQSB9Gc=9=@<>ACL[SG2PbYACB9hcG2LHACLPX<XLHTOYZ<XB9GtN.DME.YC[aB'YZ?@<>DFEHGGJL[STJQSPbYChcGJIKGJLF?ONYZt<>=9='YZLM?,YZELS<XijGOYZE[SG2PbY6eS<XPXP>G|ACEI|YC=9=9G£vLFESIKTJB9A

n=O\mTchJB@<X?¶t

FNL(n) = −β

2[U(n + 1) − 2U(n) + U(n − 1)].[U(n + 1) − U(n − 1)] .

¿ k j£¦\mGonaQSB9Gc=9=@<>ACL=9ES<XiCYZLM?9G[aE[STJQSPbYChcGJIKGJLF?2YZELS<XijGOYZE[SG2PbY6eS<XPXP>G

nN

U(n) = <e

Aω1(0)eiω1t−ik(ω1)an + Aω2(0)e

iω2t−ik(ω2)an

, ¿ k_m£

=9ACPXES?@<>ACL½[SGPd\mTcDFE.YZ?@<>ACLµ[SGQSB9ACQ.YZFYZ?@<>ACLµPX<XLHTOYZ<XB9G ¿ ^xj£[HYZLH=fP>GhOYC=f[SG[SGJEan´¤B9TcDMEHGJLHhcGc=l[]\mGonahJ<X?'YZ?@<>ACLω1GJ?

ω2NGc=@?4B9GJQ~ACB@?9TcGk[HYZLH=4Pd\mGon*QSB9Gc=9=9<>ACLº ¿ k F£o¥LLHG|B9GJ?9GJL.YZLM?VDFEHG|P>Gc=4?9GJB@IKGc= sPbYB9TcDFEHGJLHhcGl[a<¢§]TJB9GJLHhcG

Ω = ω2 − ω1NaPbY4ACB9hcGLHACLPX<XLHTOYZ<XB9G| ¿ k F£ =9G%B9TcTchJB@<X?w=9ACEH=PbY6°ACB@IKG2=@ES<XitYZLF?9G¸t

FNL = 2β<e

iAω1(0)A

∗ω2

(0)eiΩt−i(k1−k∗

2)an

[cos

(k1 − k∗

2

2a

)− cos

(k1 + k∗

2

2a

)]sin

(k1 − k∗

2

2a

). ¿ kkj£

ACB9=9DMEHGω1

?9GJLH[ijGJB9=ω2

Ω = ω2 − ω1

?9GJLH[ijGJB9=0£oN.hcGJ?@?9G%ACB9hcG2='G2=@<XIVQSPX<¢¡.G=9ACEH=PbY4°ACB@IKGxt

FNL ' −2β<eAω1(0)A

∗ω2

(0)eiΩt

e−2|k′′(ω)|an[cos(k′(ω)a

)− hcAt=@R (|k′′(ω)|a

)] =@<XLSR (|k′′(ω)|a)

, ¿ k ¿ £Ai

k′(ω) ' k′(ω1) ' k′(ω2)GJ?

k′′(ω) ' k′′(ω1) ' k′′(ω2)=9ACLF?fB9Gc=@Q0GchJ?@<XijGJIKGJLF?kP>Gc=|Q.YZB@?@<>Gc=fB9TcGJPXP>GgGJ?

<XI|YZt<XL.YZ<XB9G[aEgLHACI4eSB9G[]\mACLH[SG6[SG6Pd\mACLH[SG[SG4Q~ACIVQ.YZjG[SG4R.YZES?9GB9TcDMEHGJLHhcGω ' ω1 ' ω2

8¯2YZLH=wPbY|PX<XIV<X?9Ge.YC=9=9G¤B9TcDMEHGJLHhcG

ω ωcGJ?YZ?@?9TJLFE.YZ?@<>ACLdYZ<XeSP>G |k′′|/|k′| 1

N~ESLHG6ACB@IKGGJLHhcACB9GQSPXEH=2=@<XIVQSP>GV[SG6PbY|ACB9hcGLHACLPX<XLHTOYZ<XB9GB9Gc=@Q0ACLH='YZeSP>G2[aEQSB9A*hcGc=9=@EH=[SG2[STJIKAM[aESPbYZ?@<>ACLWYZQSQ.YZB'Y>?5t

FNL ' aβ

à(ω)

ω2

ω2c

<eAω1(0)A

∗ω2

(0)eiΩt

e− an

à(ω) . ¿ k =M£

j GJ?@?9G°ACB9hcGLHACL|PX<XLHTOYZ<XB9G,Gc=@?¦QSB9ACQ0ACB@?@<>ACLSLHGJPXP>G,YZElhcAMGr|hJ<>GJLM?v[SGLHACLa_dPX<XLHTOYZB@<X?9TwDFE.YC[aB'YZ?@<>DMEHGβN!s%Pd\ <XLFijGJB9=9G

[SGVPbYkP>ACLStEHGJESB6[]\^YZ?@?9TJLFE.YZ?@<>ACL±hOYZB'YChJ?9TJB@<>=@?@<>DMEHG|[SGVPd\ <XLM?9GJLH=@<X?9T|R.YZES?9GK¤B9TcDMEHGJLHhcGà(ω)

j GJ?@?9GV°ACB9hcGK[STchJB9A >?YcijGch2PbY[a<>=9?'YZLHhcGhcACIVIKG

e− an

à(ω)

qMGc=,QSR.YC='Gc=w[SGc=%ACLH[SGc=%[SG4Q~ACIVQ.YZjGGJLn = 0

=9ACLM?wQSB@<>='Gc=,LMESPXP>Gc=ON~P>Gc=wYZIVQSPX<X?@EH[SGc=hcACIVQSP>GonaGc=Aω1(0)GJ?

A∗ω2

(0)=9ACLF?vB9TcGJPXP>Gc=ONMGJ?vPbY2LHAC?'YZ?@<>ACL

A1NA2

Gc=@? [SACB9TJL.YcitYZLF?ES?@<XPX<>=9TcGt_qM<8[SG,QSPXEH=A1 = A2

NFP>G:?9GJB@IKGA1A2Gc=@?vTJFYZPs

A2 Ai A = A1 = A2Gc=@? Pd\^YZIVQSPX<X?@EH[SG2[]\mGonahJ<X?'YZ?@<>ACL[SG,Pd\mACLH[SGs2PbY¤B9TcDMEHGJLHhcG

ω1GJ?[SGPd\mACLH[SGsPbY

¤B9TcDMEHGJLHhcGω2.¯%YZLH=,hcGhOYC=ONSPbYACB9hcG2LHACLPX<XLHTOYZ<XB9Gf ¿ k =M£ itYZB@<>GhcACIVIKGPd\^YZIVQSPX<X?@EH[SG4[SG2Pd\mACLH[SG4[SG2Q0ACIVQ.YZjG

YZEhOYZB@B9TA2 ]GB9TJt<XIKG|wg

ω ωc£w[STchJB@<X?Q.YZB2Pd\mTcDFE.YZ?@<>ACLº ¿ k =*£wGc=@?2<>[SGJLM?@<>DFEHG|YZEB9TJt<XIKGlmo_`¯ ACLH[SG

QSPbYZLHG£[SG%Pd\^YZLF?9GJLSLHG4Q.YZB'YZIKTJ?@B@<>DMEHG[HYZLH=P>Gc=IV<XPX<>GJEanRHACIKACjUJLHGc=4pmu,Aiaxjy®z`

4! P#& P+ /x : N$&u

BIH B,H&G u]ELN¿ MQSORT M½e ]RLN¿hOR[UcdaIORTR¿:[ÁORQ À.LNMMQ WZY[U\,TRQU]R^_Q¦\^YZL.YZPX*='G[aEf[STJQSPbYChcGJIKGJLM?[STJIKAM[aESP>T

U(n)[SGe.YC=9='GB9TcDFEHGJLHhcG

ΩNjACeS?9GJLFEk=9ACEH=¦PbY2ACB@IKG[SG:Pd\mTcDME.YZ?@<>ACL

¿ nmOsj£oN~Gc=@?,B9TOYZPX<>=9TGJLgQSB9GJIV<>GJB,PX<>GJEgQ0ACESBP>Gc=wACLH[SGc=,[SG4Q~ACIVQ.YZjG6;hg´QSB9ACQ.YZFYZ?@<XijGc=2GJ?,GJLW=9GchcACLH[PX<>GJEgQ~ACESBP>Gc=kACLH[SGc=[SGgQ0ACIVQ.YZjG";ºg TJiCYZLHGc='hcGJLF?9Gc=O:¥¦Lº[SGJRHACB9=[SGPbYB9TJt<>ACL[]\mGonShJ<X?'YZ?@<>ACLN<dmGt P>ACB9=9DMEHGP>Gg?9GJB@IKGe− an

à(ω)[SGc= TcDFE.YZ?@<>ACLH=% ¿ nmOsF£ACEW ¿ k =*£?9GJLH[|ijGJB9=

0¤P>Gc=vACLH[SGc=v[SGQ~ACIVQ.YZjG,=9ACLF? =@E_r|='YZIVIKGJLM?YZ?@?9TJLMEHTcGc=£oN

PbYACB9hcG|LHACLPX<XLHTOYZ<XB9Gl[SGJiM<>GJLM?LHTJtPX<XjGOYZeSP>Gt ¦\mACLH[SGk[STJIKA*[aESP>TcGfwgGc=9?4YZP>ACB9=GJLF?@<>UJB9GJIKGJLF?jTJLHTJB9TcGkGJ?4GJPXP>G='GkQSB9ACQ.YZjGPX<XeSB9GJIKGJLF?k[HYZLH=KP>GIV<XPX<>GJE¾¯2YZLH=|hcGhOYC=NPd\mACLH[SG[STJIKA*[aESP>TcG

U(n)Gc=@?fTJFYZP>GPsWESLHGk°ACLHhJ?@<>ACL

G(ω1, ω2)IESPX?@<XQSPX<>TcGQ.YZB%ESLW?9GJB@IKGV[SG6QSR.YC=9G [

1 − eik(Ω)an−i∆kan]eiΩt−ik(Ω)an ' eiΩt−ik(Ω)an DMES<[STchJB@<X?

=YQSB9ACQ.YZFYZ?@<>ACLf[HYZLH=P>GIV<XPX<>GJE|[a<>=@Q0GJB9=@<¢@j GJ?@?9G °ACLHhJ?@<>ACLG(ω1, ω2)

B9GJQSB9Tc=9GJLF?9G[SACLHhESLHG ACLHhJ?@<>ACL|=9Q~GchJ?@B'YZP>G[SG´?@B'YZLH=@GJB@?[STchJB@<XitYZLF?P>G´QSB9AMhcGc=9=9EH="LHACL PX<XLHTOYZ<XB9G²[]\mGonShJ<X?'YZ?@<>ACL [SG´PbY½B9TcDFEHGJLHhcG²[a<¢§]TJB9GJLHhcG±Q.YZBYZES?9AZ_[STJIKA*[aESPbYZ?@<>ACL¥PXP>Gk[SACLSLHGlPd\mACQSQ~ACB@?@ESLS<X?9T[SGl?@B9ACESijGJBVPd\^YZIVQSPX<X?@EH[SGGJ?6PbYQSR.YC=9Gk[aE=@<XtL.YZP[STJIKA*[aESP>TlsPbYB9TcDMEHGJLHhcG

Ω = ω2 − ω1s6Q.YZB@?@<XB[SG2PbYVhcACLSL.YZ<>='='YZLHhcG[SGc=:ACLH[SGc=,[SG%Q~ACIVQ.YZjG4TJIV<>=9Gc=YZEanl¤B9TcDMEHGJLHhcGc=

ω1GJ?

ω2Y°ACLHhJ?@<>ACL

G(ω0,Ω) = G(ω1 = ω0, ω2 = ω0 − Ω)Q0GJES?YZEH=9=9< J?@B9GfES?@<XPX<>=9TcGfQ~ACESB6Pd\^YZL.YZPX*='Gf[aE´hOYC=

[HYZLH=%P>GcDMEHGJPESL"Q.YCDMEHGJ?[]\mACLH[SGV;ºgºGc=@?%ES?@<XPX<>=9T|hcACIVIKGV=@<XtL.YZP¦[SGQ~ACIVQ.YZjGt¯%YZLH=hcGVhOYC=ON0GJLYChchcACB9[YcijGchPd\mTcDME.YZ?@<>ACL ¿ ¿ £GJ?PbYV[STo¡HLS<X?@<>ACL[SG

UΩ(n, t, ω1 = ω0, ω2 = ω0 − Ω)[HYZLH=Pd\mTcDME.YZ?@<>ACL ¿ nmOsM£oNHGJL[SGJRHACB9=[SG

PbY6B9TJt<>ACL=9ACESB9hcG|n 1

£oN

UΩ(n 1, t, ω0, τm) = =m

∫ +∞

−∞G(ω0,Ω)e−Ω2τ2

meiΩ(t−k(Ω)

Ωan)dΩ .

¿ k À £

YZBhcACLH=9TcDFEHGJLM?ONZP>G¾QSB9A*[aES<X?2πG(ω0,Ω)e−Ω2τ2

mB9GJQSB9Tc=9GJLM?9GP>G¾=@Q0GchJ?@B9G [SG¾Pd\mACLH[SG [STJIKAM[aESP>TcG[HYZLH=P>G¾=9*=@?9UJIKG

[SGhcA*ACB9[SACLSLHTcGc==9G[STJQSPbYOYZLF?vYOijGchGJPXP>GtNCP>ACB9=9DFEHGn 1

C¥LV[]\^YZES?@B9Gc=?9GJB@IKGc=ONZP>G=@Q0GchJ?@B9G [aEV=@<XtL.YZPS[STJIKA*[aESP>TQ0GJES?J?@B9G ACeS?9GJLMEKGJLIESPX?@<XQSPX<bYZLF?¾P>G =9Q~GchJ?@B9G[SG Pd\mGJLFijGJP>ACQSQ0G[]\ <XLF?9GJLH=@<X?9T[aEQ.YCDMEHGJ?[]\mACLH[SG;ºg

e−Ω2τ2mQ.YZBPbY

°ACLHhJ?@<>ACLG(ω0,Ω)

jw\mGc=@? PbYB'YZ<>=9ACLQ0ACESBPbYCDMEHGJPXP>GG(ω0,Ω)

Gc=9? YZQSQ~GJP>TcG¦ACLHhJ?@<>ACL6=@Q0GchJ?@B'YZP>G¾[SG?@B'YZLH=°GJB@?O¡HL[SG hcACIVQSB9GJLH[aB9GPbYw[aML.YZIV<>DMEHG[SG PbY:jTJLHTJB'YZ?@<>ACL|wg[HYZLH= PbY,B9TJt<>ACLV[SGc==9ACESB9hcGc=LHACLPX<XLHTOYZ<XB9Gc=GJ?LHAC?'YZIVIKGJLF?P>GBBvCP>GW[SGPbY[a<>=@Q~GJB9=9<>ACLµ[SGiM<X?9Gc='=9GtN¾P>G?9GJB@IKGW[SGQSR.YC=9GghcACIVQSP>GJ? [

1 − eik(Ω)an−i∆kan]eiΩt−ik(Ω)an =9GJB'Y

YZL.YZPXa=9TQ.YZBfPbY=@ES<X?9Gt gYZ<>=NvGJLµQSB9GJIV<>GJBkPX<>GJENvESLHGWYZL.YZPXa=9GWGc=@?KdYZ<X?9G[SGgPbY=9GJESP>GACLHhJ?@<>ACL¸[SG?@B'YZLH=GJB@?G(ω1, ω2)

ACESB2Pd\^YZL.YZPXa=9G[aEWQSB9AMhcGc=9=9EH=w[SG[STJIKA*[aESPbYZ?@<>ACLDFES<=@ES<X?ON~P>Gc=,B9TcDFEHGJLHhcGc=2[SG6Q0ACIVQ.YZjG

ωGJ?%P>G6LHACIeSB9G

[]\mACLH[SG±YC=9='AMhJ<>T=9ACLF?WLHACB@I|YZPX<>=9Tc=gB9Gc=@Q0GchJ?@<XijGJIKGJLF?"Q.YZBWPbY²¤B9TcDMEHGJLHhcG´[SGhcACESQSESB9Gωc

GJ?WP>GdYChJ?9GJESB 2πkc

¾\^YcitYZLF?'YZjGQSB@<XLHhJ<XQ.YZPH[]\ ESLHG?9GJPXP>G LHACB@I|YZPX<>='YZ?@<>ACLNjACES?@B9GPbYw=@<XIVQSPX<¢¡.hOYZ?@<>ACLK[SG PbY,LHAC?'YZ?@<>ACLNjGc=@?DMEHGvP>Gc=i*<X?9Gc=9=9Gc=[SG%QSR.YC='G

cφGJ?[SG%tB9ACESQ0G

cg=9ACLF?,TJFYZP>Gc=>s m%P>ACB9=9DMEHG%PbY6B9TcDFEHGJLHhcG

ω?9GJLH[kijGJB9= a

ACB9=9DMEHG4hcGc=,LHACB@I|YZPX<>='YZ?@<>ACLH=2=9ACLM?%Go§]GchJ?@EHTcGc=ON8PbYV°ACLHhJ?@<>ACLW[SG?@B'YZLH=°GJB@?%[aEQSB9A*hcGc=9=@EH=%[SG[STJIKA*[aESPbYZ?@<>ACLQSB9GJLH[PbY4ACB@IKG2jTJLHTJB'YZP>G=@ES<XiCYZLM?9Gxt

G(ω1, ω2) =β

2αA2 cos(k1 − k∗

2) − cos(k1 + k∗2)

Ω2 − sin2(k1 − k∗2)

sin(k1 − k∗2) .

¿ ktj£

N)f% ¶/x+´/ +´/ +P UP U U_*!/¯%YZLH=4P>GlhOYC=4[SGc=ACLH[SGc=[SGKQ0ACIVQ.YZjGlQSB9ACQ.YZFYZ?@<XijGc=ONIsPd\ <XLMijGJB9=9Gk[SGc=4ACLH[SGc=[SGfQ~ACIVQ.YZjGlTJiCYZLHGc='hcGJLF?9Gc=

Q0ACESB2P>Gc=9DMEHGJPXP>Gc=2ESL[STJijGJP>ACQSQ0GJIKGJLF?6GonSYChJ?QSPXEH=Q~ACEH=9='TV[SGVPbYfACLHhJ?@<>ACL[SG?@B'YZLH=GJB@?GGc=@?2Q0At=9=@<XeSP>GtN0<XP¦Gc=@?

LHTchcGc='='YZ<XB9G2[SG%dYZ<XB9G2DFEHGJP>DMEHGc=:RMMQ0AC?@RHUc=9Gc=:=@<XIVQSPX<¢¡.hOYZ?@B@<>hcGc=<t|Ω| |ω1|, |ω2| ,

ω1 = ω ,

ω2 = ω − Ω , Ω > 0 . ¿ kjyt£

!¹ #º¨ªµ ¦§¢

ω = 1

ω − 1 Ω1 − ω Ω

1 − ω = Ω ω − 1 = Ω

ω

ω = 0

Ω

ω Ω ω − 1 Ω

p 1 → p 1

1 − ω Ω

h )f 8 ¢p¦5²« º¥¤B¢Ü©¦5Bª¦»µ «¯³«©:¢p¦¶²´¢¶µB· ª¨ªµ ¦¢¶²±l¤2¥¦p±µ ©ª© 5 !¹.7

YB9GJPbYZ?@<>ACL0 < Ω ω ≤ 1

Gc=@?KYZP>ACB9=B9TOYZPX<>=9TcGt¥L±QSB9GJL.YZLM?|GJL±hcACIVQS?9GhcGc=VTJFYZPX<X?9Tc=KGJ?<XLHTJFYZPX<X?9Tc=ONh9R.YCDMEHG2?9GJB@IKG[SG2PbY4°ACLHhJ?@<>ACL[SG2?@B'YZLH=@GJB@? ¿ ktF£vQ0GJES?=9G2=9<XIVQSPX<¢¡.GJBO

¥L´hcACLH=@<>[STJB'YZLF?VDMEHG|Pd\^YZ?@?9TJLME.YZ?@<>ACL²[SGc=ACLH[SGc=4QSB9ACQ.YZFYZ?@<XijGc=VGc=9?4=@E_r|='YZIVIKGJLM?6°YZ<XeSP>GfQ~ACESB6Q0GJB@IKGJ?@?@B9GP>GJESBvQSB9ACQ.YZFYZ?@<>ACLNa<dmGtMP>G,IKAM[aESP>Gw[SGPbYQ.YZB@?@<>Gw<XI|YZt<XL.YZ<XB9G%[aEfLHACI4eSB9Gw[]\mACLH[SG |k′′(ω)| 1

GJ?GJLfLHTJtPX<XjGOYZLM?PbY%[a<>=@Q0GJB9=@<>ACL|[]\^YZ?@?9TJLFE.YZ?@<>ACL" |k′′(ω)| ' |k′′(ω−Ω)| £oNtPbY2Q.YZB@?@<>G<XI|YZt<XL.YZ<XB9G[SGPbY%[a<¢§]TJB9GJLHhcG (k1−k∗

2)=\mTchJB@<X?

2i|k′′(ω)| M¥LKES?@<XPX<>='YZLM?¾P>G:[STJijGJP>ACQSQ0GJIKGJLF?[SG:¬ YOMP>ACBYZEKQSB9GJIV<>GJBvACB9[aB9G[SGPbY%B9GJPbYZ?@<>ACLf[SG[a<>=9Q~GJB9=@<>ACL|Q~ACESB¦P>G?9GJB@IKG B9TcGJP(k′

1−k′2)NCitYZPbYZeSP>G Q~ACESB |Ω| |(∂k′/∂ω)/(∂2k′/∂ω2)| NthcG ?9GJB@IKG=O\mTchJB@<X? k′

1−k′2 ' Ω∂k′

∂ω (ω) 1

¯2YZLH=:hcG2hOYC=ON

sin(k1 − k∗2) ' Ω

∂k′

∂ω(ω) + 2i|k′′(ω)| ¿ ktxj£

cos(k1 − k∗2) ' 1 ,

¿ ktsj£YZEQSB9GJIV<>GJB,ACB9[aB9GGJL

Ω∂k′

∂ω (ω)GJ?GJL |k′′(ω)| .ijGch2P>Gc=:IKJIKGc=,YZQSQSB9Acna<XI|YZ?@<>ACLH=ON

cos(k1 + k∗2) ' cos(k′

1 + k′2)

' cos(2k′(ω) − Ω∂k′

∂ω(ω)) ' cos(2

YZB9hc=@<XL(ω)) + Ω

∂k′

∂ω(ω) sin(2

YZB9hc=@<XL(ω))

' 1 − 2ω2 + 2ω√

1 − ω2

(Ω

∂k′

∂ω(ω)

).

¿ ¿ j£¥LB9GJIVQSPbYOYZLF?P>Gc=?9GJB@IKGc=6=@<XIVQSPX<¢¡.Tc= ¿ ktsF£GJ?l ¿ ¿ F£[HYZLH=Pd\mGonaQSB9Gc=9=@<>ACL[SGKPbYk°ACLHhJ?@<>ACL´[SGK?@B'YZLH=°GJB@?

¿ ktj£oN.GJ?wGJLLHAC?'YZLF?2DMEHG ∂k′

∂ω (ω) = c−1g (ω) = (1 − ω2)−1/2 NSESLHGLHACESijGJPXP>G6GonaQSB9Gc=9=@<>ACLQ0ACESBPbYACLHhJ?@<>ACLW[SG?@B'YZLH=°GJB@?

GGc=@?:ACeS?9GJLFEHGt

G(ω,Ω) ' β

αA2ω2

Ωcg(ω) + 2i|k′′(ω)|

(Ω

cg(ω) + 2i|k′′(ω)|)2

− Ω2

. ¿ ¿ m£

¥LlES?@<XPX<>='YZLF? PbYLHAC?'YZ?@<>ACLà(ω) = 1

2|k′′(ω)|

Q~ACESB PbY2P>ACLStEHGJESB[]\^YZ?@?9TJLME.YZ?@<>ACL[SG:Pd\ <XLF?9GJLH=9<X?9T%YChcACEH=9?@<>DFEHGw[SG

4! P#& P+ /x : N$&u

Q0ACIVQ.YZjGtNHGJ?:GJLIKA*[a<¢¡8YZLF?,Pd\mGon*QSB9Gc='=@<>ACL ¿ ¿ m®£oNHhcGJ?@?9G[SGJB@LS<>UJB9G=9G%B9TcTchJB@<X?¶t

G(ω,Ω) ' β

αA2ω2

Ωà(ω)cg(ω) + i

[Ω à(ω)

cg(ω) (1 − cg(ω)) + i] [

Ω à(ω)cg(ω) (1 + cg(ω)) + i

] . ¿ ¿ kj£

¥L½<XLM?@B9AM[aES<>=YZLF?P>GQ.YZB'YZIKUJ?@B9Gp′ = à(ω)

cg(ω)Ω = k(Ω)à(ω)cφ(Ω)cg(ω)

[HYZLH=khcGJ?@?9GW[SGJB@LS<>UJB9GWGon*QSB9Gc='=@<>ACLN¾PbY°ACLHhJ?@<>ACL

GQSB9GJLH[PbYACB@IKG t

G(ω,Ω) ' β

αA2ω2à(ω)

p′ + i

[p′ (1 − cg(ω)) + i] [p′ (1 + cg(ω)) + i].

¿ ¿t¿ £

¥LhcACLH=@<>[STJB'YZLM?wDMEHGQ0ACESBESLHG6YZL.YZPX*='GDME.YZPX<X?'YZ?@<XijG[aEhcACIVQ0ACB@?9GJIKGJLF?2[SG4hcGJ?@?9G2°ACLHhJ?@<>ACLGN1 + cg(ω)Gc=9?¦[aE|IKJIKGACB9[aB9G[SG:tB'YZLH[SGJESBvDMEHG

10 ≤ cg(ω) ≤ 1

£oNFPd\mGon*QSB9Gc='=@<>ACLg ¿ ¿t¿ £Gc=@?¡HL.YZP>GJIKGJLM?=@<XIVQSPX<¢¡.TcG=9ACEH=PbY4°ACB@IKGxt

G(ω,Ω) ∼ β

αA2ω2à(ω)

1

p + i,

¿ ¿ =M£

Aip = p′(1 − cg(ω)) = à(ω)k(Ω)

cφ(Ω)cg(ω) (1 − cg(ω))

Gc=@?ESLgQ.YZB'YZIKUJ?@B9GVYC[a<XIKGJLH=@<>ACLSLHGJPd0ijGchΩ ω ≤ 1

NPbY6B9GJPbYZ?@<>ACL

cφ(Ω) ' cφ(0) = 1Gc=9?ijTJB@<¢¡.TcGtN.GJ?P>G2Q.YZB'YZIKUJ?@B9G

pQ~GJES?='G%B9TcTchJB@<XB9G=9ACEH=PbY4°ACB@IKGxt

p ' à(ω)k(Ω)cφ(Ω) − cg(ω)

cg(ω).

¿ ¿tÀ £

¯2YZLH=hcGJ?@?9GGon*QSB9Gc='=@<>ACLN à(ω)cg(ω)

B9GJQSB9Tc=9GJLF?9GP>G"?9GJIVQH=IV<>=Q.YZBP>GQ.YCDFEHGJ?g[]\mACLH[SG[SG"Q0ACIVQ.YZjG;hg»Q~ACESBQ.YZB9hcACESB@<XB='YB9TJt<>ACL¸[]\^YZ?@?9TJLFE.YZ?@<>ACL hOYZB'YChJ?9TJB@<>=9?@<>DFEHGtGdYChJ?9GJESB à(ω)

cg(ω) (cφ(Ω) − cg(ω))B9GJQSB9Tc='GJLF?9GWDFE.YZLM?

sPXES<,PbY=9TJQ.YZB'YZ?@<>ACL¸[email protected]?@<bYZP>GghOYZB'YChJ?9TJB@<>=@?@<>DMEHGgGJLM?@B9GgP>GQ.YCDMEHGJ?l[]\mACLH[SGW;hg GJ?lP>G=@<XtL.YZP%wg [STJIKA*[aESP>T(sPd\mTch'RHGJPXP>G

à(ω)N.ht\mGc=@?_s®_`[a<XB9G<sVPbYPX<XIV<X?9G4[SGPbYB9TJt<>ACLg[]\ <XLF?9GJB'YChJ?@<>ACLWLHACLPX<XLHTOYZ<XB9Gt8GQ.YZB'YZIKUJ?@B9G

pGc=9?,YZP>ACB9=

QSB9ACQ0ACB@?@<>ACLSLHGJPshcGJ?@?9G[a<>=@?'YZLHhcG2IKGc=@ESB9TcGGJLLHACI4eSB9G[SG2P>ACLStEHGJESB9=,[]\mACLH[SGc=,YChcACEH=@?@<>DMEHGc=wg

N)f%U% »! &! / $% )U ACB9=9DMEHGp 1

N8hcGJ?@?9G4='TJQ.YZB'YZ?@<>[email protected]?@<bYZP>G6Gc=@?,?@B9Uc=Q0GJ?@<X?9G4[SGJitYZLF?wPbYP>ACLStEHGJESB,[]\mACLH[SG

λ(Ω)[aE=@<XtL.YZP[STJIKAM[aESP>TtN8GJ?PbY6°ACLHhJ?@<>ACL[SG%?@B'YZLH=°GJB@?:QSB9GJLH[PbY4°ACB@IKG=@ES<XitYZLF?9Gxt

G(ω,Ω) ∼ −iβ

αA2ω2à(ω) .

¿ ¿ j£¯2YZLH=%hcGVhOYC=ON8PbYliCYZB@<bYZ?@<>ACL"[SGPd\^YZIVQSPX<X?@EH[SGV[aE"=@<XtL.YZP¦[STJIKA*[aESP>TVGJLgACLHhJ?@<>ACL"[SGPbY|¤B9TcDMEHGJLHhcG[SG6Q~ACIVQ.YZjGGc=9?B9TJt<>G:Q.YZB¦Pd\^YZ?@?9TJLME.YZ?@<>ACLl[SGc=vACLH[SGc=¾[SGQ0ACIVQ.YZjG4¤Q.YZB'YZIKUJ?@B9G

ω2à(ω)£oF Y2[a<>=@Q0GJB9=@<>ACL|[SGi*<X?9Gc=9=9GLHG@ACEHG

Q.YC=¾[SGBBvCP>G<XIVQ~ACB@?'YZLM?¾=9AC<X?¦Q.YZB9hcGDFEHGPbYwP>ACLStEHGJESB [SGPd\^YZLF?9GJLSLHGGc=9??@B9ACQV°YZ<XeSP>GtNF=9AC<X?Q.YZB9hcG:DMEHGPbY%[a<¢§0TJB9GJLHhcGGJLM?@B9GwP>Gc=vi*<X?9Gc=9=9Gc=

cg(ω)GJ?

cφ(Ω)Gc=@? LHTJtPX<XjGOYZeSP>GtNaQ0ACESBGJLM?@B'Y>LHGJBESLHG%=9TJQ.YZB'YZ?@<>[email protected]?@<bYZP>G2=@E_r|='YZLM?9GwGJLM?@B9G

P>G|Q.YCDFEHGJ?[]\mACLH[SGk;ºg GJ?4P>Gl=@<XtL.YZP[STJIKAM[aESP>Tlwg sPbY=9ACB@?@<>Gl[SG|PbYB9TJt<>ACL±[]\ <XLF?9GJB'YChJ?@<>ACLYACLHhJ?@<>ACLGTJ?'YZLM?<XLH[STJQ~GJLH[HYZLM?9G[SGwPbYitYZB@<bYZeSP>G

ΩNM[HYZLH=hcGwhOYC=PX<XIV<X?9GtN*P>GwQSB9AZ¡HP~?9GJIVQ0ACB9GJP[aE=@<XtL.YZP][STJIKAM[aESP>TV[HYZLH=P>G

hOYC=,[SGPbY|[STJIKAM[aESPbYZ?@<>ACLW[]\ ESLQ.YCDMEHGJ?,[]\mACLH[SG£:B9Gc=@?9G<XLHh9R.YZLSjT6DFE.YZPX<X?'YZ?@<XijGJIKGJLM?wQ.YZB,B'YZQSQ~ACB@?sVPbYACLHhJ?@<>ACL[SG%IKA*[aESPbYZ?@<>ACL<XLS<X?@<bYZP>G[aE=@<XtL.YZP[SG2Q0ACIVQ.YZjGt

ACESBvESLHGP>ACLStEHGJESBv[]\^YZ?@?9TJLME.YZ?@<>ACLl[aEK? MQ0Gà(ω) ∼ 1/ω

?9GJPXP>G:DME\mGJPXP>G:Gc=@?¾hcACLH=@<>[STJB9TcG:Q.YZB¦PbY2=@ES<X?9G:GJ?¾ESLHGYZIVQSPX<X?@EH[SGw[]\mGonShJ<X?'YZ?@<>ACLkhcACLH=9?'YZLF?9G%[SGc=vACLH[SGc=[SGQ~ACIVQ.YZjGc=

ANFPbY2°ACLHhJ?@<>ACLk[SG:?@B'YZLH=°GJB@?

G(ω,Ω)YZEStIKGJLF?9G

PX<XLHTOYZ<XB9GJIKGJLM?wYOijGchPbY4B9TcDFEHGJLHhcG[SG%Q0ACIVQ.YZjGω

!¹ #º¨ªµ ¦§¢

N N)N) f + / $% & ACB9=9DMEHGp 1

N~hcGJ?@?9G=9TJQ.YZB'YZ?@<>ACL"[email protected]?@<bYZP>G4Gc=9?,?@B9Uc=,tB'YZLH[SG[SGJiCYZLM?wPbYP>ACLStEHGJESB[]\mACLH[SG

λ(Ω)[aE=@<XtL.YZP[STJIKA*[aESP>TtN.GJ?PbY6°ACLHhJ?@<>ACL[SG%?@B'YZLH=°GJB@?:QSB9GJLH[PbYACB@IKG2=@ES<XitYZLF?9G¸t

G(ω,Ω) ∼ β

αA2ω2à(ω)

1

p=

β

αA2 ω2cg(ω)

Ω(1 − cg(ω)),

¿ ¿ yt£ACEGJLHhcACB9GYcijGch

cg(ω) =√

1 − ω2N

G(ω,Ω) ∼ β

αA2 ω2

√1 − ω2

Ω(1 −√

1 − ω2).

¿ ¿ xj£j G,[STJijGJP>ACQSQ0GJIKGJLF?:Gc=9?vitYZPbYZeSP>GESLS<>DMEHGJIKGJLF?Q0ACESB

1−ω ΩNFhcGwDMES<0YC=9=9ESB9GPbYiCYZPX<>[a<X?9T%[aEk[STJijGJP>ACQSQ~GJIKGJLM?

[SG2¬ YOMP>ACBw[SG%PbY6B9GJPbYZ?@<>ACL[SG[a<>=@Q0GJB9=@<>ACLB9TOYZPX<>=9T%QSB9TchcTc[SGJIVIKGJLM?%YOijGchP>G%Q0GJ?@<X?Q.YZB'YZIKUJ?@B9GΩa¯\mACB9Gc=GJ?:[ST BsaN

<XP¾Gc=@?4Q~At=9=9<XeSP>GK[SG|[a<XB9GlDFEHG|[HYZLH=P>GfhOYC=4[]\ ESLHGfGonahJ<X?'YZ?@<>ACL´[SGK? MQ0GKQ.YCDMEHGJ?4[]\mACLH[SGl[SGKQ0ACIVQ.YZjGtNP>GKQSB9AZ¡HP?9GJIVQ~ACB9GJP[aEW=@<XtL.YZP[STJIKAM[aESP>T6=9GJB'Y|<XLF?9TJtB9TxsKhOYZEH='G6[aEdYChJ?9GJESB

ΩYZEg[STJLHACIV<XL.YZ?9GJESB[SG6hcGJ?@?9GGon*QSB9Gc=9=9<>ACL

¯%GJEan='ACEH=_`hOYC=PX<XIV<X?9Gc=°YZe~ACES?@<>=9=YZLF?hsVPbYIKJIKG2°ACB@IKGQ0ACESBG£ Q0GJESijGJLF?%J?@B9G2<>[SGJLF?@<¢¡.Tc=%[HYZLH=Pd\mGon*QSB9Gc=9=9<>ACL

¿ ¿ xj£<t~P>GVhOYC=2[SG6°YZ<XeSP>G[a<>=@Q0GJB9=@<>ACL[SG6iM<X?9Gc='=9GAiω 1

[HYZLH=2hcGVhOYC=cg(ω) ' cφ(Ω) ' 1

£wGJ?%P>GVhOYC=2[SGACB@?9G2[a<>=9Q~GJB9=@<>ACLAi

ωGc=9?QSB9AMh'RHG[SG

1cg(ω) cφ(Ω)

£ [HYZLH=:PbY6PX<XIV<X?9G2?9ACES?[SGIKJIKGAi1 − ω Ω

ACB9=9DMEHG

ω 1NaPbY6°ACLHhJ?@<>ACL

GY6PbY4°ACB@IKG

G(ω,Ω) ∼ β

αA2 1

Ω.

¿ ¿ sj£Gc=hcACLH[a<X?@<>ACLH=

p 1GJ?

ω 1=@<XtLS<¢¡.GJLF?¦DMEHG PbY,P>ACLStEHGJESB¾[]\ <XLF?9GJB'YChJ?@<>ACLKLHACLVPX<XLHTOYZ<XB9G

à(ω)Gc=@??@B9Uc=tB'YZLH[SG

[SGJiCYZLM?PbY:P>ACLStEHGJESB[]\mACLH[SG[aE6=9<XtL.YZP*[STJIKA*[aESP>Tλ(Ω)

NZ<dmGtI|YZPXtB9T ESLHG¾°YZ<XeSP>G [a<¢§]TJB9GJLHhcG [SG i*<X?9Gc=9=9GtNPbY:B9TJt<>ACL[]\ <XLF?9GJB'YChJ?@<>ACLGc=@? =@E_r|='YZIVIKGJLM?P>ACLStEHGQ0ACESB DMEHG,PbY=9TJQ.YZB'YZ?@<>[email protected]?@<bYZP>GwhOYZB'YChJ?9TJB@<>=@?@<>DMEHG%GJLF?@B9G%=9ACESB9hcGc= LHACLPX<XLHTOYZ<XB9Gc=%GJ?w=@<XtL.YZP[STJIKA*[aESP>T4=9AC<X?w<XIVQ0ACB@?'YZLF?9Gt0¯2YZLH=whcG6hOYC=ON.PbYV°ACLHhJ?@<>ACL

G(ω,Ω)Gc=@?,<XLH[STJQ~GJLH[HYZLM?9GV[SGPbY

¤B9TcDMEHGJLHhcG2[SGwQ~ACIVQ.YZjGωN*<dmGtSGJL[SGJRHACB9=:[SGc=='ACESB9hcGc=ONaPd\^YZIVQSPX<X?@EH[SG[aE=@<XtL.YZP[STJIKA*[aESP>T2LHG%[STJQ~GJLH[Q.YC=[SG

ωACB9=9DMEHG

ω?9GJLH[ijGJB9=

1°YOijGch?9ACEC@ACESB9=

1 − ω Ω£oNaPbY6°ACLHhJ?@<>ACL

GYVPbY4ACB@IKG=9<XIV<XPbYZ<XB9G=@ES<XitYZLF?9G

G(ω,Ω) ' β

αA2

√1 − ω2

Ω.

¿ = j£Yw[a<¢§0TJB9GJLHhcG[SG i*<X?9Gc=9=9Gc=

cg(ω)GJ?

cφ(Ω)Gc=@?<XIVQ0ACB@?'YZLF?9GtNthcGDMES<SYC='=@ESB9G ESLHG¾°ACB@?9G <XLa³HEHGJLHhcG[SG PbY,[a<>=9Q~GJB9=@<>ACL

[SGi*<X?9Gc=9=9G=@ESB|P>Gg=@<XtL.YZP%[STJIKAM[aESP>TtGdYChJ?9GJESB √1 − ω2

DMES<%YZQSQ.YZB'Y>?YZEµLFESIKTJB'YZ?9GJESB[SGPd\mGon*QSB9Gc='=@<>ACL ¿ = j£[STchJB@<X?¾PbY[STchJB9AC<>=9='YZLHhcG:[SGPd\^YZIVQSPX<X?@EH[SGw[STJIKA*[aESP>TcG:P>ACB9=9DMEHG

ω=O\^YZQSQSB9A*h9RHG[SG:PbYw¤B9TcDMEHGJLHhcG:[SG:hcACESQSESB9G

LHACB@I|YZPX<>=9TcGωc = 1

¯2YZLH=khcGc=l[SGJEanº[SGJB@LS<>GJB9==9ACEH=_`hOYC=lPX<XIV<X?9Gc=ON P>GghcACIVQ~ACB@?9GJIKGJLM?[SG

GGc=@?l<>[SGJLF?@<>DMEHGtNPd\ <XLa³HEHGJLHhcG"[SGPbY

[a<>=@Q0GJB9=@<>ACLGc=@?<XIVQ0ACB@?'YZLF?9GGJ?,=9G%?@B'YC[aES<X?:Q.YZB:ESLHG2<XLM?9TJtB'YZ?@<>ACL?9GJIVQ0ACB9GJPXP>G[aE=@<XtL.YZP [STJIKAM[aESP>TwgN N)N R%#&

ω! + !/

1 ¦\ <XLF?9TJB9J?[]\^YZL.YZPX*='GJB4hcG|hOYC=PX<XIV<X?9GtNAiPbYhcACLH[a<X?@<>ACL

Ω 1 − ωGc=@?:ijTJB@<¢¡.TcGlωGc=@?:?@B9Uc=:QSB9A*h9RHG4[SGPbY6B9TcDMEHGJLHhcG[SGhcACESQSESB9G

1£oN.Gc=@?[SGijTJB@<¢¡.GJB,PbYVhcACB@B9Gc=9Q~ACLH[HYZLHhcG6GJLF?@B9GPbY

PX<XIV<X?9GR.YZES?9G¤B9TcDMEHGJLHhcG[SG2PbY >OACLHGQSB9ACQ.YZFYZ?@<XijG6GJ?:PbYVPX<XIV<X?9Ge.YC=9=9G%B9TcDMEHGJLHhcG[SG2PbY >OACLHGTJitYZLHGc=9hcGJLF?9GlhOYC=TJ?@EH[a<>T[HYZLH=PbYV='GchJ?@<>ACL ¿ ¿ ka ¿ £o

ACB9=9DMEHGΩ 1 − ω

NaP>G2[STJijGJP>ACQSQ0GJIKGJLF?%=9ACEH=°ACB@IKG2[SG=9TJB@<>G2[SG¬Yc*P>ACB[aE?9GJB@IKGk′

1 − k′2

LHG2Q0GJES?Q.YC==9GwdYZ<XB9GYcijGch2P>G2Q0GJ?@<X?Q.YZB'YZIKUJ?@B9G

ΩNaI|YZ<>=Q~GJES?:J?@B9GB9TOYZPX<>=9TYcijGchP>G%Q~GJ?@<X?:Q.YZB'YZIKUJ?@B9G

1 − ωt

k′1 − k′

2 = k′(ω) − k′(ω − Ω) = k′(1 − (1 − ω)) − k′(1 − Ω − (1 − ω))

' k′(1 − (1 − ω)) −[k′(1 − Ω) − (1 − ω)

∂k′

∂ω(1 − Ω) +

1

2(1 − ω)2

∂2k′

∂ω2(1 − Ω) + . . .

]

' k′(1 − (1 − ω)) −[k′(1 − Ω) − (1 − ω)√

2Ω' −

√2Ω +

√2(1 − ω)

].

¿ =m£

5 4! P#& P+ /x : N$&u

¹,LHGW=@<XIVQSPX<¢¡.hOYZ?@<>ACLYZQSQ.YZB'Y>?GJL¸ES?@<XPX<>='YZLF?P>G"[STJijGJP>ACQSQ0GJIKGJLF?gPX<XIV<X?9T"[SGWPbY°ACLHhJ?@<>ACL YZB9hc=@<XL(1 − x)

DMES<<XLM?9GJB@iM<>GJLM?,[HYZLH=PbYVB9GJPbYZ?@<>ACL[SG[a<>=@Q0GJB9=@<>ACL ¿ ^sF£@£vP>ACB9=9DMEHG

x 1tHYZB9hc=@<XL

(1 − x) ' π/2 +√

2x + . . .

Y[SGJB@LS<>UJB9GGonaQSB9Gc=9=@<>ACL[SG| ¿ =mZ£oNShcACIeS<XLHTcGYcijGchhcG2[STJijGJP>ACQSQ0GJIKGJLF?%[SGYZB9hc=@<XLNS[SGJi*<>GJLF?

k′1 − k′

2 ' −√

2Ω + . . . , ¿ =´kj£

GJ?P>Gc=?9GJB@IKGc==9ES<XiCYZLM?9=ONDMES<MYZQSQ.YZB'YZ<>=9=9GJLM?[HYZLH=PbY °ACB@IKG¦jTJLHTJB'YZP>G¾[SG¦PbY°ACLHhJ?@<>ACL[SG¦?@B'YZLH=°GJB@?v ¿ ktF£0QSB9GJLSLHGJLF?P>Gc=ACB@IKGc=»t

sin(k1 − k∗2) ' sin(−

√2Ω − 2i|k′′(ω)|) ' −

√2Ω − 2i|k′′(ω)|

cos(k1 − k∗2) ' 1 − 2 sin2

(k1 − k∗

2

2

)' 1 − 1

2

(√2Ω + 2i|k′′(ω)|

)2

cos(k1 + k∗2) ' cos(k′

1 + k′2) ' −1 + Ω .

¿ = ¿ £j Gc=GonaQSB9Gc=9=@<>ACLH=PX<XIV<X?9Gc= ¿ = ¿ £=9ACLF?B9GJQ~ACB@?9TcGc=g[HYZLH=Pd\mTcDFE.YZ?@<>ACL ¿ ktj£kQ0ACESB[SACLSLHGJBgESLHGLHACESijGJPXP>G

GonaQSB9Gc=9=@<>ACL[SGPbY4ACLHhJ?@<>ACL[SG?@B'YZLH=GJB@?Gt

G(ω,Ω) ' β

αA2

√2Ω + 2i|k′′(ω)|

Ω2 −[√

2Ω + 2i|k′′(ω)|]2 .

¿ = =M£

j ACIVIKGΩ 1

N0P>G?9GJB@IKGΩ2 YZE"[STJLHACIV<XL.YZ?9GJESB4[SGVhcGJ?@?9GVGonaQSB9Gc=9=@<>ACLGc=@?2?9ACEC@ACESB9=2LHTJtPX<XjGOYZeSP>GK[SGJitYZLF? Ω

Y6LHAC?'YZ?@<>ACL

2|k′′(ω)| = 1/à(ω)Gc=@?:B9To_d<XLM?@B9AM[aES<X?9GtN8GJ?

G=9G2=@<XIVQSPX<¢¡.G¸t

G(ω,Ω) ' −β

αA2 1√

2Ω + 2i|k′′(ω)|.

¿ = À £

,EGJLHhcACB9GtN.GJL<XLF?@B9A*[aES<>='YZLM?P>G2Q.YZB'YZIKUJ?@B9G6YC[a<XIKGJLH=@<>ACLSLHGJPP =

√Ω2 à(ω)

NG=O\mTchJB@<X?=9<XIVQSP>GJIKGJLF?5t

G(ω,Ω) ' − β

2αA2à(ω)

1

P + i.

¿ =Fj£ACB9=9DFEHGPbYhcACLH[a<X?@<>ACL

1−ω ΩGc=@?¾B9TOYZPX<>=9TcGtNjhcG:DMES<HTJ?'YZ<X?¦P>G:hOYC=¦[HYZLH=¾PbY%=9ACEH=_`=9GchJ?@<>ACL|QSB9TchcTc[SGJLF?9G ¿ ¿ kanmtnmjN

PbYfR.YZES?9G4B9TcDFEHGJLHhcGhOYZB'YChJ?9TJB@<>=@?@<>DMEHGK[SG6Q~ACIVQ.YZjGGc=@? ω1+ω22 = ω+ω−Ω

2 ' ωN~P>ACB9=9DMEHG

Ω ω~ Pd\ <XLFijGJB9='GtN

[HYZLH=¾PbY2QSB9Tc=9GJLM?9G,YZL.YZPXa=9GtNjPbYhcACLH[a<X?@<>ACL1−ω Ω

<XIVQSPX<>DFEHG,DFEHGPbY2¤B9TcDMEHGJLHhcGhOYZB'YChJ?9TJB@<>=9?@<>DFEHGw[SGQ0ACIVQ.YZjGGc=9?4I|YZ<XLM?9GJL.YZLF? ω1+ω2

2 = ω+ω−Ω2 = 2+2(1−ω)−Ω

2 ' 1 − Ω2

¯2YZLH=P>GkhOYC=PX<XIV<X?9GfQSB9Tc='GJLF?1 − ω Ω

N P>GQ.YZB'YZIKUJ?@B9G

pES?@<XPX<>=9T2Q0ACESBPd\^YZL.YZPX*=9G[aEhOYC=

1 − ω ΩQSB9TchcTc[SGJLM?Q0GJES?:J?@B9G2B9TcTchJB@<X?hcACIVIKGt

p = à(ω)Ω

cΦ(Ω)

cΦ(Ω) − cg(ω)

cg(ω)' à(ω)

Ω

cg(1 − Ω/2),

¿ =Myt£

YOijGchcΦ(Ω) ' 1

GJ?cg(ω) ' cg(1−Ω/2) → 0

g<XL.YZP>GJIKGJLM?ONjhcACIVIKGcg(1−Ω/2) '

√1 − (1 − Ω/2)2 '

√ΩN

P>GkQ.YZB'YZIKUJ?@B9GQSB9TchcTc[SGJLF?pGc=@?KTcDMES<XitYZP>GJLF?fYZE±LHACESijGOYZE²Q.YZB'YZIKUJ?@B9G

P¦\ <XLM?9GJB@QSB9TJ?'YZ?@<>ACL½QSRM*=@<>DMEHG[SGhcGc=

Q.YZB'YZIKUJ?@B9Gc=,Gc=@?:[SACLHh2<>[SGJLM?@<>DFEHGtACB9=9DMEHG

P 1NMP>G,[STJQSR.YC=YZjGwGJLM?@B9G%=9ACESB9hcGc= LHACLfPX<XLHTOYZ<XB9Gc=GJ? =@<XtL.YZP~[STJIKA*[aESP>TsPbYPX<XIV<X?9Gw[SGPbYB9TJt<>ACL

[]\ <XLM?9GJB'YChJ?@<>ACLGc=@?:°YZ<XeSP>GtNSGJ?:PbY4°ACLHhJ?@<>ACL[SG2?@B'YZLH=@GJB@?QSB9GJLH[PbY4ACB@IKGxt

G(ω,Ω) ' −iβ

2αA2à(ω) .

¿ =Fxj£

!¹ #º¨ªµ ¦§¢

j GJ?@?9G°ACLHhJ?@<>ACL"Gc=@?wQSB9ACQ~ACB@?@<>ACLSLHGJPXP>Gxsà(ω)

GJ?,LHG6[STJQ0GJLH[Q.YC=%[SGΩj GhOYC=,PX<XIV<X?9GhcACB@B9Gc=@Q~ACLH[geS<>GJLusKPbY

PX<XIV<X?9Gp 1

N1 − ω Ω

NSGonaQSB9Gc=9=@<>ACL ¿ ¿ M£oACB9=9DMEHG

P 1NFP>Gw[STJQSR.YC='YZjG%GJLF?@B9G%=9ACESB9hcGc= LHACLlPX<XLHTOYZ<XB9Gc=GJ?v=@<XtL.YZP][STJIKA*[aESP>ThsPbYPX<XIV<X?9Gw[SG,PbYB9TJt<>ACL

[]\mGonahJ<X?'YZ?@<>ACLGc=@?<XIVQ~ACB@?'YZLM?ONHPbY6°ACLHhJ?@<>ACL[SG2?@B'YZLH=@GJB@?QSB9GJLH[PbYACB@IKG t

G(ω,Ω) ' − β

2αA2 1√

Ω.

¿ =Fsj£

¯2YZLH=¦hcGhOYC=ONtP>G dYChJ?9GJESB1/√

ΩYC='=@ESB9GPd\ <XLF?9TJtB'YZ?@<>ACLlQ.YZB@?@<>GJPXP>G[]\mACB9[aB9G¶m*okj£[]\ ESLKQSB9AZ¡HPH?9GJIVQ0ACB9GJPdNj[STJIKAM[aESP>T

s6Q.YZB@?@<XB[]\ ESLHGACLH[SG[SGQ~ACIVQ.YZjG[SG%? MQ0G2Q.YCDMEHGJ?[]\mACLH[SGtN N)N *³+ /x$&u +´/¸/ ´´/¸ "$% ´´/

¦\^YZL.YZPXa=9G¦[aE2B9Tc=@ESPX?'YZ? ¿ nmOsF£~[HYZLH=PbY B9TJt<>ACL[SGc= =9ACESB9hcGc=]LHACLPX<XLHTOYZ<XB9Gc=NOht\mGc=@?_s®_`[a<XB9GP>ACB9=9DMEHG |eik(Ω)an−i∆kan|L\mGc=@?Q.YC=?@B9Uc=<XLa°TJB@<>GJESB.s1Nt='GEH=@?@<¢¡.GQ0ACESB¦[SGc=ACLH[SGc=¦[SGQ0ACIVQ.YZjGQSB9ACQ.YZFYZ?@<XijGc=OjACESB¾[SGc=ACLH[SGc=¦[SGQ0ACI_

Q.YZjGKTJitYZLHGc=9hcGJLF?9Gc=ONYZ?@?9TJLFEHTcGc=6=@ESB2ESLHGK[a<>=@?'YZLHhcGf[SGKDMEHGJP>DFEHGc=2eS<XPXP>Gc=N]PbYkB9TJt<>ACL='ACESB9hcGKGc=@?=@<PX<XIV<X?9TcGf[HYZLH=Pd\[email protected]|DMEHGKPd\mACL´=9GV?@B9ACESijGfQSB9Gc=9DMEHGK?9ACEC@ACESB9=GJL[SGJRHACB9=O G|B9Tc=9ESPX?'YZ?l ¿ nmOsF£TJ?'YZLF?6P>G|QSB9A*[aES<X?4[aE?9GJB@IKG[SGQSR.YC='G [

1 − eik(Ω)an−i∆kan]eiΩt−ik(Ω)an GJ?[SG,PbY°ACLHhJ?@<>ACLk[SGw?@B'YZLH=°GJB@?

G(ω,Ω)TJ?@EH[a<>TcGwQSB9TchcTc[SGJIVIKGJLM?

Q~ACESBP>Gc=¦ACLH[SGc=¾[SGvQ0ACIVQ.YZjGQSB9ACQ.YZFYZ?@<XijGc=ONj<XPSB9Gc=@?9G P>G?9GJB@IKG[SGQSR.YC=9Gs2YZL.YZPX*='GJBOtG?9GJB@IKGeiΩt−ik(Ω)an [STo_hJB@<X?PbYQSB9ACQ.YZFYZ?@<>ACLV[SGPd\mACLH[SG wg[STJIKAM[aESP>TcGv[HYZLH=PbYh9R.Y>LHGtZ`PFB9Gc=@?9GRsYZL.YZPXa=9GJB P>G?9GJB@IKG

1−eik(Ω)an−i∆kan NGJ?Q.YZBhcACLH=9TcDMEHGJLF?ºs°YZ<XB9G2TJIKGJB@jGJB,P>G2hcACIVQ~ACB@?9GJIKGJLM?w[SG%PbYQSR.YC='G%LHAC?9TcGΦ=@ES<XitYZLF?9Gxt

Φ = (∆k − k(Ω))an . ¿ À j£

¯2YZLH=¾PbYPX<XIV<X?9G,AifPd\^YZ?@?9TJLFE.YZ?@<>ACL[SGc= ACLH[SGc=[SG:Q0ACIVQ.YZjG,Gc=@?¾°YZ<XeSP>GtNFP>GIKA*[aESP>G[aEl?9GJB@IKGeik(Ω)an−i∆kanGc=@?¦QSB9A*h9RHGw[SG

1*¯%YZLH=vPd\^YZL.YZPXa=9GDMES<8=@ES<X?ONFPbY2Q.YZB@?@<>G,<XI|YZt<XL.YZ<XB9G,[aElLHACIeSB9Gw[]\mACLH[SG,QSB9ACQ.YZFYZ?@<¢ Gc=@?¾LHTJtPX<XjTcGt

∆k = k′(ω1) − k′(ω2)Q~GJES?J?@B9G4=@<XIVQSPX<¢¡.T=9<

k′(ω1 = ω + Ω/2)GJ?

k′(ω2 = ω − Ω/2)=9ACLF?w[STJijGJP>ACQSQ~Tc=wGJL

=9TJB@<>Gc=w[SG6¬Yc*P>ACBK=9ACEH=wPbY|hcACLH[a<X?@<>ACL1 − ω Ω

£o8¯2YZLH=%hcG4hOYC=N∆k ' Ω/cg(ω)

EH=9DME\^YZE?@B9AC<>=9<>UJIKG4ACB9[aB9GGJL

ΩNSAi

cg(ω) = ∂ω∂k (ω)

Gc=@?ONaB'YZQSQ~GJP>ACLH=@_dP>GtNSPbYi*<X?9Gc=9=9G2[SG%tB9ACESQ0G[aEQ.YCDMEHGJ?:[]\mACLH[SG[SG%Q0ACIVQ.YZjG;ºgG4LHACIeSB9G[]\mACLH[SG

k(Ω)Gc=@?wTJFYZP,s

Ω/cΦ(Ω)N~Ai

cΦ(Ω)Gc=@?,PbYKi*<X?9Gc=9=9G4[SGQSR.YC='G<s|PbYVe.YC=9='GB9TcDFEHGJLHhcG

Ωj GJ?@?9GViM<X?9Gc='=9G[SGVQSR.YC=9GVGc=@?ACeS?9GJLMEHG¸slQ.YZB@?@<XB[SGPbYlB9GJPbYZ?@<>ACL[SGV[a<>=@Q0GJB9=@<>ACL· ¿ ^sM£wGJ?=O\mGon*QSB@<XIKG|=9ACEH=%PbY

ACB@IKGcΦ(Ω) = Ω/ arcsin(Ω) ' 1

Q0ACESBΩ 1

]YiM<X?9Gc=9='GV[SGtB9ACESQ~GK='GV[STJB@<XijG|YZEH=9=@<¦[SGPbYkB9GJPbYZ?@<>ACL[SG[a<>=@Q0GJB9=@<>ACL ¿ ^sF£GJ?wGc=@?TJFYZP>G<s

cg(ω) = cΦ(0)√

1 − ω2.YZB%hcACLH=9TcDMEHGJLF?ONHPbYKQSR.YC=9Gf ¿ À M£bsKYZL.YZPXa=9GJBQ0GJES?

=9G%B9TcTchJB@<XB9GYOijGch[SGc=DME.YZLF?@<X?9Tc=LHACB@I|YZPX<>=9TcGc=¶t

Φ = −2Ω

(1 − 1

cg(ω)

)n .

¿ À m£

qM<8hcGJ?@?9GwQSR.YC=9GwGc=@?vTJFYZP>Gs −2π`¤B9Gc=@Q~GchJ?@<XijGJIKGJLM? −(2π +1)`

£¾YOijGch` ∈ N

NFP>G?9GJB@IKG1− eik(Ω)an−i∆kan [aE

B9Tc=@ESPX?'YZ?f ¿ nmOsF£wGc=9?IV<XLS<XIESI¤B9Gc=@Q~GchJ?@<XijGJIKGJLM?4I|Y®na<XIESIf£oGc=%B9TcDMEHGJLHhcGc=[SGVQ0ACIVQ.YZjGVQ0ACESBP>Gc=9DMEHGJPXP>Gc=ESLIV<XLS<XIESI Gc=@?ACeS?9GJLFEN.='YZ?@<>=°ACLF?,YZP>ACB9=PbY6B9GJPbYZ?@<>ACL=@ES<XitYZLF?9GtN

ω =

√

1 −(

1 − π`

Ωn

)−2

, ¿ À kj£

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n =π`

Ω(1 − (1 − ω2)−1/2)−1 .

¿ Àt¿ £

4! P#& P+ /x : N$&u

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eik(Ω)an−i∆kan L\mGc=@?QSPXEH=TJFYZP s1I|YZ<>=[SGJi*<>GJLF?:<XLa°TJB@<>GJESBO

q*ACLhcACIVQ0ACB@?9GJIKGJLF?w[STJQ~GJLH[[SGwPbY[a<¢§]TJB9GJLHhcG2GJLF?@B9G2PbY4i*<X?9Gc=9=9G%[SG%QSR.YC=9G%wg[aE=@<XtL.YZP[STJIKA*[aESP>TcΦ(0) ' 1GJ?PbYViM<X?9Gc=9='G2[SG2tB9ACESQ~G;ºg±[SGc=:='ACESB9hcGc=:LHACLPX<XLHTOYZ<XB9Gc=

cg(ω)H¯2YZLH=PbY6B9TJt<>ACL[]\mGona<>=@?9GJLHhcG[SGc=='ACESB9hcGc=:LHACL

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e−i∆kan [SG%Pd\mTcDFE.YZ?@<>ACL´ ¿ nmOsF£ Gc=@??@B9Uc=<XLa°TJB@<>GJESBhs1£o

j ACIVIKG"[HYZLH=Pd\^YZL.YZPX*='G"[SGWPbY´[STJIKAM[aESPbYZ?@<>ACL[]\mACLH[SGc=QSB9ACQ.YZFYZ?@<XijGc=ON<XPGc=@?Q0At=9=@<XeSP>GW[SGWB9GJQ.YZB@?@<XB[SGPd\mGonaQSB9Gc=9=@<>ACLjTJLHTJB'YZP>Gk[SG|PbY°ACLHhJ?@<>ACL[SGl?@B'YZLH=°GJB@?k ¿ ktM£%Q0ACESB4Pd\^YZL.YZPXa=9Gk[SG|PbY[STJIKA*[aESPbYZ?@<>ACL´[SGc=ACLH[SGc=TJitYZLHGc=9hcGJLM?9Gc=O_gSB9TcDFEHGJLHhcGc=GJ? LHACIeSB9Gc=[]\mACLH[SGc=='ACLF?[SACLHh,LHACB@I|YZPX<>='Tc=B9Gc=@Q~GchJ?@<XijGJIKGJLM?Q.YZB

ωcGJ?

2kc/πaY

B9GJPbYZ?@<>ACL[SG[a<>=@Q0GJB9=@<>ACLLHACB@I|YZPX<>=9TcGtNHQ0ACESBP>Gc=IKA*[SGc=:TJitYZLHGc=9hcGJLF?9=6ω > 1

£oNH=O\mTchJB@<X?YZP>ACB9=¶t

k(ω) =π

2− i

YZB9hcAt=@R(ω) .

¿ À =M£¥¦LgQ0At='YZLM?

ω2 = ωGJ?

ω1 = ω + ΩYcijGch

Ω > 0N~PbY|ACLHhJ?@<>ACL"[SG6?@B'YZLH=°GJB@?V ¿ ktF£:Q0GJES?%='GB9TcTchJB@<XB9GVhcACIVIKG

ESLHGw°ACLHhJ?@<>ACL[SGωGJ?

ΩSGc=[STJijGJP>ACQSQ0GJIKGJLF?9=wGonSYChJ?9=:=9ES<XiCYZLM?9=

sin(k1 − k∗2) = iω

√(ω + Ω)2 − 1 + i(ω + Ω)

√ω2 − 1

cos(k1 − k∗2) = ω(ω + Ω) +

√(ω + Ω)2 − 1

√ω2 − 1

cos(k1 + k∗2) = −ω(ω + Ω) +

√(ω + Ω)2 − 1

√ω2 − 1 ,

='ACLF?%=@ESeH=@?@<X?@EHTc=2[HYZLH=wPd\mTcDFE.YZ?@<>ACL· ¿ ktM£Q0ACESBw[SACLSLHGJB2ESLHG°ACB@IKG6GonHYChJ?9G[SGPbY|°ACLHhJ?@<>ACL"[SG?@B'YZLH=°GJB@?%Q~ACESB[SGc=ACLH[SGc=:[SG2Q0ACIVQ.YZjGTJiCYZLHGc='hcGJLF?9Gc=¶t

G(ω,Ω) = −iβ

αA2

ω(ω + Ω)[ω√

(ω + Ω)2 − 1 + (ω + Ω)√

ω2 − 1]

Ω2 +[ω√

(ω + Ω)2 − 1 + (ω + Ω)√

ω2 − 1]2 .

¿ ÀtÀ £

¥¦LQSB9GJL.YZLM?%GJLWhcACIVQS?9G6P>GdYZ<X?,DMEHGΩ ω

N8GJ?,Q0ACESBw[SGc=,B9TcDFEHGJLHhcGc=w[SG4Q~ACIVQ.YZjG=@E_r|='YZIVIKGJLM?2TJP>AC<XtLHTcGc=[SG%PbY¤B9TcDMEHGJLHhcG2[SGhcACESQSESB9G|

ω − 1 Ω£ hcGJ?@?9GGon*QSB9Gc=9=9<>ACL[SGJiM<>GJLM?¶t

G(ω,Ω) ' −iβ

αA2 ω

2√

ω2 − 1.

¿ À j£ P.Gc=@?v<XLF?9TJB9Gc=9=YZLF?[SG:B9GJI|YZB9DMEHGJBDFEHG,[HYZLH=vhcG,hOYC=ONjP>ACB9=9DMEHG

ω[SGJiM<>GJLM?vtB'YZLH[SGwQ.YZB¾B'YZQSQ0ACB@?bs2PbY%B9TcDMEHGJLHhcG,[SG

hcACESQSESB9Gωc = 1

NHP>G4IKAM[aESP>G |G(ω,Ω)| ∼ βα

A2

2

Gc=@?,<XLH[STJQ~GJLH[HYZLM?%[SG4PbYB9TcDMEHGJLHhcG4[SGQ~ACIVQ.YZjGωYZ<XLH=@<DFEHG

[SGPbYk¤B9TcDMEHGJLHhcGK[a<¢§]TJB9GJLHhcGΩ]Gc=='ACESB9hcGc=2LHACLPX<XLHTOYZ<XB9Gc=='ACLF?P>A*hOYZPX<>=9TcGc==@ESBPbYk=@ESB°YChcG|hcGVDFES<¦<XLSRS<Xe0GP>Gc=

Go§]GJ?9=w[SG[a<>=@Q0GJB9=@<>ACLW[SGi*<X?9Gc=9=9GGJ?%[]\^YZ?@?9TJLFE.YZ?@<>ACL]Pd\ <XLMijGJB9=9GtN8?9ACECACESB9=%GJLWhcACLH=@<>[STJB'YZLF?2DMEHGΩ ω

N.I|YZ<>=Q0ACESB[SGc=¤B9TcDMEHGJLHhcGc=w[SGQ0ACIVQ.YZjG?@B9Uc=,QSB9AMh'RHGc=,[SGPbYB9TcDFEHGJLHhcG4[SGhcACESQSESB9Gk

ω − 1 Ω£oNHPbYACLHhJ?@<>ACLW[SG

?@B'YZLH=@GJB@?itYZES?¶tG(ω,Ω) ' −i

β

αA2

√2√Ω

. ¿ À yt£

!¹ #º¨ªµ ¦§¢

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Ω,EhcACLF?@B'YZ<XB9GtN PbYPX<XIV<X?9Gg ¿ À yj£oNYC='=@ESB9GKDMEHGVP>G

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ω2 = ωGJ?ω1 = ω + Ω

YOijGchΩ > 0

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ω1 = ωGJ?

ω2 = ω − ΩYcijGch

Ω > 0DFES<Y|TJ?9TES?@<XPX<>=9TcGt0Y

hcACIVQ.YZB'YZ<>=9ACL"[aEB9Tc=9ESPX?'YZ?V ¿ À yF£:ACeS?9GJLME[HYZLH=,PbY|PX<XIV<X?9GTJiCYZLHGc='hcGJLF?9Gω → 1

Nω > 1

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ω → 1Nω < 1

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ΩQ.YZB −Ω

j GchJ<adYZ<X?¾YZQSQ.YZB'Y>?@B9GESLdYChJ?9GJESB −i[HYZLH=PbY,°ACLHhJ?@<>ACLK[SG?@B'YZLH=°GJB@?¦Ai −Ω

Gc=@?<XLF?@B9A*[aES<X?GJLkB'YZ<>=9ACL[aEk°YChJ?9GJESB

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sPbYk?@B'YZLH=@<X?@<>ACL´[SGω < 1

sω > 1

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ω

ΩG[STJQSPbYChcGJIKGJLM?,[STJIKA*[aESP>TGc=@?:[HYZLH=:P>GhOYC=jTJLHTJB'YZPQSB9ACQ0ACB@?@<>ACLSLHGJPs

UΩ ∼ FNL(n = 0, ω,Ω)

Ω2 − sin2(∆k),

¿ À xj£

AiFNL

Gc=@?kPbY"°ACB9hcGgLHACLºPX<XLHTOYZ<XB9G<XIVQSPX<>DMEHTcG"[HYZLH=lP>GgQSB9AMhcGc=9=9EH=l[]\^YZES?9AZ_`[STJIKA*[aESPbYZ?@<>ACL ¤PbY[STJQ0GJLH[HYZLHhcG?9GJIVQ~ACB9GJPXP>G

eiΩt Gc=@?:ACIV<>=9G£o.G2[STJLHACIV<XL.YZ?9GJESBw[SG%? MQ0G ¨ B9Tc=9ACL.YZLM?ªGc=@?R.YZeS<X?@EHGJPQ0ACESB:P>Gc=QSB9A*hcGc=9=@EH= °ACB9hcTc=O¯2YZLH=:P>GhOYC=,PX<XIV<X?9Gω 1

¤P>GQ.YZB'YZIKUJ?@B9GVYC[a<XIKGJLH=@<>ACLSLHGJPp 1

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à(ω)t

1

Ω2 − sin2(∆k)∼ à(ω)λ(Ω) .

¿ À sj£ Pd\ <XLFijGJB9=9GtN[HYZLH=P>GfhOYC=PX<XIV<X?9G

ω 1N<XP¦L\ YQ.YC=4[]\mGo§]GJ?9=[]\^YChchJESI4ESPbYZ?@<>ACL²[aE=@<XtL.YZPdN=9<XIVQSP>GJIKGJLF?

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sin2(∆k) ' − sinh2(2|k′′(ω)|) ∼ e4|k′′(ω)| ∼ ω4 ¿ ^ j£[SACIV<XLHG°ACB@?9GJIKGJLF?¾[HYZLH=P>G[STJLHACIV<XL.YZ?9GJESBvB9Tc=9ACL.YZLF?ONMGJ?P>Gc=hcACLH[a<X?@<>ACLH=¾[]\^YChchJESI4ESPbYZ?@<>ACLl[aEK=@<XtL.YZP8[a<XIV<XLFEHGJLF?B'YZQS<>[SGJIKGJLF?%YcijGchPd\^YZEStIKGJLM?'YZ?@<>ACLg[SG

ω

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nGc=9?vTJFYZP>GsPbY4[a<¢§]TJB9GJLHhcGwGJLM?@B9GwP>Gc=¦°ACB9hcGc=YZQSQSPX<>DFEHTcGc==@ESB =9Gc=[SGJEanfh§vC?9Tc=O*ºQ.YZB@?@<XB[SG

Pd\mTcDFE.YZ?@<>ACL ¿ ryj£oNFNL(n) ∼ [Uω(n + 1) − Uω(n)]2 − [Uω(n) − Uω(n − 1)]2 ,

¿ ^_m£[HYZLH=PbYPX<XIV<X?9G2e.YC='=9Gw¤B9TcDMEHGJLHhcG|

ω 1£oNHh'R.YCDFEHG[STo°ACB@I|YZ?@<>ACL

[Uω(n + 1) − Uω(n)] /aGJ?

[Uω(n) − Uω(n − 1)] /a ,

Gc=@?:<XLMijGJB9=9GJIKGJLF?%QSB9ACQ0ACB@?@<>ACLSLHGJPXP>G³sVPbYVP>ACLStEHGJESB%[]\mACLH[SG4YChcACEH=@?@<>DMEHG4;hgNHhcGDMES<Gc=@?QSB9ACQ~ACB@?@<>ACLSLHGJPEsω. Y

[a<¢§0TJB9GJLHhcG6GJLF?@B9G6P>Gc=ACB9hcGc=%YZQSQSPX<>DMEHTcGc= sKPbYKeS<XPXP>G6[SGh'R.YCDFEHGh§vC?9T6Gc=@?,hcACLM?@BBvCP>TcG6Q.YZB,Pd\^YZeH=9ACB@QS?@<>ACL"[SGPd\mACLH[SG

4! P#& P+ /x : N$&u

;hgN8ACEgQ.YZB%PbYl[a<¢§0TJB9GJLHhcGGJLF?@B9GP>Gc=wR.YZES?9Gc=wB9TcDFEHGJLHhcGc=N~ht\mGc=9? sl[a<XB9G6Q.YZBΩfj GchJ<<XIVQSPX<>DMEHGtN~[HYZLH=%PbY|PX<XIV<X?9G

ω 1N

FNL ∼ ω2 maxk(Ω), k′′(ω)

.

¯%YZLH= P>G%hOYC=[]\ ESLHG2YZLF?9GJLSLHGYOijGchà(ω) λ(Ω)

°YZ<XeSP>G2YZ?@?9TJLME.YZ?@<>ACL[SGc=ACLH[SGc=[SGwQ~ACIVQ.YZjG£wpûwAOiSxjyz`N

FNL ∼ ω2k(Ω) ¿ ^kj£

YZBhcACLH='TcDFEHGJLM?ONs6Pd\^YZ<>[SG[SGc=:TcDME.YZ?@<>ACLH=4 ¿ À xM£oN ¿ À sj£GJ? ¿ ^kF£oN

UΩ ∼ ω2à(ω) , ¿ ^ ¿ £

hcG2DMES<Gc=@?P>G2B9Tc=9ESPX?'YZ?hJPbYC=9=@<>DMEHG%Q~ACESBPd\^YZLM?9GJLSLHGQ.YZB'YZIKTJ?@B@<>DMEHG¸mo_`¯[HYZLH=P>Gc=IV<XPX<>GJEanRHACIKACjUJLHGc=pmuwAOiSxjy®z`¯%YZLH=PbYkPX<XIV<X?9G

ω 1N0PbY=@<[email protected]?@<>ACLGc=@?[a<¢§]TJB9GJLF?9Gt Yl°ACB9hcG

FNL[SGVPd\mTcDME.YZ?@<>ACLµ ¿ ^_m£%Gc=9?hcACLF?@BBvCP>TcG

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FNL(n = 0, ω,Ω) ∼ −U 2ω(−1) ∼ e4|k′′(ω)| ∼ ω4 .

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ω 1NUΩ

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ω

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ω ∼ 1ACEN,GJL jTJLHTJB'YZPdN,P>ACB9=9DMEHG"P>Gc=

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n¤LHACIeSB9G

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Ω ' 8.10−4 NGJ?C = 2.10−7 j ACIVIKG Ω ω1, ω2

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hcACIVIKGω ' 8 10−2, 3 10−1, 6 10−1, 0.98

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p 1N

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0 ≤ cg(ω) ≤ 1GJ?P>GK=@<XtL.YZP [STJIKAM[aESP>TfDFES<¾='GQSB9ACQ.YZjGskPbYki*<X?9Gc=9=9G

cΦ(Ω) ' 1 q*GJP>ACLPd\mTcDME.YZ?@<>ACL

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` = −1NΩ ' 8 10−4 GJ? ω ' 0.98

[SAC<X?YZQSQ.YZB'Y>?@B9G sKPbYf[a<>=@?'YZLHhcGn ' 1100

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p ' 13=@ESB:PbY6¡HtESB9G

¿ ¿ ACESBhcACIVQ.YZB'YZ<>=9ACLN2Pd\^YZIVQSPX<X?@EH[SG±[aE =@<XtL.YZPV[STJIKA*[aESP>T´Q.YZBW[SGJEanACLH[SGc="[SGQ~ACIVQ.YZjG±TJitYZLHGc=9hcGJLF?9Gc=

ω ' 1, 083

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4! P#& P+ /x : N$&u

100

101

102

103

104

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100

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p ' 0.65

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n=200 n=500

n=3000 n=7000

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cg(ω) < cΦ(Ω)£o jw\mGc=@?hcGVDMES<¦Gc=@?2<XPXPXEH=@?@B9T

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p 1GJ?

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p 1ω ' 0.97

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GJ?n = 7000

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p 1Nω ' 0.083

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−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t/τm

!

1 % '* #$1 *%)+*& ' *,,*) 0 #1) (% 02 /43(35

h N N ¤Zª¨¦p«%©%«%¬¨'ª¦¦¬*«¥¢ 3(µnª²«&¦£¢¤¦«%¬¨d²´¢ .«%©:¢p¦¦¢5£_¬±!¤ µ ¢¶£¤2¬$2Eµb²´¢ .«%©:¢p¦p¦§¢l²!¥¯x¬²±µn¥B¢ 3£_ª¤©%«¤² · ¬¨²´¢p¦x²´¢º£_¬¯>£ª1;´¢£¤2¬p£_ª1;ª©%«/.¢¦§¹8 ¢º£_ª¤2ª¯ ©%¤B¢

p.ª±_©º²±u¨¬«&¤ ª± ;¤«&¦ µnª«¤³¤2¢p¦:£¢2§©%« .¢¯¸¢¨©

0.16 0.731.6 2.76 6.6 12.2

%C = 2 10−7 %.¢© Ω ' 1.7 10−3 ¹

BIHEH&G hYLN]RM½ E½a ] LNMMa,^½:[Q"¿L"OR½:M`bQY!M½:a ]¾OEQ E½ EQUMMQ

qaYZLH=:h9R.YZLSjGJBwPbY[a<>=@?'YZLHhcG[]\mACeH='GJB@iCYZ?@<>ACL[aE=@<XtL.YZP[STJIKA*[aESP>TtNH<XPGc=@??9ACES?,[SG2IKJIKG2Q0At=9=@<XeSP>G[]\mACeS?9GJLS<XBESLHG6?@B'YZLH=@<X?@<>ACL"[HYZLH=%PbY|°ACB@IKG6[aEWQSB9AZ¡HP?9GJIVQ0ACB9GJP[SG4i*<X?9Gc=9=9G6[STJIKA*[aESP>TcGVGJLWh'R.YZLSjGOYZLF?4ESLgQ.YZB'YZIKUJ?@B9GV[aE=9<XtL.YZP[SG6Q0ACIVQ.YZjG6hcACIVIKG6PbY|B9TcDFEHGJLHhcG6hcGJLF?@B'YZP>GV[aEWQ.YCDMEHGJ?%[]\mACLH[SGACEWGJLHhcACB9GP>G?9GJIVQH=2hOYZB'YChJ?9TJB@<>=9?@<>DFEHG[SGWIKA*[aESPbYZ?@<>ACL

τm ∼ 1/ΩºqMESBPbY´¡HtESB9G ¿ À NESLHG?9GJPXP>G?@B'YZLH=@<X?@<>ACL Gc=9?B9GJQSB9Tc=9GJLF?9TcGtNQ~ACESBPbYCDFEHGJPXP>G"P>G

Q.YZB'YZIKUJ?@B9GYC[a<XIKGJLH=9<>ACLSLHGJPpiCYZB@<>GW[SG

p ' 0.16s

p ' 12.2GJLh'R.YZLSjGOYZLF?PbYB9TcDFEHGJLHhcGgQ0ACB@?9GJEH=9G"[aE

Q.YCDMEHGJ?[]\mACLH[SG A YZEH=9=@<>GJLGJLF?@B9Gω ' 0.083

GJ?ω ' 0.97

,j Gc=4hOYC=PX<XIV<X?9Gc=6hcACB@B9Gc=@Q~ACLH[SGJLM?4B9Gc=9Q~GchJ?@<XijGJIKGJLM?YZEan|GonaQSB9Gc=9=@<>ACLH=2 ¿ ¿ F£¾GJ?2 ¿ ¿ xj£¦[SG,PbY2°ACLHhJ?@<>ACL[SG?@B'YZLH=@GJB@?

G(ω,Ω)[aElQSB9A*hcGc=9=@EH= []\^YZES?9AZ_`[STJIKAM[aESPbYZ?@<>ACL

ACESBp 1

NG(ω,Ω) ∼ ω2à(ω)

NHYZP>ACB9=DMEHGwQ~ACESBp 1

NG(ω,Ω) ∼ ω2cg(ω)/Ω(1 − cg(ω))

N*hcG%DMES<YC=9=@ESB9GESLHG4<XLF?9TJtB'YZ?@<>ACL[aEW=@<XtL.YZP[STJIKAM[aESP>Tt~,EIKACIKGJLM?%[SG4PbY|?@B'YZLH=@<X?@<>ACL²¤<XLF?9TJtB'YZ?@<>ACL[aEgQSB9AZ¡HP[SGiM<X?9Gc='=9G£oN.P>GQ.YZB'YZIKUJ?@B9GKYC[a<XIKGJLH=9<>ACLSLHGJP

pGc=@?[SG6Pd\mACB9[aB9GV[SG

1N]hcG6DMES<Q~GJB@IKGJ?[SGV[STJ?9GJB@IV<XLHGJBP>G4Q.YZB'YZIKUJ?@B9GKI|YZP¦hcACLSLFE

à(ω) ' cg(ω)/k(Ω)(1 − cg(ω))NSQSES<>=9DMEHG2P>Gc=YZES?@B9Gc=,=9ACLF?hcACLM?@BBvCP>Tc=4

ωNΩ£o

`PGc=@?<XIVQ~ACB@?'YZLM?w[SG2LHAC?9GJBwDFEHGhcGJ?@?9G?@B'YZLH=@<X?@<>ACLp 1 → p 1

B9TOYZPX<>=9TcGGJLWYZEStIKGJLF?'YZLF?wPbYB9TcDMEHGJLHhcGQ0ACB@?9GJEH=9G

ωGc=@?"TcDFES<XitYZP>GJLF?9G±DME.YZPX<X?'YZ?@<XijGJIKGJLF? sµPbY½IKJIKGítYZB@<bYZ?@<>ACL Q~ACESBWP>GQ.YZB'YZIKUJ?@B9G

pB9TOYZPX<>=9TcG±GJL

YZEStIKGJLM?'YZLF?:PbYPbYZB@jGJESB=@Q0GchJ?@B'YZP>G%YC='=9AMhJ<>TcG sΩACEGJLl[a<XIV<XLME.YZLF?:P>G,?9GJIVQH=hOYZB'YChJ?9TJB@<>=@?@<>DMEHG%[SG,IKA*[aESPbYZ?@<>ACL

τm ∼ 1/Ω£oYB9TOYZPX<>='YZ?@<>ACL[SGghcGJ?@?9Gg?@B'YZLH=@<X?@<>ACL¸tB ChcGus

τmQ~ACESBk[a<¢§]TJB9GJLF?9Gc=itYZP>GJESB9=k[SGPbYB9TcDFEHGJLHhcG

Q0ACB@?9GJEH=9GωQ0ACESB@B'YZ<X?KQ~GJB@IKGJ?@?@B9G[]\mGon*?@B'YZ<XB9GPbYW[STJQ0GJLH[HYZLHhcGkB9TcDFEHGJLHhJ<>GJPXP>G[SGkPd\^YZ?@?9TJLFE.YZ?@<>ACLº[SGc=KACLH[SGc=K[SG

Q0ACIVQ.YZjGà(ω)

!¹ ¤Zª«©:¢¯¸¢¨©E¨±¯¸¥¤p«°±¢ ¢©.¤2¥p¦±!µ ©ª©¦

B,HEHCB hYLN]RM½*E½:a ] `RYa`.LfeL E½:W → [ LN]EQM^_Q] EACB9=9DMEHGP>Gc=lACLH[SGc=k[SGgQ0ACIVQ.YZjGLHGg=9ACLM?lQSPXEH=lQSB9ACQ.YZFYZ?@<XijGc=kI|YZ<>=k=9ACLM?lP>A*hOYZPX<>=9TcGc=k[HYZLH=lPd\[email protected]"YZE

ijAC<>=@<XL.YZjGk[SG|Pd\mTJIKGJ?@?9GJESBON¦P>Gc=4QSRHTJLHACIKUJLHGc=VhJESI4ESPbYZ?@<¢=fYC=9=9A*hJ<>Tc=<sPbY[a<>=@Q~GJB9=9<>ACL´[SGfi*<X?9Gc=9=9Gl[a<>[email protected]'YZ<>=9=9GJLM?Oj G2eSB@EH=9DMEHGh'R.YZLSjGJIKGJLF?[HYZLH=:P>G?@B'YZLH=@Q0ACB@?,[SGc=ACLH[SGc=w[SGQ0ACIVQ.YZjGGc=9?ESLHG='ACESB9hcG[]\ <XLa°ACB@I|YZ?@<>ACLH=w=9ESB:P>GIV<XPX<>GJENHGJ?QSPXEH=:Q.YZB@?@<>hJESPX<>UJB9GJIKGJLM?w=9ESB:='Y6IV<>hJB9At=@?@B@EHhJ?@ESB9Gt

N65 ¶/¸+ + :´ ! !/x+ P # Yg°ACB@IESP>GhcACIVQSP>UJ?9G ¿ nmOsF£VGc=@?KES?@<XPX<>=9TcGQ0ACESB|?@B'YChcGJB|Pd\^YZIVQSPX<X?@EH[SG[aE·=9<XtL.YZP[STJIKAM[aESP>TGJL±°ACLHhJ?@<>ACL

[SG|PbYB9TcDFEHGJLHhcGl[SGlQ~ACIVQ.YZjGk=@ESB6PbY¡HtESB9G ¿ â Yg[a<>=@?'YZLHhcGk[]\mACeH=9GJB@itYZ?@<>ACL²Gc=@?n = 5000

NPbYhcACLH=9?'YZLF?9G[]\^YZ?@?9TJLFE.YZ?@<>ACL

C = 5.10−7 NGJ?P>Gc=4¤B9TcDMEHGJLHhcGc=[STJIKAM[aESP>TcGc= Ω ' 1.7 10−4, 8.3 10−4, 3.3 10−3, 8.3 10−3 ¯2YZLH=hcGc=DFE.YZ?@B9G%hOYC=ONaESLHG%h9RMES?9G[]\^YZEIKAC<XLH=ESLACB9[aB9G2[SGwtB'YZLH[SGJESBGJLYZIVQSPX<X?@EH[SG[aE=@<XtL.YZP][STJIKA*[aESP>T2Gc=@?ACeH=9GJB@ijTcGxskPbYl?@B'YZLH=@<X?@<>ACLGJLF?@B9GKACLH[SGc=[SGKQ~ACIVQ.YZjGVQSB9ACQ.YZFYZ?@<XijGc=4GJ?TJiCYZLHGc='hcGJLF?9Gc=DMES<¾YZQSQ.YZB'Y>?<s

ω = 1

¯2YZLH=ESLHG hcACLa¡HtESB'YZ?@<>ACLKAi4P>G [STJQSR.YC='YZjG GJLF?@B9G=9ACESB9hcGc=LHACL6PX<XLHTOYZ<XB9Gc=GJ?=@<XtL.YZPa[STJIKAM[aESP>TGc=@? <XIVQ~ACB@?'YZLM?=@ESBESLHGtB'YZLH[SG:Q.YZB@?@<>G,[SGPbY%e.YZLH[SGQSB9ACQ.YZFYZ?@<XijG6

Ω ' 8.3 10−3 £oNtPbY2h'RFES?9Gw[]\^YZIVQSPX<X?@EH[SGºs%PbY2?@B'YZLH=@<X?@<>ACLfGc=@?¾QSPXEH=°YZ<XeSP>GVGJLB'YZ<>=9ACL[SGPbY='YZ?@ESB'YZ?@<>ACL[SGVPd\mGo§0GJ?[SG|[STJIKAM[aESPbYZ?@<>ACLQ~ACESBP>Gc=ACLH[SGc=2QSB9ACQ.YZFYZ?@<XijGc=Oj GJQ0GJLH[HYZLF?ONP>ACB9=9DMEHG%P>G[STJQSR.YC='YZjGGJLM?@B9G=9ACESB9hcGc=LHACLPX<XLHTOYZ<XB9Gc=GJ?=@<XtL.YZP [STJIKAM[aESP>TGc=9?°YZ<XeSP>G=9ESBPbYI|Y®GJESB9GQ.YZB@?@<>G[SGPbY6e.YZLH[SG2QSB9ACQ.YZFYZ?@<XijGf

Ω ' 1.7 10−4 £oNShcGJ?@?9Gh'RFES?9G4[]\^YZIVQSPX<X?@EH[SG6YZ?@?9GJ<XLM??@B9AC<>=:ACB9[aB9Gc=,[SG%tB'YZLH[SGJESBOj ACIVIKG[HYZLH=P>GhOYC=

p 1[SGvPbY¡HtESB9G ¿ ¿ N®ESLHG =9EHhchcGc=9=@<>ACLV[SG¾IV<XLS<XI|Y%GJ?[SGvI|Y®n*<XI|Y%Gc=9?ACeH=9GJB@ijTcG Q0ACESB

P>Gc=QSPXEH= tB'YZLH[SGc=iCYZP>GJESB9=[SGΩ=9ESBPbY¡HtESB9G ¿ â®G¦IKJIKGvQSRHTJLHACIKUJLHGGc=@?B9Gc=@Q0ACLH='YZeSP>G [SG¾hcG¾hcACIVQ0ACB@?9GJIKGJLF?ON

<dmGtcPbY[a<>=@Q0GJB9=@<>ACL6[SGiM<X?9Gc='=9G¦DMES<FGJLF?@B'Y>LHG¾ESL6YC=@*LHh9RSB9ACLS<>=@IKGvGJLF?@B9G =9ACESB9hcGc=LHACLPX<XLHTOYZ<XB9Gc= GJ?=@<XtL.YZPF[STJIKA*[aESP>TtNTJ?'[email protected]?@<XijGJIKGJLF?GJL½QSR.YC=9GgGJ?fGJLºACQSQ~At=@<X?@<>ACLº[SGQSR.YC='GtvY°ACB@IESP>G´ ¿ À kF£Q0GJB@IKGJ?l[]\mTJitYZPXEHGJBkhcGc=IV<XLS<XI|YaN0hcG6DMES<Q~ACESB

n = 5000N]GJ?

Ω ' 8.3 10−3 [SACLSLHGESLQSB9GJIV<>GJB2IV<XLS<XI4ESI sω ' 0.37

j G6B9Tc=@ESPX?'YZ?hcACLHhcACB9[SG2eS<>GJLYOijGchPbYhcACESB@e0GhcACB@B9Gc=@Q0ACLH[HYZLF?9G=9ESBPbY6¡HtESB9G ¿ âNSPX<XtLHG2LHAC<XB9Gt

PbYf?@B'YZLH=@<X?@<>ACLNω = 1

N0GJ?k(ω) = 2/a

NhcGDFES<Q0GJB@IKGJ?[]\mACeS?9GJLS<XB[SGc=2Q.YZB'YZIKUJ?@B9Gc=hcACIVIKGKPbY|B'YZ<>[SGJESB[SGc=%hcACLF?'YChJ?9=

αN8PbYK°ACB9hcG6=9?'YZ?@<>DFEHGGonSGJB9hcTcG6=9ESB,PbYlh9R.Y>LHGtN0P>G[a<bYZIKUJ?@B9G

a[SGc=%eS<XPXP>Gc=sfQ.YZB@?@<XB2[SGc=2TcDFE.YZ?@<>ACLH=

Zmtnmkj£oNZmt^sF£ GJ?Zmtnm ¿ £[aEh9R.YZQS<X?@B9G¸mt

N65 ¶/¸+ + + 4 //*!'+ #

¥LkhcG,DMES<8hcACLHhcGJB@LHGwP>GQSB9AZ¡HP0[aEl=@<XtL.YZP0[STJIKAM[aESP>Ts2Q.YZB@?@<XB[SG,Pd\mTJIV<>=9=@<>ACLk[]\ ESLfQ.YCDMEHGJ?[]\mACLH[SG A YZEH='=@<>GJL[SGgQ~ACIVQ.YZjGtN='Y°ACB@IKGWQ~GJES?TJijACPXEHGJBP>ACB9=[SGgPbY?@B'YZLH=@<X?@<>ACL GJLM?@B9G"ACLH[SGc=[SGgQ~ACIVQ.YZjGWQSB9ACQ.YZFYZ?@<XijGc=GJ?TJiCYZLHGc='hcGJLF?9Gc=O qMESBPbYl¡HtESB9G ¿ ryaN0?@B9AC<>=?@B'YZLH=@<X?@<>ACLH==9ACLM??@B'YChcTcGc=ONh'R.YChJESLHGKQ0ACESBESLACB9[aB9GK[SGVtB'YZLH[SGJESB6[aEQ.YZB'YZIKUJ?@B9G6YC[a<XIKGJLH=9<>ACLSLHGJP

pNH[SG

p 1GJLQ.YC='='YZLF?,Q.YZB

p ' 1EH=9DME\s

p 1H¬B9AC<>=,B9TcDFEHGJLHhcGc=Q~ACB@?9GJEH=9Gc=

=9ACLF?ES?@<XPX<>=9TcGc=Q0ACESBVh9R.YCDMEHGk?@B'YZLH=@<X?@<>ACLNω ' 0.67

GJ?0.83

DFES<hcACB@B9Gc=@Q0ACLH[SGJLF?sg[SGc=VACLH[SGc=QSB9ACQ.YZFYZ?@<XijGc=GJ?

ω ' 1.083DMES<¦hcACB@B9Gc=@Q0ACLH[sk[SGc=2ACLH[SGc=TJiCYZLHGc='hcGJLF?9Gc=O YliCYZP>GJESB[SGPbYkhcACLH=@?'YZLM?9GK[]\^YZ?@?9TJLFE.YZ?@<>ACLB9Gc=@?9G

<XLHh9R.YZLSjTcG4GJ?:Gc=@?TJFYZP>G¶sC = 10−6 ACESBdYZ<XB9G|Q.YC='=9GJBP>G|Q.YZB'YZIKUJ?@B9G

p[SG

p 1s

p 1NP>G|?9GJIVQH=hOYZB'YChJ?9TJB@<>=@?@<>DFEHGl[SGfIKAM[aESPbYZ?@<>ACL´Gc=@?

iCYZB@<>T6[SGτm ' 8.3 10−2 s

τm ' 3.3 10−5 YcijGch ω ' 0.67N]GJ?

C = 10−6 ~ACB9=9DMEHG p 1N8P>G6h9R.YZLSjGJIKGJLM?

[SGlQSB9AZ¡HP[STJIKA*[aESP>TYC=9=9A*hJ<>T sPbY?@B'YZLH=9<X?@<>ACL²GJLF?@B9G[SGc=VACLH[SGc=V[SGlQ~ACIVQ.YZjGkQSB9ACQ.YZFYZ?@<XijGc=KijGJB9=V[SGc=VACLH[SGc=[SGKQ0ACIVQ.YZjGkTJiCYZLHGc=9hcGJLM?9Gc=Gc=@?4B9GJQSB9Tc='GJLF?9Tk=@ESBPbY¡HtESB9G ¿ ry*N°GJLHJ?@B9Gl4,j ACIVIKGfP>Gc=6Go§]GJ?9=4[SGl[a<>=@Q0GJB9=@<>ACL[SGi*<X?9Gc=9=9G4[HYZLH=PbY >OACLHGQSB9ACQ.YZFYZ?@<XijG=9ACLM?<XIVQ0ACB@?'YZLF?9=K

p 1N8hOYC=PX<XIV<X?9Gk ¿ ¿ xF£@£oN.P>GQSB9AZ¡HP[SGi*<X?9Gc=9=9GGc=@?

<XLF?9TJtB9T:GJ?¦Gc=9?QSB9ACQ0ACB@?@<>ACLSLHGJPs,PbY,°ACLHhJ?@<>ACL A YZEH=9=9<>GJLSLHGttGQSB9AZ¡HPH[SGiM<X?9Gc='=9G[STJIKAM[aESP>TcGswQ.YZB@?@<XBv[]\mACLH[SGc=¦[SGQ~ACIVQ.YZjGTJiCYZLHGc=9hcGJLM?9Gc=vGc=9?ONjDFE.YZLM?.s%PXES<dNtQSB9ACQ0ACB@?@<>ACLSLHGJPNs%PbY2[STJB@<XijTcGQSB9GJIV<>UJB9G,[]\ ESLHG A YZEH=9=9<>GJLSLHGP>ACB9=9DMEHGPbY¤B9TcDMEHGJLHhcG%[SG%Q~ACIVQ.YZjG2Gc=9?=9E_r|='YZIVIKGJLF?TJP>AC<XtLHTcG[SG%PbYB9TcDMEHGJLHhcG2[SG2hcACESQSESB9GK¤ijAC<XBP>G2hOYC=PX<XIV<X?9GK ¿ À F£@£oY6?@B'YZLH=@<X?@<>ACL[SG%PbY >OACLHG2QSB9ACQ.YZFYZ?@<XijGijGJB9=PbY >OACLHG2TJiCYZLHGc='hcGJLF?9G2Q0ACESBP>Gc=:ACLH[SGc=:[SG%Q0ACIVQ.YZjG2=9G%I|YZLS<¢°Gc=@?9G

5 4! P#& P+ /x : N$&u

0 0.2 0.4 0.6 0.8 1 1.2 1.410

−4

10−3

10−2

10−1

100

ω/ωc

!

"#$% & ' ()+*) (,& -./%) 0 #1) (% 02 / 3(35

h N N ¤2ª¨¦p«%©%«¬¨¾¢¨©%¤2¢ µnª ;!¥¨¥¤2ª©%«¬¨ ²± ¦p« ;¨ªµ¶²´¥¯x¬²±µn¥l£ª¤ ²´¢p¦ ¬¨²´¢p¦#²´¢ £_¬¯º£_ª ;´¢l£¤2¬p£_ª1;ª ©%« .¢p¦ ¢©£_ª¤ ²!¢p¦#¬¨²´¢p¦#²´¢ £_¬¯º£_ª ;´¢ ¥ .ª¨f¢p¦¢Ü©:¢p¦§¹ 8ªd²«&¦©ª¨p¢ ² · ¬¦§¢¤ .ª©%«¬¨ ¢p¦© 29)´¥B¢ 3

n = 5000%

µnª B¬¨¦©ªÜ©:¢'² · ª©©:¥¨±ª©%«¬¨C = 5 10−7 % ¢§© µn¢p¦49p¤B¥2°±¢¨p¢p¦"²´¥¯x¬²±µ ¥B¢p¦-.ªµ ¢Ü© ¤B¢p¦£¢2§©%« .¢¯¸¢Ü©

Ω '1.7 10−4, 8.3 10−4, 3.3 10−3 ¢©

8.3 10−3 ²± ;¤p«&¦¶µ ª«&¤5ª±l¨¬«¤§¹

!¹ ¤Zª«©:¢¯¸¢¨©E¨±¯¸¥¤p«°±¢ ¢©.¤2¥p¦±!µ ©ª©¦

−2 −1 0 1 2 3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−2 −1 0 1 2 3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−2 −1 0 1 2 3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.5 1 1.510

−2

10−1

100

0.5 1 1.510

−3

10−2

10−1

100

0.5 1 1.510

−3

10−2

10−1

100

C

B

A

p 1

p ' 1

p 1

ω

ω

ω

'"% % '* ! ' '$ % ! "%''%( * )#)+

t/τm

+!

!

+!

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p¹ 8 ¢p¦b«¨¦§¢¤©%«¬¨¦w¤2¢:£¤2¥p¦¢Ü©:¢¨© µ ¢p¦>ª¯º£µ «©%±²´¢p¦>²´¢p¦w¦« ;¨ª± )

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ω = 1/12N8P>GJESB2YZ?@?9TJLFE.YZ?@<>ACL"Gc=@?,[SACLHhB9GJPbYZ?@<XijGJIKGJLF?%dYZ<XeSP>G

C = 10−6 £YZ<XLH=@< DFEHG2P>Gc=Go§0GJ?9=,[SG[a<>=@Q0GJB9=@<>ACL[SG2i*<X?9Gc=9=9G| p ' 0.14

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n = 5000eS<XPXP>Gc=ONvPd\^YZIVQSPX<X?@EH[SGW[aE·QSB9AZ¡HP2[STJIKA*[aESP>TgYZEStIKGJLM?9G?9ACEC@ACESB9=kGJL²°ACLHhJ?@<>ACLµ[SGPbY[a<>=9?'YZLHhcGtN

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p ' 0.8£oNGc=@?='YZ?@ESB9TcG sQ.YZB@?@<XBV[SGfPbY[a<>=9?'YZLHhcG

[SGn = 1000 − 2000

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p 1 → p 1YZL.YZPX*='TcGf[HYZLH=hcGkh9R.YZQS<X?@B9GtGfQSRHTJLHACIKUJLHG[SGl[a<>=@Q~GJB9=9<>ACL´[SGfi*<X?9Gc=9=9G|='G

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ω = 1/12¹p ' 0.14

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σ = bnε3/2H(ε) ,©aDzQ®«

gfveEHGwy@fJVP=GVdORbUOTVM3GVADUZUOGIC¡bZg-U]dOEHGIC¯TVE@fC=PSNHcadDGICUOGICONQEQEHGICu\GVP

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»@ZwygfADJVPSNHgfAUOG£KGu@Ir-NHCSNHUOGH(ε)

NQADU]NHcWdDGecWdDGeC=GVdOEHCxEHGIC5JIgfAaP*@fJVP=C7JIgfbeMORSNQbZTICYJIgfAaPSRSNQOdDGVAWP@Bd]jªJIgfAaPSR*@BNQAaP=GICUD@BADC>EHGxbo@BP=TVRSN@Bdz0b%@BRFJIgfAarGVAWPSNHgfA\<UD@BADCE@ZCSdONQP=G5UOGIC>UOTVrGVEHgfMOM3GVbZGVAaP=Cu\1dOADGYJIgfbeMOR=GIC=CSNHgfA²JIgfRSR=GIC=MlgfADU²n

σ\

ε > 0z :@>UOTkwygfRSbo@BPSNHgfA7CSP*@BPSNHcWdDG

ε0U]d(bo@BP=TVRSN@Bd©TIcWdONQrf@BEHGVAWP=GnE@FUOTkwgfRSbo@BPSNHgfACSP*@BPSNHcWdDGbZgu°GVAOADGUOGIC»JIgfAaP*@fJVP=C

UD@BADC9dOAbeNQEQNHGVd¨R=TV|sdOEQNHGVR«9UOTVP=GVRSbeNQADGpEHGmbZgWU]dOEHGTVE@fCSPSNHcWdDGEQNQADTu@BNQR=Gdσ(ε0)/dε

M3gfdORUOGICUOTkwygfRSbo@BPSNHgfADCZGVPJIgfAWPSR*@BNQAaP=GICx@fJIgfdDCSPSNHcWdDGICFUOGKw_@BNQOEHG(@BbeMOEQNQPSdDUOG |ε| |ε0|

GVP |σ| σ0 ≡ σ(ε0)\OR=GICSM3GIJVPSNQrGVbZGVAaPuz

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σ(ε)z(bgfdOR¡dOADG/UOTVbZgfADCSPSR*@BPSNHgfAmcWd1@BEQNQP*@BPSNQrG

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σ0 + σ = bn0(ε0 + ε)3/2H(ε0 + ε) + bn1(µε0 + ε)3/2H(µε0 + ε) .©aDz`X«

:GIC>ADgfb(OR=GIC/bZgu°GVADCn0GVP

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GeJIg-GkoJVNHGVAaP7@fU]NQbZGVADCSNHgfAOADGVE |µ| 1Ju@BR*@fJVP=TVRSNHC=GeEHGeJ=L1@BRS|GVbZGVAWP9UOGE@mwR*@fJVPSNHgfA@fUOU]NQPSNHgfAOADGVEQEHGoUOG9w_@BNQOEHGIC

JIgfAaP*@fJVP=Cuz :@oMORSNHC=G9GVAiJIgfbeMOP=GUOGICPSR=gfdDC(gfd|a@BMDC(NQAONQPSN@Bd]j\lC*GYR=GkwyGVRSbo@BAaPYC=gfdDC/E_^`@fJVPSNHgfA#UOG(E_^qgfADUOGe@fJIgfdDCSPSNHcWdDG

U^`@BbeMOEQNQPSdDUOG>UOGUOTkwgfRSbo@BPSNHgfA |ε| > |µ|ε0MlgfdORwygfRSbZGVR¡UOGICJIgfAaP*@fJVP=Cu\aGICSP%MlgsC=C=NQOEHGGVAeJIgfADCSNHUOTVR*@BAWPcWdDG

µ < 0z

b»gfdORFEHGIC>JIgfADCSNHUOTVR*@BPSNHgfADCKcad1@BEQNQP*@BPSNQrGICFMOR=TIC=GVAaP=GICu\<JIGICPSR=gfdDCxADGYC=gfAWPM1@fC>MORSNHC>GVA²JIgfbeMOP=Gsz?5@BADC¡EHG>Ju@fCUOGIC¯JIgfAWP*@fJVP=C¡w_@BNQOEHGVbZGVAaPMOR=TkJIgfAaPSR*@BNQAWP=C

0 < µ 1GVP |ε| |µ|ε0

\sEHGICUOTVRSNQrTIGIC¯CSdDJIJIGIC=C=NQrGICM1@BR»R*@BMOM3gfRSPn

εUOGE_^qTIcad1@BPSNHgfA©aDz`[a«:CSdONQrs@BAaP=GJu@BR*@fJVP=TVRSNHC=GVAWPEHGIC:bZg-U]dOEHGIC%TVE@fCSPSNHcadDGICEQNQADTu@BNQR=GICGVPADgfA7EQNQADTu@BNQR=GIC

U]dbo@BP=TVRSN@Bd§dmσ

dεm(ε0) ∼ bn0(1 +

n1

n0µ3/2−m)ε

3/2−m0 .

©aDz`[« G7bZgWU]dOEHGTVE@fCSPSNHcWdDG7EQNQADTu@BNQR=G9JIgfRSR=GIC=MlgfADUªn

m = 1\lGVP/EHG7bZgWU]dOEHGTVE@fCSPSNHcWdDG7ADgfAhEQNQADTu@BNQR=GcWd1@fU]R*@BPSNHcadDG

JIgfRSR=GICSM3gfADU²nm = 2

z<GICbZg-U]dOEHGICFCu^qTIJVRSNQrGVAWPFUOgfADJ5R=GICSM3GIJVPSNQrGVbZGVAaP§dσ

dε(ε0) ∼ bn0(1 +

n1

n0µ1/2)ε

1/20

©aDz aW«d2σ

dε2(ε0) ∼ bn0(1 +

n1

n0µ−1/2)ε

−1/20 .

©aDz`«gfbebZG

n1 ∼ n0\1E@JIgfAaPSRSNQOdOPSNHgfA#R=GVE@BPSNQrGUOGICxJIgfAaP*@fJVP=CKw_@BNQOEHGIC/M1@BRKR*@BMOM3gfRSP5@Bd]j²JIgfAWP*@fJVP=CxwygfRSP=CFM3gfdOR

EHGYbZg-U]dOEHG(U^qTVE@fCSPSNHJVNQP=T7EQNQADTu@BNQR=Gp©m = 1

«>GICSPFUOgfADJ9UOGYE_^qgfR=U]R=G9UOGµ1/2 1

z1 GICKJIgfAaP*@fJVP=C/w_@BNQOEHGIC>ADG ¥ gfdDGVAaPUOgfADJeM1@fC7dOAtR[ZfEHGZNQbeM3gfRSP*@BAaP7MlgfdOR7EHGICYMOR=gfMORSNHTVP=TIC9TVE@fCSPSNHcadDGIC(EQNQADTu@BNQR=GIC7U]dbeNQEQNHGVd|sR*@BAWdOE@BNQR=Gsz! E_^ NQAarGVR=C*Gs\E@eJIgfAaPSRSNQOdOPSNHgfAªR=GVE@BPSNQrG7UOGICFJIgfAaP*@fJVP=C/w_@BNQOEHGIC>M1@BR>R*@BMOM3gfRSPK@Bd]jJIgfAWP*@fJVP=CFwygfRSP=C>M3gfdOR>EHG5bZg-U]dOEHG(U^qTVE@fC=PSNHJVNQP=TADgfApEQNQADTu@BNQR=G5cad1@fU]R*@BPSNHcWdDGZ©

m = 2«¯GIC=PUOGKE_^qgfR=U]R=G5UOG

µ−1/2 1z- :GICJIgfAaP*@fJVP=Cw_@BNQOEHGIC ¥ gfdDGVAWP>UOgfADJxdOApR[ZfEHG

MOR=TVMlgfADUOTVR*@BAWP/UD@BADCEHGICFMOR=gfMORSNHTVP=TICFTVE@fCSPSNHcWdDGICADgfAEQNQADTu@BNQR=GIC/UOGICbeNQEQNHGVd]j|sR*@BAadOE@BNQR=GICuz3-47684" #.L AES%L>MNX-EL>C DNA ?5=@BCD EJF8B EA

GIC7U]NHCSPSRSNQOdOPSNHgfADCUOGZJIgfAaP*@fJVP=C9R=GVMOR=TIC=GVAWP=GVAaPEHGZADgfb7OR=GmUOGZJIgfAWP*@fJVP=CcadON@BMOM1@BR*@BNHC*C=GVAaPUD@BADC(EHGZbeNQEQNHGVd|sR*@BAadOE@BNQR=GGVA7wgfADJVPSNHgfAUOGEHGVdOR%UOTkwygfRSbo@BPSNHgfAeCSP*@BPSNHcWdDGs\µUOGEHGVdORJIgfAaPSR*@BNQAaP=GC=P*@BPSNHcadDGgfd9UOGEHGVdORwgfR=JIGCSP*@BPSNHcWdDGsz GMOEQdDC¡C*gfdOrGVAaPu\EHGICTIJ=LDGVEQEHGICUOG>UOTkwygfRSbo@BPSNHgfADCu\UOG>JIgfAWPSR*@BNQAaP=GICgfdoUOGwygfR=JIGIC¯CSP*@BPSNHcWdDGIC¡C=gfAWP¡ADgfRSbo@BEQNHC=TIGICM1@BRR*@BMOMlgfRSP5noEHGVdOR/rs@BEHGVdORxbZgu°GVAOADGUD@BADCKEHG(beNQEQNHGVdz3?5@BADCKE@oCSdONQP=Gs\<E@mUOTkwgfRSbo@BPSNHgfA#CSP*@BPSNHcWdDG7UOGICxJIgfAWP*@fJVP=C5GICSPADgfRSbo@BEQNHC=TIGM1@BRxR*@BMOMlgfRSPYnmE@mrf@BEHGVdORxbZg°GVAOADG

ε0\3GVP5GICSPKUOgfADJeJu@BR*@fJVP=TVRSNHC*TIGM1@BRKE@mrs@BRSN@BOEHG

µcWdON»PSR*@fU]dONQP

E_^qTIJu@BRSP>UOGKE@7UOTkwygfRSbo@BPSNHgfACSP*@BPSNHcWdDG5U]dJIgfAaP*@fJVP>JIgfADCSNHUOTVR=TYn7E@UOTkwygfRSbo@BPSNHgfACSP*@BPSNHcadDGKbZg°GVAOADGszD GICwygfRSbZGICUOGICpU]NHCSPSRSNQOdOPSNHgfADCUOGiJIgfAaP*@fJVP=CC=gfAWPgfOP=GVAadDGICM1@BRpbZg-UOTVEQNHC*@BPSNHgfA½AadObZTVRSNHcWdDG#JIgfbeMOEHVP=G#UOGIC@fC=C=GVb(OE@B|GIC|sR*@BAadOE@BNQR=GICq³K@fUDJÒ\» dDUD¬]\%gfM1J`D\%¾WNQEÀsXO\c@BR*ÀsÀBgfdM1@BReUOGICGkj-M3TVRSNHGVADJIGICCSP*@BPSNHcWdDGIC9UD@BADC9EHGIC=cWdDGVEQEHGIC9E@UOTVP=GIJVPSNHgfAmGICSP¡GkGIJVPSdDTIG/nKE_^`@BNHUOG/UOGM1@BMONHGVR¯UOG>Ju@BRS3gfADGY]\hdDGusÓ\cE@sÀ]®s\RSNÀsXD\ gurOsBzWL1@fcWdDGPSR*@fJIGE@BNHC*C=TIGM1@BR7EHGJu@BRS3gfADGC=dOR7dOADGmwyGVdONQEQEHGmUOGmM1@BMONHGVReU]NHCSMlgsC*TIG@BdwygfADUgfdCSdOR9EHGIC7M1@BR=gfNHCE@BP=TVR*@BEHGICU^ dOA¨JIgfAaP=GVADGVdORRSNQ|sNHUOGZJIgfRSR=GICSM3gfADUnpdOAJIgfAaP*@fJVPUOGeONQEQEHGsz :@²CSdORwy@fJIGoUOGZJ=L1@fcWdDGZPSR*@fJIGZGIC=PYEQNHTIGonpE@wygfR=JIG²©JIgfAWPSR*@BNQAaP=GpgfdUOTkwgfRSbo@BPSNHgfA<«YCSP*@BPSNHcWdDGZU]dJIgfAWP*@fJVPuz:¼/AtPSR*@BNQP=GVbZGVAaPU^ NQbo@B|GZM3GVRSbZGVP7UOGZUOTIJIgfbeMOP=GVR9EHGICYP*@fJ*LDGIC7C*GVEHgfAtEHGVdORCSdORw_@fJIG5GVP>UOgfADJYUOG5R=GVbZgfAaP=GVRxn9E@eU]NHCSPSRSNQOdOPSNHgfAiUOGIC>JIgfAaP*@fJVP=C/UD@BADC>EHGxbeNQEQNHGVdz GICKbZTVPSLDg-UOGICKGkj]M3TVRSNQbZGVAaP*@BEHGICu\3MORSNQADJVNQM1@BEHGVbZGVAWP7@µj]TIGICKrGVR=CKE@mJu@BR*@fJVP=TVRSNHC*@BPSNHgfAtUOGICxJ=L1@µºHADGIC5UOG(wgfR=JIG9GVP

UOgfADJUOGICJIgfAaP*@fJVP=CwygfRSP=Cu\fC=gfAWP%Ju@BM1@BOEHGIC¯UOGUOTVP=GIJVP=GVR¡UOGICwgfR=JIGICUOGIC=JIGVADUD@BAWP ¥ dDC=cad^`n0.1wgfNHCE@/wygfR=JIGCSP*@BPSNHcadDG

bZgu°GVAOADGmqRSNÀsXO\O gr]s1\cE@sÀ]®Iz GICCSNQb(dOE@BPSNHgfADC%AWdObZTVRSNHcadDGICUOGbeNQEQNHGVd]j(|sR*@BAWdOE@BNQR=GIC%bZgfAaPSR=GVAaP¡cWdDG¡EHGADgfb(OR=GUOGJIgfAaP*@fJVP=Cw_@BNQOEHGIC

n1GICSP

EHTV|VR=GVbZGVAaPKCSdOMlTVRSNHGVdORK@BdADgfb(OR=G7UOG5JIgfAaP*@fJVP=CFwgfRSP=Cn0cWdONADGxR=GVMOR=TIC=GVAWP=GVAaPxcad^qGVAWrWNQR=gfA!aaÀ%$ UOGIC>JIgfAaP*@fJVP=C

U]dbeNQEQNHGVdq³K@fUDsÓ\<gfM1J`Bz< :@ZJIgfAWPSRSNQOdOPSNHgfAiUOGICKJIgfAaP*@fJVP=CFw_@BNQOEHGIC/@BdhC=NQ|sA1@BE:ADgfA²EQNQADTu@BNQR=G7UOgfNQP/UOgfADJs\<C=GVEHgfAE_^qTIcad1@BPSNHgfA©aDz`[«7GVP9E@²U]NHC=JVdDC*CSNHgfAcWdONEQdONGIC=P@fC=C*gWJVNHTIGs\UOgfbeNQADGVRZnM1@BRSPSNQReU^`@BbeMOEQNQPSdDUOGICZUOGUOTkwgfRSbo@BPSNHgfADCU^qGkj]JVNQP*@BPSNHgfApUOG/E_^qgfR=U]R=GKUOG

µ ≤ 0.1− 0.01zD?xG>P=GVEQEHGICUOTkwgfRSbo@BPSNHgfADCJIgfRSR=GIC=MlgfADUOGVAWPn(UOGICwgfR=JIGICGVADJIgfR=GFMOEQdDC

!"#%$&('*) &('+&,-)./0!&!$('211 .%

110−110−2

b! "#%$

& ')(*)+-,/.021,354602,/7"8:96;=<7"80><?0.

n(µ)

+A@BC7"1DE>02,/7"8F.0>0,/G4H;I96;J.K<%780>"<%0.µ

10

L MN O/P= ')QPI?R

SUTVEW !X2Y[Z \^] %_ _`N` a [` 5_ b]dc egf [` 5_ f%d`[h µ > 1 ikj fAc Jel_ 5_ j f N` a

_J` ma _ ne 5] j N`]%_po ?_JqfJrs_ igt µ < 1 igtuj b`_ ` e t _Jel_ d`v` %_ ?]J` Hf[` j fj ] f% %_

w_@BNQOEHGICUOGFE_^qgfR=U]R=GxUOGf/f0 ≤ 0.03− 0.001

\DcadON3C=GFPSR=gfdOrGVAaPONHGVApGVA]UOGIC=C=gfdDCUOGICrf@BEHGVdOR=CcWdON<C=gfAWPgfOP=GVAWdDGICUD@BADC>EHGIC>Gkj-M3TVRSNHGVADJIGICFC=P*@BPSNHcadDGICF@fJVPSdDGVEQEHGIC7]\²dDGus´D\cE@sÀ]®a\ORSNÀsXBz?x@BADCFEHGICFJIgfADCSNHUOTVR*@BPSNHgfADCKcWdON:CSdONQrGVAaPu\1E@ZU]NHCSPSRSNQOdOPSNHgfAiUOG(UOTkwygfRSbo@BPSNHgfADC9©GVPFADgfAhUOGxwygfR=JIGIC«UOG7JIgfAaP*@fJVP=C

rs@h¢VPSR=GJIgfADC=NHUOTVR=TIGs\JIGcWdONwy@fJVNQEQNQP=GpE_^ NQAWP=GVRSMOR=TVP*@BPSNHgfA·UOGICR=TICSdOEQP*@BP=CegfOP=GVAWdDCu\GVAM1@BRSPSNHJVdOEQNHGVRoJIgfADJIGVRSA1@BAaPZEHGMOLDTVADgfbZVADGmUOGZJVE@BMOM3GVbZGVAaPuzGVPSP=GoU]NHCSPSRSNQOdOPSNHgfAGICSPYADgfP=TIG

n(µ)\:@urGIJZE@UOTkwygfRSbo@BPSNHgfACSP*@BPSNHcWdDGebZg°GVAOADG

cWdONJIgfRSR=GICSM3gfADU²nµ = 1

z?x@BADCZP=gfdDCeEHGICZJu@fCu\%dOADGUOTIJVR=gfNHC*C*@BADJIGGkj]MlgfADGVAWPSNHGVEQEHG²UOGJIGVPSP=GU]NHCSPSRSNQOdOPSNHgfAMlgfdOReEHGICewgfR=JIGICZC=P*@BPSNHcadDGIC

C=dOMlTVRSNHGVdOR=GICKnE@9wygfR=JIGYCSP*@BPSNHcWdDG5bZgu°GVAOADGp©µ = 1

«GICSPFR=GVMlgfRSP=TIG7UD@BADC>E@EQNQPSP=TVR*@BPSdOR=Gszl?xGKwy@wIgfAh|TVADTVR*@BEHGs\1JIGJIgfbeM3gfRSP=GVbZGVAaP/MlGVdOP/¢VPSR=G5UOTIJVRSNQPFM1@BRE@9wygfRSbZG5CSdONQrs@BAaP=GYUOG5E@U]NHC=PSRSNQOdOPSNHgfAhUOGYUOTkwygfRSbo@BPSNHgfA§

n(µ) ∝ e−α1(µ−1)α2µ ≥ 1 ,

©aDz `«@urGIJ

α1GVP

α2UOGVd]jJIgfADC=P*@BAaP=GICMOR=g-J=LDGICUOG®xcadON3UOTVM3GVADUOGVAaP>M1@BRGkjOGVbeMOEHGs\OgfdOPSR=G/EHGICU]NTVR=GVAaP=GIC>bZTVPSLDgWUOGIC

Gkj]M3TVRSNQbZGVAaP*@BEHGIC>gfdAWdObZTVRSNHcadDGICdOPSNQEQNHC*TIGICu\OUOGKE@YwRSNHJVPSNHgfAUD@BADCEHGKbeNQEQNHGVdzO GICrs@BEHGVdOR=CdOPSNQEQNHC=TIGICUD@BADCE@9CSdONQP=GC*gfAaP9¦Oj]TIGICmn

α1 = 1.4GVP

α2 = 3/2¾WNQEÀsXBz% ¡^ NQA]¿DdDGVADJIG²UOGJIGICeJIgfADCSP*@BAaP=GICmCSdORE@²bZg-UOTVEQNHC*@BPSNHgfA·GVPEHGIC

JIgfADC=NHUOTVR*@BPSNHgfADCcWdON1CSdONQrGVAWPGIC=P¡beNQAONQbZGszW¡AGkGVPu\aUOGIC¡wgfRSbZGICcad1@BEQNQP*@BPSNQrGICUOG>U]NHC=PSRSNQOdOPSNHgfADCPSR=ICU]NTVR=GVAaP=GIC@Bd]UOGIC*CSdDC%UOGE@/rs@BEHGVdOR%bZgu°GVAOADG

µ = 1©n(µ) = 0

M3gfdORµ ≥ 1

M1@BR¡GkjOGVbeMOEHG«:A^qgfAaP¡cWdDGPSR=IC%M3GVdU^ NQA]¿DdDGVADJIGC=dOREHGICR=TICSdOEQP*@BP=C>gfOP=GVAWdDC/UD@BADCE@ZCSdONQP=Gsz¾WdOR/E@o¦D|sdOR=G aDzQ®GICSP/R=GVMOR=TIC*GVAaP=TIGdOADG9U]NHCSPSRSNQOdOPSNHgfAtUOGUOTkwgfRSbo@BPSNHgfADC5UOG9JIgfAaP*@fJVP=Cuz /EHgfR=CxcadDG7E@owgfRSbZG

cWd1@BEQNQP*@BPSNQrGZUOGeJIGVPSP=GeU]NHCSPSRSNQOdOPSNHgfAGICSP5ONHGVAtUOTIJVRSNQP=GeM3gfdORµ > 1

\3MOEQdDCSNHGVdOR=CYJIgfbeM3gfRSP=GVbZGVAaP=C9U]NHCSPSNQADJVP=C(C*gfAaPR=GVADJIgfAWPSR=TICFC=GVEHgfA²EHGICPSR*@Irs@Bd]jR=GVMlgfRSP=TICFMlgfdOR

µ < 1z

GIC:GICSPSNQbo@BPSNHgfADCUOG¡E@>UOTkwgfRSbo@BPSNHgfACSP*@BPSNHcWdDGbZgu°GVAOADGU]d(beNQEQNHGVdε0gfAWP»TVP=T¡bZGVADTIGIC%nE_^`@BNHUOGUOGICwygfRSb(dOEHGIC

UOG7bZg-UOTVEQNHC*@BPSNHgfAiUOGICKMOR=gfMORSNHTVP=TICxbZTIJu@BAONHcWdDGIC5GVP/TVE@fCSPSNHcWdDGICxUOGIC5@fC=C=GVb(OE@B|GICx|sR*@BAWdOE@BNQR=GICxNHC=CSdDGICKUOGm]cGVE_as©~rgfNQR9EHGoJ*L1@BMONQPSR=Gª®«k\»U]dbZgWUOVEHGUOGm£KGVRSPedhyxagfL1´sf\:GVPUOGZbZGIC=dOR=GIC9U]NQR=GIJVP=GICUOGUOTkwgfRSbo@BPSNHgfACSP*@BPSNHcWdDGmUOGE_^`@fC*C=GVb(OE@B|GYC*gfdObeNHC/n9dOADGxwygfR=JIG5Gkj-P=TVRSNHGVdOR=Gsz :@ZJIgfAWPSR*@BNQAaP=GbeNQAONQb(dOb @BMOMOEQNHcWdDTIG9CSdORFEHG(beNQEQNHGVdªUD@BADC/EHGICKGkj-M3TVRSNHGVADJIGICKMOR=TIC=GVAaP=TIGICKM1@BRKE@ZCSdONQP=G9GICSP/UOG

∼ 7 kPaJIGKcadONJIgfRSR=GICSM3gfADUn7dOADGKUOTkwgfRSbo@BPSNHgfAC=P*@BPSNHcadDG/bZgu°GVAOADG5GICSPSNQbZTIG5n

ε0 ' 0.43 10−4 ± 0.15 10−4 z

:@JIgfAWPSR*@BNQAaP=Gbo@µj]NQb(dOb\cad1@BAWPn²GVEQEHGs\:GICSP(UOG ∼ 72 kPa\:JIGmcadON¡R=GVMOR=TIC*GVAaP=GdOADGoUOTkwygfRSbo@BPSNHgfAC=P*@BPSNHcadDG

bZgu°GVAOADGÜOGε0 ' 2.0 10−4 ± 0.15 10−4 z bgfdOR#dOADGJIgfAWPSR*@BNQAaP=G·dOAON@µj-N@BEHGGkj]P=TVRSNHGVdOR=GÜOG ∼ 29kPa

\P°-MONHcadDGUOGIC#Gkj]MlTVRSNHGVADJIGICªMOR=TIC*GVAaP=TIGICtUD@BADCªE@ CSdONQP=Gs\xE@ UOTkwygfRSbo@BPSNHgfA¸CSP*@BPSNHcadDGbZg°GVAOADG GICSP#GICSPSNQbZTIG¨nε0 ' 1.1 10−4 ± 0.15 10−4 z

!"#%$&('*) &('+&,-)./0!&!$('211 .%

&7' %0l;,98 354 "Y476 #.L7A C AL>EJL7I =S(L7V =0DEJF8<

?KGVd]jJIgfA]¦D|sdOR*@BPSNHgfADCtGkj-M3TVRSNQbZGVAaP*@BEHGIC#gfAWP#TVP=TR=GVP=GVAWdDGICuz(LOR=gfADgfEHgf|sNHcadDGVbZGVAWPu\5E@·MOR=GVbeNHVR=G¨GICSPhdOADGJIgfA]¦D|sdOR*@BPSNHgfAUOGmMOR=gfM1@B|a@BPSNHgfA @fJIgfdDCSPSNHcadDGprGVRSPSNHJu@BEHGs\Js^qGIC=PonhU]NQR=GpC=GVEHgfAË@iU]NQR=GIJVPSNHgfA U^`@BMOMOEQNHJu@BPSNHgfA UOGmE@JIgfAWPSR*@BNQAaP=GC=P*@BPSNHcadDGdOAON@µj]N@BEHGs\GVPZGICSPeUOTIJVRSNQP=GCSdOR7E@h¦D|sdOR=G aDz`XOz% GpbeNQEQNHGVd|sR*@BAWdOE@BNQR=GADgfA]JIgfADC=gfEQNHUOTGICSPJIgfbeM3gsC=TUOG¯ONQEQEHGICUOG¯rGVRSR=GUOG¯U]N@BbZVPSR=G

2±0.1 mmz¹ EWGICSPJIgfAaP=GVAWdZUD@BADC:dOA9R=TIJVNQMONHGVAaP%MOE@fCSPSNHcWdDG¯JV°-EQNQADU]RSNHcadDG

UOG50 cm

UOGL1@BdOPGVPUOG46 cm

UOGU]N@BbZVPSR=Gs\BwyGVRSbZTM1@BRdOAeU]NHC=cWdDGUOGMOEHGkj-NQ|sE@fC=CM3GVRSbZGVPSP*@BAaP¯UOGPSR*@BADCSbZGVPSPSR=GE@JIgfAWPSR*@BNQAaP=GoC=P*@BPSNHcadDGGkj]P=TVRSNHGVdOR=Gsz:GVPSP=GeJIgfAaPSR*@BNQAaP=GmGICSPY@BMOMOEQNHcadDTIGM1@BR5dOADGr-NHCKbZTVP*@BEQEQNHcadDGbeNQEQEQNQbZTVPSRSNHcWdDGGVPYGIC=PKbZGIC=dOR=TIGonE_^`@BNHUOGZU^ dOAtJu@BMOP=GVdOR7UOG9wygfR=JIGeCSP*@BPSNHcWdDGeUOG1@BADUOGedOPSNQEHG

1 → 10 kNz:GIJVN¡JIgfRSR=GICSM3gfADUn

UOGIC(JIgfAaPSR*@BNQAaP=GICe@[email protected] kPa

n72 kPa

MlgfdOR(JIGVPSP=GmJIgfA]¦D|sdOR*@BPSNHgfAGkj]MlTVRSNQbZGVAWP*@BEHGs\»gfdGVADJIgfR=GmnpUOGICr-NQP=GIC=C=GICUOGYMOL1@fC=G c¤UOGYE_^qgfR=U]R=GYUOG

200 − 350 m/sz

Onde depompageHF

B

C D

Onde démodulée BF

A

SUTVEW XkY ! #" f% 5_ j$ _% t ] _ _ 5_ t t f!"Hf q_ %f j _& _` j _ %f t _ 5_Fc _ i _` j$ ]Jel_ _ j '"uf ji Z _` j($ ]Jel_ _ 5_ N` f jNj _Jel_ _ ) _J` j _ %]_ t _ j *" uf j ,+ f N` f _b_ t t f-"Hf _` 5_

16 cm

>¦DA·U^qTVbZGVPSPSR=G²EHGICogfADUOGICUOGM3gfbeM1@B|G²UOGL1@BdOP=GICow~R=TIcWdDGVADJIGICu\dOAPSR*@BADC=U]dDJVP=GVdOREHgfAO|sNQPSdDU]NQA1@BEKGVPZdOAPSR*@BADC*U]dDJVP=GVdORKUOGYJVNHC@BNQEQEHGVbZGVAaP5C=gfAaP/MOE@fJITICK@BdwgfADUhU]d²R=TIJVNQMONHGVAaPxGVP/gfRSNHGVAWP=TIC/C*GVEHgfAhC=gfAh@µjOGYrGVRSPSNHJu@BE»rGVR=CFEHGL1@BdOPK©~PSR*@BADC=U]dDJVP=GVdOR=CGVP? C=dOR¡E@K¦D|sdOR=G aDz`X«kz :GIC¡PSR*@BADC=U]dDJVP=GVdOR=CC=gfAaP¯E@BRS|G1@BADUOGs\aUOGwR=TIcadDGVADJIGFJIGVAaPSR*@BEHG100 kHz

\U^qgfdOrGVRSPSdOR=G ∼ 3.8 cmM3gfdORxEHGPSR*@BADC=U]dDJVP=GVdOR5EHgfAO|sNQPSdDU]NQA1@BE/©2b%@BA1@BbZGVPSRSNHJIC/.5[sÀssX«KGVP ∼ 2.6 cmM3gfdORYEHGoPSR*@BADC=U]dDJVP=GVdOReUOGoJVNHC*@BNQEQEHGVbZGVAWPp©2b%@BA1@BbZGVPSRSNHJICrWNHUOGIgsC*Ju@BA0.®u_aa´«kz:AR=TIJIGVMOPSNHgfA\dOAPSR*@BADC=U]dDJVP=GVdOR

EHgfAO|sNQPSdDU]NQA1@BEYGIC=PmdOPSNQEQNHC=Ts\NHUOGVAWPSNHcadDG#nJIGVEQdONxdOPSNQEQNHC=ThGVA½TVbeNHC=C=NHgfA\GIC=PmMOE@fJITªGVA L1@BdOPpU]dR=TIJVNQMONHGVAWPGVPGICSPgfRSNHGVAWP=T9rGVR=CxEHG(1@fCZ©~PSR*@BADC=U]dDJVP=GVdOR c CSdORKE@Z¦D|sdOR=G aDz`Xa«kzl :@U]NHCSP*@BADJIGUOG9MOR=gfM1@B|a@BPSNHgfAtGVAaPSR=GEHGICxTVbZGVPSP=GVdOR=CGVPKEHG9R=TIJIGVMOP=GVdORYGICSP5@BEHgfR=CKUOG

16 cmz3¼/Aª@BA1@BEQ°]C=GVdOR(¾WP*@BA]wygfR=Ui³/GIC=Gu@BR=J=L¾W°-P=GVbZCZ©_¾-³ ¬f´s«/GICSP/dOPSNQEQNHC*T7M3gfdOR

E_^`@fJIcWdONHCSNQPSNHgfAtUOGICxCSNQ|sA1@Bd]j#GVPxE@obZGIC=dOR=GeU^`@BbeMOEQNQPSdDUOGonmE@ZwR=TIcWdDGVADJIG9U]dtC=NQ|sA1@BE%UOTVbZg-U]dOEHTsz GIC5GVRSR=GVdOR=CYUOGbZGIC=dOR=G@fC=C*gWJVNHTIGICen²E_^qTVEHGIJVPSR=gfAONHcadDGU]dU]NHCSM3gsCSNQPSNw/Gkj]MlTVRSNQbZGVAWP*@BE>C=gfAaPNQA]wyTVRSNHGVdOR=GICZnE@²P*@BNQEQEHGpUOGIC9C=°Wb7lgfEHGICdOPSNQEQNHC*TIC>UD@BADCE@R=GVMOR=TIC=GVAWP*@BPSNHgfAhUOGICFR=TICSdOEQP*@BP=C>CSdONQrs@BAaP=Cuz

:o% t ] _ _`

−40 −30 −20 −10 0−90

−80

−70

−60

−50

−40

−30

−20

−10

P*)'y ^' *s 6JP J !$

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"

+-,."

SUT2VEW !X/!Y &ê tuj 5_ b_` ` " uf % 5]Je j ]`10 ∼ 5 kHz 2 _ c 5_ j($ fe tuj 5_ b_ t e t f!"b_0 ∼ 80 kHz 2lt _ ! #" ?f b_ t t f!"5f q_ %f j _ + $ _(% f% _` j *" uf j _ t j _`_ j _` i _ ?f[`q_ ` _ t j _` j ` f *"5_`+!f ?f _ ` f 43[_ f tHtuj 53 []_ _` 5_26 kPa

+ _`7698Jrs_` 43[_ j _` f [` [` _ ?_ e t _Jel_ 3 [f%?f 53 [_l_ e t _Jel_ =_ t `` f_;:=<

3-4 "Y4" #?>-D5FTV LA@X5=@?5= BSC L7AA_F8D5B= ?-=0A!AL>U D5FTX ? 0V CY?NX5< 0A

GICgfADUOGIC²UOGªMlgfbeM1@B|GIC£K¤ UOG#JVNHC*@BNQEQEHGVbZGVAaPiGVPEHgfAO|sNQPSdDU]NQA1@BEHGIC²C=gfAWP²UOTVbZgWU]dOEHTIGIC|sR*fJIGnE@ADgfA]EQNQADTu@BRSNQP=TYUOGICFJIgfAWP*@fJVP=CFMOR=TIC*GVAaP=CFUD@BADCFEHGxbeNQEQNHGVd²|sR*@BAWdOE@BNQR=Gsz<¹ EGICSPNQbeM3gfRSP*@BAaP/UOGYbZGVAaPSNHgfAOADGVRxcadDG5EHGICFgfADUOGICUOGoM3gfbeM1@B|GUOGmJVNHC*@BNQEQEHGVbZGVAWPZC=gfAaPJu@BM1@BOEHGICeUOGmbZGVPSPSR=GmGVA ¥ GVddOADGmADgfA]_EQNQADTu@BRSNQP=TUOGm£KGVRSPedGVAR*@BNHC=gfAUOGE_^ NQADJVEQNQA1@BNHC=gfApUOGIC¯CSdORw_@fJIGICUOGICJIgfAaP*@fJVP=CNQAWP=GVR_|sR*@BNQADCx©U]dDGK@BdmUOTIC=gfR=U]R=G/MOR=TIC=GVAWPUD@BADCEHG>beNQEQNHGVd<«M1@BRR*@BMOM3gfRSPn9E@eU]NQR=GIJVPSNHgfAªU]d²JVNHC*@BNQEQEHGVbZGVAaPuzl¹ EMlGVdOPFUOgfADJ(Gkj-NHCSP=GVRFdOADG(JIgfbeM3gsC*@BAaP=G7UOGYJIgfAaPSR*@BNQAWP=G(ADgfRSbo@BEHG9nE@eCSdORw_@fJIGU]dJIgfAaP*@fJVPu\<GVPFUOgfADJ5dOAJIgfbeM3gfRSP=GVbZGVAaPxUOG5£/GVRSPed\1@urGIJ5dOADGYGkj]NQP*@BPSNHgfAPSR*@BADCSrGVR=C=Gsz :@EHgfAO|sdDGVdORJu@BR*@fJVP=TVRSNHCSPSNHcWdDG#U^`@BPSP=TVAad1@BPSNHgfAUOGICgfADUOGICUOGhM3gfbeM1@B|G#£K¤ GICSPGICSPSNQbZTIGªUOGiE_^qgfR=U]R=G#UOG

cadDGVEHcWdDGIC%JIGVAaPSNQbZVPSR=GICF©~P °WMONHcWdDGVbZGVAaP2−4 cm

«kzf :@>w~R=TIcWdDGVADJIGMlgfRSP=GVdDC*GdOPSNQEQNHC*TIGUD@BADCE_^qGkj]MlTVRSNHGVADJIGMOR=TIC*GVAaP=TIGGICSPYUOG

80.192 kHzz»GVPSP=GewR=TIcadDGVADJIGZMlgfRSP=GVdDC=GoGICSPYbZg-U]dOEHTIGZGVA@BbeMOEQNQPSdDUOGmn

5.12 kHz\JIGZcWdON¦OjOGZnpE@

bZ¢VbZGxwR=TIcadDGVADJIG5E_^qgfADUOG c¤UOTVbZg-U]dOEHTIGsz?5@BADCmJIGVPSP=GªGkj]MlTVRSNHGVADJIGs\E_^`@BAaP=GVAOADGiM1@BR*@BbZTVPSRSNHcWdDG#gfMlVR=GiUD@BADCmEHG²R=TV|sNQbZGªUOGCBtGIC=P=GVRSrGVEQPªEDKgurO¬µ\¯EHGIC

gfADUOGICoUOGM3gfbeM1@B|G²C*gfAaP@BPSP=TVAWdDTIGIC@urf@BAWPU^q¢VPSR=GhU]NlR*@fJVP=TIGICz¡?5@BADCoJIG²Ju@fC\¡GVPZEHgfR=C=cWdDGE_^ NQA]¿DdDGVADJIGiUOGE@U]NHCSM3GVR=CSNHgfAtUOGer-NQP=GIC=C=GeCSdOR5EHGwygfADJVPSNHgfAOADGVbZGVAaPUOGeE_^`@BAWP=GVAOADGoGICSPxw_@BNQOEHGs\EHGICYMOR=gB¦DEHC7UOGICYCSNQ|sA1@Bd]jtUOTVbZg-U]dOEHTICC=gfAaPMOR=gfM3gfRSPSNHgfAOADGVEHCxn9E@eUOTVRSNQrTIGYC=GIJIgfADUOGYUOGYE_^qGVAarGVEHgfMOM3G(U]dCSNQ|sA1@BE:UOGYMlgfbeM1@B|Gsz¾WdOR(E@¦D|sdOR=GaDz`[²C=gfAWP(MOR=TIC=GVAWP=TIGICEHGIC7UOTVM3GVADUD@BADJIGICUOGoE_^`@BbeMOEQNQPSdDUOGpU]dCSNQ|sA1@BEUOTVbZg-U]dOEHTGVAtwygfADJVPSNHgfA

UOGIC/@BbeMOEQNQPSdDUOGICxUOG7MlgfbeM1@B|G7£/¤ UOG7JVNHC*@BNQEQEHGVbZGVAaP5GVP/EHgfAO|sNQPSdDU]NQA1@BEHGICKgfOP=GVAWdDGICxMlgfdOR/dOADGYMOR=GIC=C=NHgfAhCSP*@BPSNHcWdDG@BMOMOEQNHcadDTIG(UOG

26 kPaz

:@pR=TkwyTVR=GVADJIGedOPSNQEQNHC=TIGZMlgfdOR5E_^qTIJ*LDGVEQEHGoU^`@BbeMOEQNQPSdDUOGoUOGZMlgfbeM1@B|GZGVAUOTIJVNQlGVEHCo© ¦D|sdOR=G+aDz`[a«KJIgfRSR=GICSM3gfADUnE_^`@BbeMOEQNQPSdDUOGeTVEHGIJVPSRSNHcadDG9bo@µj]NQbo@BEHGe@BMOMOEQNHcadDTIGe@Bd]j²PSR*@BADC=U]dDJVP=GVdOR=C(TVbZGVPSP=GVdOR=Cuz FNQADCSN_\<EHG7bo@µj-NQb7dObU^`@BbeMOEQNPSdDUOGoU^qTVbeNHC=C=NHgfAGICSP

0 dBz :@R=TkwyTVR=GVADJIGZdOPSNQEQNHC*TIGZM3gfdOR(E_^qTIJ=LDGVEQEHGU^`@BbeMOEQNQPSdDUOGUOTVbZgWU]dOEHTIGGVAUOTIJVNQ3GVEHC(GICSP

@BRSONQPSR*@BNQR=Gsz¼/ADG9GIC=PSNQbo@BPSNHgfAgfOP=GVAadDGeM1@BR5dOADGbZGIC=dOR=GZ@Bd#rWNQOR=gfbZVPSR=GE@fC=GVRYUOGE@r-NQP=GIC=C=GenmE@pCSdORw_@fJIGUOGICPSR*@BADC=U]dDJVP=GVdOR=CTVbZGVPSP=GVdOR=CUOgfAOADGFdOADGFrf@BEHGVdORUOGFE@(UOTkwgfRSbo@BPSNHgfA@fJIgfdDCSPSNHcWdDGFNQADU]dONQP=G/bo@µj-NQbo@BEHGKGVAmTVbeNHC*CSNHgfA\ε maxa ' 1 10−4 n 80 kHz

©~PSR*@BADC=U]dDJVP=GVdOR=C5TVbZGVPSP=GVdOR=C5UOG9JVNHC*@BNQEQEHGVbZGVAaP5GVPKEHgfAO|sNQPSdDU]NQA1@BE «kz G7Ju@BR*@fJVP=VR=G9MOd]R=GVbZGVAaPxPSR*@BADCSrGVR=C=GUOG7E_^qGkj]JVNQP*@BPSNHgfA#M1@BRKE_^qTVbZGVPSP=GVdOR5dOEQPSR*@fC=gfADgfR=GUOG9JVNHC*@BNQEQEHGVbZGVAaP(@mTVP=T9rTVRSN¦1T7M1@BRKE@obZ¢VbZG

# !"#%$&('*) &('+&,-)./0!&!$('211 .%

−35 −30 −25 −20 −15 −10 −5 0−80

−70

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−50

−40

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−20

−10

P*)'y "# U'J *s " 6JP !$

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SUTVEW !X !Y &ê tuj 5_ 5_` ` "uf % 5]Je j ]J` 0 ∼ 5 kHz 2 _ c 5_ j$ f6e tj b_ 5_ t e t f!"5_0 ∼ 80 kHz 2pt _ ! #" f% 5_ t t f!"Hf q_ f j _ + $ _(%% f _` j *" uf j _ t j _`_ j _` i _ f `q_ `_ t j _` j ` f*"b_` +!f f_ ` f 53 _ f tHtuj 43 ]%_ _` 5_41 kPa

+ _`768 rs_`53 _ j _` ?f[` [` _ %_ 6e t _ eg_ 3 sf f%43 _g_ 6e t _ eg_ =_ t N``f_;:=<

P=GIJ*LOAONHcadDGsz¹E:wy@BdOPKR=GVbo@BR=cadDGVR5cWdDG(E@mUOTkwygfRSbo@BPSNHgfAtCSP*@BPSNHcadDG7bZgu°GVAOADGeUOGICKJIgfAaP*@fJVP=C(GICSP/GICSPSNQbZTIG9PSR=ICKMOR=g-J=LDGUOG(E@

UOTkwygfRSbo@BPSNHgfA@fJIgfdDC=PSNHcadDG>UOG>MlgfbeM1@B|Gbo@µj]NQbo@BEHGsz /NQADCSN_\fE_^`@BbeMOEQNQPSdDUOG/U^qGkj]JVNQP*@BPSNHgfAUOG0 dB

JIgfRSR=GICSM3gfADUZADgfAC*GVdOEHGVbZGVAaPnxE_^`@BbeMOEQNQPSdDUOG/UOG/UOTkwgfRSbo@BPSNHgfA@fJIgfdDCSPSNHcWdDGFUOGFMlgfbeM1@B|GFbo@µj-NQbo@BEHGs\Wbo@BNHC@BdDC=C=NlnxE@(UOTkwgfRSbo@BPSNHgfAC=P*@BPSNHcadDGxbZg°GVAOADG(UOGIC>JIgfAWP*@fJVP=C9©~M3gfdOR

P0 ' 29 kPa«kz

¾WdORoE@ª¦D|sdOR=GaDz`[O\EHG²MORSNQADJVNQM1@BE/R=TICSdOEQP*@BPpGICSPZE@JVR=gfNHC=C*@BADJIGhNQAONQPSN@BEHGVbZGVAaPcWd1@fU]R*@BPSNHcadDGhUOGE_^`@BbeMOEQNQPSdDUOGUOTVbZg-U]dOEHTIGp@IrGIJmdOADGZPSR*@BADCSNQPSNHgfArGVR=C9dOADGoJVR=gfNHC=C*@BADJIGCSdONQrs@BAaP7dOADGoEHgfNMOdONHC=C*@BADJIGp[±fXJIgfRSR=GICSM3gfADUD@BAaPZnE@ADgfA]_EQNQADTu@BRSNQP=T7UOG5£/GVRSPed9@IrGIJYJVE@BMOM3GVbZGVAaP©~rgfNQR>EHGYJ=L1@BMONQPSR=Gm®«kz1GYJIgfbeM3gfRSP=GVbZGVAaPFA^`@eM1@fC>[email protected]@BNQP=C=GVr _ #f j 3@BNsBD\cadONDgfAWP¯PSR*@urf@BNQEQEHT/UD@BADCdOAZbeNQEQNHGVdo|sR*@BAWdOE@BNQR=G/JIgfbeM3gsC=T>UOGFC*@BOEHGA1@BPSdOR=GVE_za?5@BADC¯JIGP °WM3G¯UOGbeNQEQNHGVd\f@IrGIJdOADGU]NHCSM3GVR=CSNHgfA7wgfRSP=GUOG¯E@>P*@BNQEQEHGUOGIC»|sR*@BNQADC%GVP»dOADG¯|sR*@BADUOGNQRSR=TV|sdOE@BRSNQP=TUOGEHGVdOR=C:wgfRSbZGICu\EHGICKJIgfAaP*@fJVP=C/w_@BNQOEHGIC/C*gfAaPKMOEQdDCFADgfb7OR=GVd]j3\3GVP/C*GVdOEHG(dOADG9U]°WA1@BbeNHcWdDG(UOG9JVR=gfNHC=C*@BADJIG9U]dhC=NQ|sA1@BEUOTVbZg-U]dOEHT7GVAMOdONHC*C*@BADJIGF[±fX(@(TVP=T>gfDC=GVRSrTIGsz/dOPSR=G>E_^qgfDC=GVRSrs@BPSNHgfApUOG>JIGVPSP=G>PSR*@BADCSNQPSNHgfApUD@BADC¡ADgfPSR=G/Ju@fCu\EHG>MlgfNQAWPNQbeM3gfRSP*@BAaPGIC=PKcWdDG7JIGVPSP=G9PSR*@BADCSNQPSNHgfAt@oEQNHGVdªMlgfdORxUOGIC5@BbeMOEQNQPSdDUOGIC5UOG9UOTkwygfRSbo@BPSNHgfAt@fJIgfdDCSPSNHcWdDG9UOG9M3gfbeM1@B|GUOG(E_^qgfR=U]R=GUOG

15 − 25 dB@Irs@BAaP/E_^`@BbeMOEQNQPSdDUOG(U^qGkjOJVNQP*@BPSNHgfA²bo@µj]NQb(dOb ©gfd²UOGxw_@wIgfA²TIcadONQrs@BEHGVAaP=G

15 − 25 dB@Irs@BAaP/E@

UOTkwygfRSbo@BPSNHgfA²CSP*@BPSNHcWdDG5bZg°GVAOADG(UOGIC>JIgfAWP*@fJVP=C«kz GICFJIgfAWP*@fJVP=C/UOG7£/GVRSPedYMOR=g-U]dONHC=GVAWP/dOADG5R=TVM3gfADC=G5ADgfAhEQNQADTu@BNQR=G(cWdON:UOTVM3GVADUwygfRSP=GVbZGVAaPxU]dR*@BMOMlgfRSPKGVAaPSR=G

EHGZM1@BR*@BbZVPSR=GZADgfRSbo@BEQNHC=TmUOGZUOTkwgfRSbo@BPSNHgfACSP*@BPSNHcWdDGZU]dJIgfAaP*@fJVPµGVPYE_^`@BbeMOEQNQPSdDUOGUOGeMlgfbeM1@B|GoADgfRSbo@BEQNHC=TIG

εa/ε0@BMOMOEQNHcWdDTIGsz bgfdORhdOAJIgfAaP*@fJVPt@IrGIJ

µ > εa/ε0\EHGCSNQ|sA1@BEUOTVbZg-U]dOEHTUOTVM3GVADU UOGE_^`@BbeMOEQNQPSdDUOGUOG

M3gfbeM1@B|GJIgfbebZG ∼ ε 2a

\/@BEHgfR=CªcWdDGtM3gfdORµ < εa/ε0

Js^qGICSPiE@ÄDgfA]_EQNQADTu@BRSNQP=T @fC=C=g-JVNHTIG@BdJVE@BMOM3GVbZGVAaP∼ (µε0 + ε) 3/2H(µε0 + ε)

cadONUOgfbeNQADGs\1MOR=g-U]dONHC*@BAaPKdOAhC=NQ|sA1@BE»UOTVbZg-U]dOEHT7UOgfAWP/E_^`@BbeMOEQNQPSdDUOG7rf@BRSNHG7JIgfbebZG∼ ε

3/2az /NQADCSN_\sEHG/J=L1@BAO|GVbZGVAWP>UOGFJIgfbeM3gfRSP=GVbZGVAaPcad1@fU]R*@BPSNHcWdDG → 3/2

GVAar-NQR=gfA20 dB

GVAoUOGIC*C=gfdDCUOGε0GICSP

dOADGKCSNQ|sA1@BPSdOR=GxUOGKE@YMOR=TIC*GVADJIGKUOGICw_@BNQOEHGICJIgfAaP*@fJVP=CUD@BADCEHG/beNQEQNHGVd\]GVPNQADU]NHcWdDGKdOADGFEHg-Ju@BEQNHC*@BPSNHgfApNQbeM3gfRSP*@BAaP=GUOG(E@wygfADJVPSNHgfAiUOG7U]NHCSPSRSNQOdOPSNHgfA

n(µ)EHgfR=C=cWdDG

µP=GVADUhrGVR=C

0z3G(M3gfNQAaPxGICSPKU]NHC=JVdOP=TYMOEQdDCxGVAhUOTVP*@BNQEHCxUD@BADC/E@

C*GIJVPSNHgfAbZgWUOTVEQNHC*@BPSNHgfA!aDz`[]z gfR=C=cWdDG5E@JIgfAaPSR*@BNQAWP=G7CSP*@BPSNHcWdDG(@BMOMOEQNHcWdDTIG(GICSPF@BdO|sbZGVAWP=TIGs\©

41 kPaCSdORE@9¦D|sdOR=G aDz aW«k\OE@ZUOTkwgfRSbo@BPSNHgfA

C=P*@BPSNHcadDGbZg°GVAOADGmUOGICYJIgfAWP*@fJVP=C9@BdO|sbZGVAWP=Gs\»GVP(E@PSR*@BADCSNQPSNHgfAGVAWPSR=GZdOAJIgfbeM3gfRSP=GVbZGVAaPcad1@fU]R*@BPSNHcWdDGZGVPYdOA

:o% t ] _ _` 0#

P0 P0

Γ(v)2 ∼ 1

ε(v)0

Γ(h)2 ∼ 1

ε(h)0

SUT2VEW !X kY jNj 5` f% 5_ j$ f ` t _ t f?fe 8 %_ b_ j ]%f]3 [f%?f 53 _+ ` 3 $ _ f _

` f 53 _ f!%f j _ P0_` f t5tuj 53 ]%_mq_ %f j _Jel_ ` j$ f`` _Jela j f-"5_ "?f j f%_ i _ jNj _ % _ j _ k_

f N` t _ 5_` Jr f_` 5_c _ i 3 u` _ ]%_J` t %]dc ] ?_ _ jNj _Jel_ q_ %f j _Jel_ 0698Jrs_` t jNj ]%_` 2 +!f 5] c Jegf ` f 53 _me 5_ _

ε(v)0

5_` f%d` _ ]` q_ %f j _Jel_ _` % tuj 5` ] j _ q]%_ 3 [_j f 5]dc Je:f% ` f 53 _ e H_ _

ε(h)0

5_J` f%d` _ ]` r f j _Jel_ 3 _ _v` Hf[` e j _ ?] " j _ 0 & 2 5]` % ] 0 ) 2 _de tj 53 _ _ f N` t _ t f fe 8 %_ b_ j ]?f ]3 [f%?f 43[_

Γ(v)2 < Γ

(h)2

JIgfbeMlgfRSP=GVbZGVAWPmGVA MOdONHC*C*@BADJIG[±fX#UOGpE_^`@BbeMOEQNQPSdDUOGiUOTVbZgWU]dOEHTIG²@iEQNHGVd·MlgfdORZUOGICo@BbeMOEQNQPSdDUOGICUOGpM3gfbeM1@B|GMOEQdDCxTVEHGVrTIGICZ©~P°-MONHcadDGVbZGVAWP

+5 dBM1@BRxR*@BMOMlgfRSPYn

26 kPaCSdOR/E@o¦D|sdOR=GHaDz`[«kz3A#Gk3GVPu\<EHG9MOLDTVADgfbZVADGeUOG

JVE@BMOMlGVbZGVAWPR=GICSM3gfADC*@BOEHGxUOG/E@9U]°-A1@BbeNHcadDGxUOGxJVR=gfNHC=C*@BADJIG5U]dCSNQ|sA1@BEUOTVbZg-U]dOEHTxGVAMOdONHC*C*@BADJIGY[±fX]\O@BMOM1@BR*@µºHPMlgfdOR¡UOGIC@BbeMOEQNQPSdDUOGIC¯U^qGkjOJVNQP*@BPSNHgfAoMOEQdDC¡TVEHGVrTIGICnxJu@BdDC=G>U^ dOADG>UOTkwygfRSbo@BPSNHgfAoCSP*@BPSNHcWdDGbZgu°GVAOADG/UOGIC¡JIgfAaP*@fJVP=CMOEQdDC>TVEHGVrTIGsz?5@BADCZEHGICmUOGVd]j·Ju@fCoUOGICe¦D|sdOR=GICaDz`[tGVP aDz a1\E_^`@BbeMOEQNQPSdDUOGiU]dCSNQ|sA1@BEKUOTVbZgWU]dOEHT²MOR=gurGVA1@BAaPU^qgfADUOGICUOG

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P µs$

fp = 80kHz

fp = 50kHz

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fp = 80 kHz_

fp = 50 kHz

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50 kHzGVP

80 kHzz

bgfdOR:dOADGw~R=TIcWdDGVADJIGM3gfRSP=GVdDC=G¡UOG50 kHz

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10 cmz1¾WdOR>JIGVPSP=G(U]NHCSP*@BADJIGYUOGxMOR=gfM1@B|a@BPSNHgfA\DEHG5C=NQ|sA1@BE:UOTVbZgWU]dOEHT(GICSP MlgfRSP=T YnE@9r-NQP=GIC=C=GxUOGIC

gfADUOGICUOGM3gfbeM1@B|GUOGpJVNHC*@BNQEQEHGVbZGVAWPu\GVPZ@Bd]UOGVEnhUOGpJIGVPSP=GpU]NHCSP*@BADJIGs\NQEC=GoMOR=gfM1@B|GnhC*@²rWNQP=GIC*C=GmUOGMOL1@fC=GEHgfAO|sNQPSdDU]NQA1@BEHGs\>MOEQdDCTVEHGVrTIGsz b»gfdORdOADGhwR=TIcadDGVADJIGªM3gfRSP=GVdDC=GªUOG

80 kHz\E@EHgfAO|sdDGVdORU^`@BPSP=TVAad1@BPSNHgfAUOGIC

gfADUOGICUOGM3gfbeM1@B|GUOGJVNHC*@BNQEQEHGVbZGVAWPZGICSPeU^qGVAar-NQR=gfA2 − 4 cm

\MOEQdDC(w_@BNQOEHGpcadDGmMOR=TIJITIUOGVbebZGVAaPuz¯ GpCSNQ|sA1@BEUOTVbZg-U]dOEHTGICSP@BEHgfR=C M3gfRSP=T mnE@²r-NQP=GIC=C=GoUOGICgfADUOGICUOGmMlgfbeM1@B|GUOGmJVNHC@BNQEQEHGVbZGVAaPZCSdOR7dOADGU]NHC=P*@BADJIGoMOEQdDCw_@BNQOEHGecWdDG7M3gfdOR

50 kHz\GVP5EHGR=GIC=P=GeUOGE@U]NHCSP*@BADJIGeUOGMOR=gfM1@B|a@BPSNHgfAP=gfP*@BEHGoGICSPxM1@BR=JIgfdORSdnE@r-NQP=GIC=C=GUOG

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80 kHz@BRSRSNQrGFMOEQdDCP[ZfPcWdDG>JIGVEQdONlNHC=CSdoU^qgfADUOGICUOG>M3gfbeM1@B|GKnYE@xwR=TIcadDGVADJIGFMlgfRSP=GVdDC*G

50 kHzz-G

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UD@BADCdOA9R=TV|sNQbZGU]°WA1@BbeNHcWdDGszfA7M1@BRSPSNHJVdOEQNHGVRu\fE@/U]NQE@BP*@BADJIGcWdON-CSdONQPdOADGEHgfN]GVA9MOdONHC=C*@BADJIG[±fXKGICSP%U]NQR=GIJVP=GVbZGVAaP@fC*C=gWJVNHTIG@Bd JVE@BMOM3GVbZGVAaPÜOGICJIgfAWP*@fJVP=CuzZGICJIgfAaP*@fJVP=CJVE@BMOM1@BAWP=CMlGVdOrGVAWP¢VPSR=GNQAaP=GVRSMOR=TVP=TICJIgfbebZG·EHG

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¦D|sdOR=G aDz`]z] :@U]NQR=GIJVPSNHgfA²UOGKMOR=gfM1@B|a@BPSNHgfA²GICSPbo@BNQAaP=GVA1@BAWPFLDgfRSN dugfAWP*@BEHGs\1UOGVd]jmPSR*@BADC*U]dDJVP=GVdOR=CFUOGxJVNHC*@BNQEQEHGVbZGVAaPNHUOGVAaPSNHcWdDGIC5C=gfAaPxdOPSNQEQNHC=TICKM3gfdORxE@Z|TVADTVR*@BPSNHgfAtUOGICxgfADUOGIC5UOG7MlgfbeM1@B|GGVPxdOAhPSR*@BADC*U]dDJVP=GVdOR5EHgfAO|sNQPSdDU]NQA1@BE¡GICSPdOPSNQEQNHC=T5GVAR=TIJIGVMOPSNHgfAz¾WdORE@7¦D|sdOR=G aDzQ®uÀO\]EHGIC>@BbeMOEQNQPSdDUOGIC/UOGIC>CSNQ|sA1@Bd]jUOTVbZgWU]dOEHTICNHC*CSdDCU^qgfADUOGICFUOGxM3gfbeM1@B|G5UOG5JVNHC*@BNQEQEHGVbZGVAaP

MlgfE@BRSNHC*TIGICrGVRSPSNHJu@BEHGVbZGVAaP>©C=GVEHgfA7EHGIC:J*L1@µºHADGIC»UOG¡wgfR=JIG«3GVP»LDgfRSN dugfAaP*@BEHGVbZGVAWPF©gfRSPSLDgf|gfA1@BEHGVbZGVAaP¡@Bd]j(J=L1@µºHADGICUOGwgfR=JIG«:C=gfAWP»PSR*@fJITIGICuzB :@FwR=TIcadDGVADJIGUOG¯M3gfbeM1@B|GGICSP»JIGVAWPSR=TIGCSdOR

80.192 kHzGVP»E@>wR=TIcadDGVADJIGUOTVbZgWU]dOEHTIGrs@BdOP

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UOG1 dB

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n(µ)ADG5UOTVMlGVADUM1@fC>UOG

ε0«kz

¾WdORtE@¦D|sdOR=G aDzQ®s®s\5E@ bZ¢VbZG·Gkj]MlTVRSNHGVADJIG@TVP=T¨bZGVADTIGs\5bo@BNHC@urGIJ dOADG¨JIgfAWPSR*@BNQAaP=G CSP*@BPSNHcadDG MOEQdDCTVEHGVrTIGs\¡TV|a@BEHGn

24 kPaz?5@BADCZJIG²Ju@fCu\%EHGICPSR*@BADCSNQPSNHgfADC

2 → 3/2UD@BADCeE_^`@BbeMOEQNQPSdDUOGhU]d CSNQ|sA1@BE>UOTVbZgWU]dOEHT

GVA·wygfADJVPSNHgfAUOG²E_^`@BbeMOEQNQPSdDUOGªUOGhMlgfbeM1@B|GªC=GhCSNQPSdDGVAaPGVAar-NQR=gfAn −22 dBGVP −15 dB

MlgfdORE_^qGkj]JVNQP*@BPSNHgfAMlgfE@BRSNHC*TIGLDgfRSN dugfAaP*@BEHGVbZGVAWPFGVP¡M3gfdOR¯E_^qGkjOJVNQP*@BPSNHgfAmM3gfE@BRSNHC=TIG>rGVRSPSNHJu@BEHGVbZGVAWPu\-R=GICSMlGIJVPSNQrGVbZGVAWPuzW¼/ADG/@BAONHC*gfPSR=gfMONHGU]dM1@BR*@BbZVPSR=GADgfAEQNQADTu@BNQR=GGkGIJVPSNwFUOG

7 dBGIC=P7gfDC=GVRSrTIGsz :@²PSR*@BADCSNQPSNHgfA

2 → 3/2U]dC=NQ|sA1@BEUOTVbZg-U]dOEHT

NHC=CSd U^qgfADUOGICUOGM3gfbeM1@B|GM3gfE@BRSNHC=TIGICorGVRSPSNHJu@BEHGVbZGVAWPª©~EHgsC@BAO|GICmCSdORoE@i¦D|sdOR=G,aDzQ®s®µ«A^qGICSPmJIGVM3GVADUD@BAaPmM1@fCONHGVAiUOTk¦DAONHG9GVAiR*@BNHC=gfAiU^ dOAªJIgfbeM3gfRSP=GVbZGVAaP5M1@BRSPSNHJVdOEQNHGVR5UOG(E@mU]°-A1@BbeNHcadDG9UOG7JVR=gfNHC*C*@BADJIG7GVA#@BbeMOEQNQPSdDUOGU]dCSNQ|sA1@BE1UOTVbZg-U]dOEHT/@BdOP=gfdORUOGFJIGVPSP=G>PSR*@BADC=NQPSNHgfAz-GFJIgfbeM3gfRSP=GVbZGVAaPC=GR=GVPSR=gfdOrG/UD@BADC¯E@5bZgWUOTVEQNHC*@BPSNHgfAcadON<CSdONQPJIGVPSP=G/C*GIJVPSNHgfAmM3gfdORJIGVRSP*@BNQADGICwygfRSbZGICUOGFE@(U]NHC=PSRSNQOdOPSNHgfAUOGKJIgfAaP*@fJVP=CUD@BADCEHG>beNQEQNHGVdz] ¡^`@BbeMOEQNQPSdDUOGxU]dpCSNQ|sA1@BEUOTVbZgWU]dOEHT/NHC=CSdU^qgfADUOGICUOG>M3gfbeM1@B|GFM3gfE@BRSNHC=TIGICLDgfRSN dugfAaP*@BEHGVbZGVAWP>GICSPP=gfd ¥ gfdOR=CCSdOMlTVRSNHGVdOR=GxnYJIGVEQEHGKU]dpCSNQ|sA1@BEUOTVbZgWU]dOEHTYNHC=CSdU^qgfADUOGICFUOGxM3gfbeM1@B|G5M3gfE@BRSNHC=TIGIC>rGVRSPSNHJu@BEHGVbZGVAWPKU^qGVAWrWNQR=gfA

10 dBz

¾WdORE@¦D|sdOR=G aDzQ®uXD\VE@JIgfAaPSR*@BNQAWP=GCSP*@BPSNHcWdDG¡@GVADJIgfR=G¡TVP=T@BdO|sbZGVAaP=TIGs\uM3gfdOR»@BPSP=GVNQADU]R=G64 kPa

zI :@U]NTVR=GVADJIGGVAaPSR=GoEHGIC7@BbeMOEQNQPSdDUOGIC9UOGIC(CSNQ|sA1@Bd]jUOTVbZg-U]dOEHTIC(NHC*CSdDCYU^qgfADUOGIC9UOGeMlgfbeM1@B|GoMlgfE@BRSNHC*TIGICYLDgfRSN dugfAaP*@BEHGVbZGVAWPZGVPrGVRSPSNHJu@BEHGVbZGVAaPrs@BRSNHG/GVAWPSR=G

10 dBGVP

15 dBCSdORP=gfdOP=GFE@Y|a@BbebZG/U^`@BbeMOEQNQPSdDUOGxUOG>M3gfbeM1@B|Gsz- GIC¯PSR*@BADC=NQPSNHgfADC

2 → 3/2C=gfAaP#CSNQPSdDTIGIC#n −23 dB

GVPtn −8 dBMlgfdORiEHGICªGkj]JVNQP*@BPSNHgfADC#M3gfE@BRSNHC=TIGICiLDgfRSN dugfAaP*@BEHGVbZGVAWPGVP

#' !"#%$&('*) &('+&,-)./0!&!$('211 .%

−35 −30 −25 −20 −15 −10 −5 0−50

−40

−30

−20

−10

0

10

P*)'y "# U'J *s " 6JP !$

# %('*) +.,."

"

SUTVEW XkY &ê tj b_ 5_` ` "uf % 5] e j ]` _ c 5_ j($ fe tuj 5_ 5_ t e t f!"b_ 0 t t f!"Hf r f j _ 2 #+ $ _(%% f 5_ `f jdj _Jel_ _` t j fN`]_ `_ j j _` Jr f _J` 5_ c _ 0q_ %f j _Jel_ 2 t j _` j `f*"5_` i _ =_` t j f ` ]%_ dr "% uf j _Jel_ f % Jrsf k_` 5_-c _&0r f j _Jel_ 2Et j _` _ j _`+ f ?f _E` f 53 [_ f!%f j _ f t5tuj 53 ]%_g_` ∼ 64kPa

rGVRSPSNHJu@BEHGVbZGVAWPu\¡R=GICSM3GIJVPSNQrGVbZGVAaPuz ¡^`@BAONHC=gfPSR=gfMONHG²ADgfAËQNQADTu@BNQR=GNQADU]dONQP=G²@BPSP=GVNQADU·@BEHgfR=C15 dB

z »@iPSR*@BADCSNQPSNHgfA@fC*C=gWJVNHTIGYneE@eM3gfE@BRSNHC*@BPSNHgfA²rGVRSPSNHJu@BEHG9@EQNHGVdineUOGYMOEQdDCwygfRSP=GIC/@BbeMOEQNQPSdDUOGICxUOG5M3gfbeM1@B|GYM3gfdOR>dOADG(JIgfAWPSR*@BNQAaP=GC=P*@BPSNHcadDGUOG

64 kPacWdDG¡M3gfdOR:dOADGJIgfAaPSR*@BNQAWP=GCSP*@BPSNHcadDGUOG

24 kPagfdGVADJIgfR=G

8 kPazµ GIC%JIgfAaP*@fJVP=CrGVRSPSNHJu@Bd]j\

MOEQdDCwgfRSP=GVbZGVAWPFMOR=TkJIgfAaPSR*@BNQAWP=Cu\3C=gfAaP/GVAGk3GVP>MOEQdDC/U]NoJVNQEHG(n9w_@BNQR=GYJVE@BMOM3GVRu\DGVP>dOADG(@BbeMOEQNQPSdDUOG7U^qGkj]JVNQP*@BPSNHgfAMOEQdDCwgfRSP=G5GICSPFADTIJIGIC=C*@BNQR=Gsz ¡^qGVADC=GVb(OEHGmUOGoJIGIC5R=TIC=dOEQP*@BP=C(GICSP(JIgfLDTVR=GVAWP@urGIJZEHGICYUOGVd]j#MORSNQADJVNQM1@BEHGIC9gfDC=GVRSrs@BPSNHgfADC7GVPYNQAWP=GVRSMOR=TVP*@BPSNHgfADC

cWdONEHGVdOR=CZC=gfAaPZ@fC=C*gWJVNHTIGIC§E_^qGVbeMOE@fJIGVbZGVAaPUOGE@²PSR*@BADC=NQPSNHgfA2 → 3/2

UD@BADCE@iU]°-A1@BbeNHcadDGUOGJVR=gfNHC*C*@BADJIGU]dCSNQ|sA1@BEFUOTVbZg-U]dOEHT²GICSPZNQAWP=GVRSMOR=TVP=ThJIgfbebZG²dOADGbo@BAONwyGICSP*@BPSNHgfAU]d·MOLDTVADgfbZVADGhUOGJVE@BMOM3GVbZGVAaPpUD@BADCZEHGbeNQEQNHGVdGVPwgfdORSAONQPdOADGxGICSPSNQbo@BPSNHgfAUOGKE_^`@BAONHC=gfPSR=gfMONHGxNQADU]dONQP=GxUD@BADCEHG/beNQEQNHGVdpM1@BREHGKJ=L1@BRS|GVbZGVAaPFdOAON @µj-N@BE_zO :@U]NTVR=GVADJIGU^qGkoJu@fJVNQP=TZU]dªMOR=gWJIGIC*CSdDCxUOG9UOTVbZg-U]dOE@BPSNHgfAC=GVEHgfAªE@mM3gfE@BRSNHC*@BPSNHgfA#rGVRSPSNHJu@BEHGegfdªLDgfRSN dugfAaP*@BEHGZUOGICgfADUOGIC/UOGYM3gfbeM1@B|G(UOGYJVNHC@BNQEQEHGVbZGVAaP5GICSP>dOADG(bo@BAONwGICSP*@BPSNHgfAiUOGYE_^`@BAONHC=gfPSR=gfMONHG7ADgfAEQNQADTu@BNQR=G7MOR=TIC=GVAaP=G9UD@BADC>EHGbeNQEQNHGVd\1GVPMlGVdOP>M3GVRSbZGVPSPSR=GYTV|a@BEHGVbZGVAWP/UOG5E@Ju@BR*@fJVP=TVRSNHC=GVRuz

>¦DAUOG¯beNHGVd]j9JIgfbeMOR=GVADU]R=GE_^ NQA]¿DdDGVADJIGUOGICJIgfAWP*@fJVP=Cwy@BNQOEHGIC%UD@BADCEHGbeNQEQNHGVdC=dOR»EHGMOR=gWJIGIC*CSdDCADgfA7EQNQADTu@BNQR=GUOG(UOTVbZgWU]dOE@BPSNHgfA\3GVPK@BNQADC=N:UOGYEHGICKJu@BR*@fJVP=TVRSNHC=GVRKnZPSR*@IrGVR=CKE@ZU]NHC=PSRSNQOdOPSNHgfAiUOGYUOTkwygfRSbo@BPSNHgfADCxUOG(JIgfAWP*@fJVP=Cu\1dOAbZg-UOVEHGCSNQbeMOEHGmU^`@BAWP=GVAOADGM1@BR*@BbZTVPSRSNHcadDG@urGIJoMORSNHC=GGVAJIgfbeMOP=GU]dMOLDTVADgfbZVADGpUOGmJVE@BMOM3GVbZGVAaPeGVPUOGoE@wygfRSbZG5UOG5E@U]NHCSPSRSNQOdOPSNHgfAiUOG5JIgfAaP*@fJVP=C

n(µ)GIC=Pbo@BNQAaP=GVA1@BAaPxUOTVrGVEHgfMOM3Tsz

: 5] j `f 0.

&(' 254 =; .<;*2>,¼/ADG²bZg-UOTVEQNHC*@BPSNHgfAnUOGV|sR=TªUOGiJIgfbeMOEHGkj-NQP=TªJVR=gfNHC=C@BAaP=G#UOGhE_^`@BAaP=GVAOADGªM1@BR*@BbZTVPSRSNHcWdDG#UD@BADCmEHGICbeNQEQNHGVd]j

|sR*@BAadOE@BNQR=GIC7MOR=GVA1@BAaPGVAJIgfbeMOP=GZEHGZMOLDTVADgfbZVADGoUOGZJVE@BMOM3GVbZGVAaPGICSPYMOR=TIC=GVAWP=TIGsz: GZMlgfNQAWP7UOGZUOTVM1@BRSP9UOGZJIGbZgWUOVEHG(GICSPE@R=GVE@BPSNHgfAhJIgfAWPSR*@BNQAaP=G(¯UOTkwgfRSbo@BPSNHgfAiUOGY£KGVRSPed\DcadONC^qTIJVRSNQPFUD@BADC>EHGYJu@fCFCSP*@BPSNHcadDG(C=gfdDCE@9wygfRSbZGUOT ¥ n9R=GVADJIgfAWPSR=TIGoyxgfL1´ss¯§

σ0 = B(ε0)3/2 .

©aDz¬s« gfR=C=cWd^ dOADGMlGVRSPSdO1@BPSNHgfA¶U]°-A1@BbeNHcadDG½©gfADUOG¨@fJIgfdDCSPSNHcWdDG«²rWNHGVAWPiCu^`@ ¥ gfdOP=GVRtn E@JIgfAaPSR*@BNQAWP=G CSP*@BPSNHcWdDGs\

UD@BADCKE@oEQNQbeNQP=G7wR=TIcadDGVAWPSNHGVEQEHGeUOG9MlGVRSPSdORS1@BPSNHgfAtcWd1@fCSNCSP*@BPSNHcWdDG(M1@BRxR*@BMOM3gfRSPY@Bd]jhP=GVbeMDCxJu@BR*@fJVP=TVRSNHCSPSNHcWdDGICYU]dMOR=gfOEHVbZGs\OE@R=GVE@BPSNHgfA²JIgfAWPSR*@BNQAaP=G(UOTkwygfRSbo@BPSNHgfA²UOGY£KGVRSPedYC=GxR=TITIJVRSNQP/C=gfdDC>E@7wgfRSbZGZ3@BNss@D\cGVEs[f¯§

σ0 + σ = B(ε0 + ε)3/2 .©aDz`´«

F¦DAtUOGeMOR=GVADU]R=GZGVAJIgfbeMOP=GeE@M3gsC=CSNQONQEQNQP=TZMlgfdOR5EHGICYJIgfAWP*@fJVP=Cu\:UOGeJVE@BMOM3GVRu\:Js^qGIC=P(npU]NQR=GZUOGZCu^qgfdOr-RSNQRYGVPUOGxC=GKR=GkwGVRSbZGVRFC*gfdDCE_^`@fJVPSNHgfAUOGxE_^qgfADUOGY@fJIgfdDCSPSNHcadDGs\DE@YwygfADJVPSNHgfAUOG5£KGu@Ir-NHCSNHUOGxPSR*@fU]dONHC*@BAaPFEHGKwy@BNQPcWd^`@BdDJVdOADGJIgfAaPSR*@BNQAWP=G(A^qGkj]NHCSP=GxEHgfR=C=cWdDGxEHGYJIgfAaP*@fJVPKGICSP>gfdOrGVRSPu\DGICSPNQAWPSR=gWU]dONQP=Gm]cGVE_as¯§

σ0 + σ = B(ε0 + ε)3/2H(ε0 + ε) ,©aDz`«

gfdGVADJIgfR=Gs\σ0 + σ = Bε

3/20 (1 +

ε

ε0)3/2H(1 +

ε

ε0) .

©aDzQ®uÀ« M1@BRSPSNQRUOGbo@BNQAaP=GVA1@BAaPu\WE@xUOTkwygfRSbo@BPSNHgfAoCSP*@BPSNHcWdDG

ε0GICSP¡CSdOMOM3gsC=TIGR=GVMOR=TIC=GVAWP=GVR¯E@xUOTkwygfRSbo@BPSNHgfAoCSP*@BPSNHcWdDG

bZgu°GVAOADGiUOGICoJIgfAWP*@fJVP=Cu\GVPoE@#JIgfADCSP*@BAaP=GBMlGVdOPmTVrGVAaPSdDGVEQEHGVbZGVAaP¢VPSR=G²bZgWU]N¦1TIG²MlgfdORoP=GVAONQRmJIgfbeMOP=G²UOGIC

M1@BR*@BbZVPSR=GICFGkGIJVPSNwCFU]dbeNQEQNHGVd©~rgfNQR>EHG5J*L1@BMONQPSR=Go®µ«kz?5@BADC>dOAª@BRSR*@BAO|GVbZGVAaPxUOTIC=gfR=UOgfAOADTs\1EHGIC/JIgfAaP*@fJVP=CKA^qgfAaP/M1@fC>dOADG7UOTkwgfRSbo@BPSNHgfAªCSP*@BPSNHcWdDGYTV|a@BEHGYbo@BNHC/dOADG

U]NHCSPSRSNQOdOPSNHgfAeCSP*@BPSNHCSPSNHcWdDGUOGUOTkwygfRSbo@BPSNHgfADCUOGJIgfAaP*@fJVP=C¡Gkj]NHCSP=GszsGVPSP=GU]NHCSPSRSNQOdOPSNHgfAeUOT ¥ n/MOR=TIC=GVAWP=TIGGVPU]NHC*JVdOP=TIGMOR=TIJITIUOGVbebZGVAaPxGICSP>ADgfP=TIG

n(µ)gfv

µR=GVMOR=TIC*GVAaP=GYEHGYR*@BMOMlgfRSPKGVAaPSR=GYE@ZUOTkwgfRSbo@BPSNHgfAiU]d²JIgfAaP*@fJVPKJIgfADCSNHUOTVR=T(GVP

E@UOTkwgfRSbo@BPSNHgfAhbZgu°GVAOADG7UOGIC>JIgfAaP*@fJVP=Cε0zO?5@BADC>JIGYJu@fC\]E@JIgfAaPSR*@BNQAaP=G7P=gfP*@BEHGYGICSP>TV|a@BEHG9np§

σ0 + σ = B0

∫ +∞

0n(µ)(µ +

ε

ε0)3/2H(µ +

ε

ε0)dµ ,

©aDzQ®s®«

@IrGIJB0 = Bε

3/20 F¦DA9UOGADGJIgfADCSNHUOTVR=GVRcadDGE@/M1@BRSPSNHGU]°-A1@BbeNHcWdDGUOGE@KJIgfAaPSR*@BNQAWP=G

σ\E@FM1@BRSPSNHGCSP*@BPSNHcadDG

σ0GICSPR=GVPSR*@BADJ=LDTIG

UOGxM1@BRSPFGVPFU^`@BdOPSR=GYU]dC=NQ|sADGYTV|a@BE:UOGxE_^qTIcWd1@BPSNHgfA²MOR=TIJITIUOGVAaP=G©aDzQ®s®µ«K§

σ = B0

∫ +∞

0n(µ)

[(µ +

ε

ε0)3/2H(µ +

ε

ε0) − µ3/2

]dµ .

©aDzQ®uX«¾WN]E_^qGkj]JVNQP*@BPSNHgfAZUOGUOTkwygfRSbo@BPSNHgfAo@fJIgfdDCSPSNHcWdDGGICSPCSNQAadDC=g_UD@BEHG

ε = εa sin(θ)\BE@/JIgfAWPSR*@BNQAaP=GF@fC*C=gWJVNHTIGCu^qTIJVRSNQP§

σ = B0

∫ +∞

0n(µ)

[(µ +

εa sin(θ)

ε0)3/2H(µ +

εa sin(θ)

ε0) − µ3/2

]dµ .

©aDzQ®u[«?5@BADCFEHG7Ju@fCFU^ dOADG9Gkj]JVNQP*@BPSNHgfAªCSNQAWdDC=g_UD@BEHGs\1EHG9CSNQ|sA1@BEUOTVbZgWU]dOEHT9GICSPFUOG(w~R=TIcWdDGVADJIGYAWdOEQEHGs\<UOgfADJ9CSP*@BPSNHcWdDGsz

GVMlGVADUD@BAWPu\OJIGVPSP=G5@BMOMOR=gWJ*LDGxdOPSNQEQNHC*@BAaPdOADGxGkj]JVNQP*@BPSNHgfACSNQbeMOEHGKM3GVRSbZGVPUOG/R=GVADU]R=GxJIgfbeMOP=GxUOGKE_^`@BbeMOEQNQPSdDUOG5U]dCSNQ|sA1@BEOUOTVbZgWU]dOEHTMOdONHC*cad^ dOA9w_@fJVP=GVdOR

2C=GVdOEHGVbZGVAWP¯Gkj-NHC=P=GGVAWPSR=GdOADG>@BbeMOEQNQPSdDUOGUOTVbZgWU]dOEHTIGU]°-A1@BbeNHcadDG5©~NHC=C=dDG

UOGMOEQdDCSNHGVdOR=C¯w~R=TIcWdDGVADJIGICUOG>M3gfbeM1@B|G«¡GVP¡dOADGK@BbeMOEQNQPSdDUOGxUOTVbZgWU]dOEHTIG/CSP*@BPSNHcadDG9©~NHC=CSdDG/U^ dOADGFC=GVdOEHG>w~R=TIcWdDGVADJIG

!"#%$&('*) &('+&,-)./0!&!$('211 .%

UOGªM3gfbeM1@B|G«kz bgfdORi@urgfNQRi@fJIJIICdOAONHcWdDGVbZGVAaP#nE@¨JIgfbeM3gsC*@BAaP=GUOG#1@fC=C=GiwR=TIcWdDGVADJIGtUOG#E@JIgfAaPSR*@BNQAWP=Gs\dOAbZgu°GVAOA1@B|GP=GVbeMlgfR=GVECSdORdOADGMlTVRSNHg-UOGUOGmE_^qgfADUOGpUOGM3gfbeM1@B|GGIC=PGkGIJVPSdDTsz%¹EM3GVRSbZGVPU^qTVEQNQbeNQADGVReE@JIgfbeM3gsC*@BAWP=G(nE@9wR=TIcadDGVADJIG(UOGxMlgfbeM1@B|G©GVAbZgu°GVAOADG(AadOEQEHG(C=dORdOADG5MlTVRSNHg-UOG«@BNQADCSNcWdDG5EHGICL1@BRSbZgfAONHcWdDGICC=dOMlTVRSNHGVdOR=C>TVrGVAWPSdDGVEQEHGVbZGVAaPx|TVADTVR=TICuz

〈σ〉 ∼ B0

∫ 2π

0

∫ +∞

0n(µ)

[(µ +

εa

ε0sin(θ))3/2H(µ +

εa

ε0sin(θ)) − µ3/2

]dµ dθ ,

©aDzQ®PaW«gfv 〈 . . . 〉 R=GVMOR=TIC=GVAaP=G7E@bZg°GVAOADG7CSdORFdOADGYM3TVRSNHgWUOGYUOGYE_^qgfADUOG(UOGYM3gfbeM1@B|G(£K¤z1GVPSP=G(TIcWd1@BPSNHgfA©2aDzQ®Pa-«GICSPdOPSNQEQNHC*TIG/UD@BADCE@YC=dONQP=GFM3gfdOREHGICCSNQb7dOE@BPSNHgfADCAadObZTVRSNHcWdDGICUOG>E_^`@BAWP=GVAOADGxM1@BR*@BbZTVPSRSNHcadDGxUD@BADCUOGICbeNQEQNHGVd]j@IrGIJU]NTVR=GVAaP=GIC/U]NHCSPSRSNQOdOPSNHgfADCFC=P*@BPSNHCSPSNHcadDGIC/UOG5JIgfAaP*@fJVP=C

n(µ)z

¼FADG@BdOPSR=GTVP*@BMlG9M3GVRSbZGVPxUOG9MOR=GVADU]R=GGVA#JIgfbeMOP=GE_^ NQA]¿DdDGVADJIGeUOG(E_^`@BPSP=TVAWd1@BPSNHgfAUOGIC5gfADUOGIC5UOG7MlgfbeM1@B|GC=dOR5EHGZMOR=g-JIGIC=CSdDCYUOGmUOTVbZgWU]dOE@BPSNHgfAbZgWUOTVEQNHC=ToM1@BRYE_^qTIcWd1@BPSNHgfA ©aDzQ®Pa-«kz¼/ADGZUOTIJVR=gfNHC=C@BADJIGmGkj]MlgfADGVAWPSNHGVEQEHGoUOGICgfADUOGIC7UOGeM3gfbeM1@B|GoCSdOR5dOADGoEHgfAO|sdDGVdOR7Ju@BR*@fJVP=TVRSNHCSPSNHcWdDG

àGIC=P5NQAaPSR=g-U]dONQP=G@urGIJZEHGw_@fJVP=GVdOR(U^`@BbeMOEQNQPSdDUOG

εaz

¯^ NQAaP=TV|sR*@BPSNHgfAªCSdORE@JIgWgfR=UOgfAOADTIG7CSM1@BPSN@BEHGxGIC=PF@BEHgfR=CFGkGIJVPSdDTIGZ§

〈σ〉 ∼ B0

∫ +∞

0

∫ 2π

0

∫ +∞

0n(µ)

[(µ +

εa

ε0sin(θ)e−

xà )3/2H(µ +

εa

ε0sin(θ)e−

xà ) − µ3/2

]dµ dθ dx .©aDzQ®u«

bEQdDCSNHGVdOR=CR=TICSdOEQP*@BP=C»bZgfAaPSR=GVAaP¡cWdDG¡E@FU]NHCSPSRSNQOdOPSNHgfAeUOG¡wygfR=JIGICUOG¯JIgfAaP*@fJVPn(f)\uM3gfdOR»UOGIC:wgfR=JIGIC»NQA]wyTVRSNHGVdOR=GIC

n#E@#rs@BEHGVdORobZgu°GVAOADGªU]dbeNQEQNHGVdf0MOR=TIC=GVAWP=G²dOAMOE@BP=Gu@Bd

n(f) = const.z5bgfdORmEHGICJIgfAaP*@fJVP=CpUOGh£/GVRSPed\

f ∼ ε3/2 \OJIG5cadONC=NQ|sAON¦1GYcad^ dOADG5U]NHC=PSRSNQOdOPSNHgfA n(f)JIgfADCSP*@BAWP=GYM3gfdOR

f < f0JIgfRSR=GICSM3gfADU²n7dOADGYU]NHCSPSRSNQOdOPSNHgfA

UOTIJVR=gfNHC*C*@BAaP=GiUOGhUOTkwygfRSbo@BPSNHgfAn(µ) ∼ µ−1/2 z b»gfdORpdOADG²U]NHC=PSRSNQOdOPSNHgfA n(f) ∼ f−1/3 GVA]UOGIC=C=gfdDCpUOG f0

\U]NQrGVRS|GVAWP=G9MlgfdOR/EHGIC>w_@BNQOEHGIC>wygfR=JIGIC/bo@BNHCFNQAWP=TV|sR*@BOEHGs\<E@oU]NHCSPSRSNQOdOPSNHgfA#UOG(UOTkwygfRSbo@BPSNHgfA

n(µ)JIgfRSR=GIC=MlgfADUD@BAWP=G

GIC=P>JIgfADCSP*@BAaP=GYGVA]UOGIC*C=gfdDC>M3gfdORµ < 1

z ¡^ NQbeMOEHTVbZGVAaP*@BPSNHgfAAadObZTVRSNHcWdDG#UOGªE@wgfRSb7dOEHG·©aDzQ®PaW«p@TVP=TiGkGIJVPSdDTIG#M3gfdORU]N3TVR=GVAWP=GICpwgfRSbZGIC²UOGhE@

U]NHC=PSRSNQOdOPSNHgfAtCSP*@BPSNHCSPSNHcWdDG9UOGUOTkwygfRSbo@BPSNHgfADCYUOGJIgfAaP*@fJVP=Cn(µ)z3?5@BADCxdOAiMOR=GVbeNHGVR5P=GVbeMDC\lC*GVdOEHG7dOADGM3gfRSPSNHgfA

UOG5JIgfAWP*@fJVP=C>w_@BNQOEHGIC>GICSPK@ ¥ gfdOP=TIGxwygfRSbZGVEQEHGVbZGVAaPKGVA]UOGIC=C=gfdDC/U^ dOADG5rf@BEHGVdOR/UOGµTV|a@BEHG(n

0.1zlGVPSP=GYgfM3TVR*@BPSNHgfA

M3GVRSbZGVPYUOGebZgfAWPSR=GVR7GVAaPSR=Gm@BdOPSR=GICu\cWdDGEHGADgfb(OR=GoUOGeJIgfAWP*@fJVP=CYw_@BNQOEHGICYGICSP5MOR=TVM3gfADUOTVR*@BAaP9CSdOR5E_^`@BMOM1@BRSNQPSNHgfAUOG9E@mPSR*@BADCSNQPSNHgfAtcWd1@fU]R*@BPSNHcadDG → MOdONHC=C*@BADJIGe[±fXpUD@BADCxE@UOTVM3GVADUD@BADJIGeGVA#@BbeMOEQNQPSdDUOGoU]dªC=NQ|sA1@BE%UOTVbZg-U]dOEHTs\GVP>cWdDGxE@9wygfRSbZG5GVEQEHGk_bZ¢VbZG(UOG5E@U]NHC=PSRSNQOdOPSNHgfA²rGVR=CEHGICw_@BNQOEHGICFJIgfAWP*@fJVP=CFGICSPbZgfNQADCFNQbeMlgfRSP*@BAWP=Gsz¾WdOR5E@¦D|sdOR=G*aDzQ®u[O\C=gfAWPYR=GVMOR=TIC=GVAWP=TIGIC7CSNQb7dOEQP*@BADTVbZGVAaP9E@pU]NHCSPSRSNQOdOPSNHgfAUOGeJIgfAWP*@fJVP=C

n(µ)dOPSNQEQNHC=TIGeM3gfdOR

EHGJu@BEHJVdOEUOGoE_^`@BbeMOEQNQPSdDUOGUOTVbZg-U]dOEHTIGpGVAwygfADJVPSNHgfAÜOGµ\:GVP9E_^`@BbeMOEQNQPSdDUOGUOTVbZg-U]dOEHTIGp@fC*C=gWJVNHTIGmGVAwygfADJVPSNHgfA

UOGYE_^`@BbeMOEQNQPSdDUOG7UOG5M3gfbeM1@B|G(£K¤\ eεaε0

zD GICKTIJ=LDGVEQEHGICKGVA²UOTIJVNQ3GVEHCFM3gfdOR>EHGICK@BlJVNHC=C*GICµGVP eεa

ε0

C=gfAaPFJIgfA]wygfADU]dDGICGVA½@fJIJIgfR=U½@IrGIJiEHGICpGICSPSNQbo@BPSNHgfADCGkGIJVPSdDTIGICuz GICTIJ=LDGVEQEHGICpM3gfdORpEHGICgfR=UOgfAOADTIGICC*gfAaP@BdDC=CSNKMOR=TIC=GVAaP=TIGICGVAUOTIJVNQ3GVEHCu\OE@ZU]NHCSPSRSNQOdOPSNHgfAhUOG7JIgfAaP*@fJVP=CKTVP*@BAaPFADgfRSbo@BEQNHC*TIGYM1@BRFR*@BMOM3gfRSP/@BdhADgfb(OR=G(UOG(JIgfAaP*@fJVP=CxU]dbeNQEQNHGVdhGVPE@R=TkwTVR=GVADJIGGVA¨@BbeMOEQNQPSdDUOGmU]dCSNQ|sA1@BEUOTVbZgWU]dOEHTTVP*@BAaPcadDGVEHJIgfADcWdDGsz% :@U]NHC=PSRSNQOdOPSNHgfAUOGUOTkwygfRSbo@BPSNHgfADC9UOGJIgfAWP*@fJVP=C>JIgfADCSNHUOTVR=TIGs\OCSdORE@(¦D|sdOR=G aDzQ®u[]\O@(E@7M1@BRSPSNHJVdOE@BRSNQP=TYU^q¢VPSR=GKdOAONwygfRSbZG5GVA]UOGIC=C=gfdDCUOGKE@7rf@BEHGVdOR

µ = 1z

/d]UOGIC=CSdDCmUOG²JIGVPSP=G²rf@BEHGVdORu\¯E@tUOTIJVR=gfNHC=C*@BADJIGiGICSPGkj-M3gfADGVAaPSNHGVEQEHGª@IrGIJhEHGM1@BR*@BbZVPSR=Gµz¡ :@twgfRSbZGhGkjO@fJVP=Gs\

Js^qGIC=PnYU]NQR=GFEHGIC¡rs@BEHGVdOR=CUOGICJIgWGkoJVNHGVAWP=CgfdmMOdONHC=C*@BADJIGIC¯NQAaP=GVRSrGVA1@BAaP=CUD@BADCE_^qGkj]MlgfADGVAWPSNHGVEQEHGKUOTIJVR=gfNHC*C*@BAaP=GKC=gfAaPU]NTVR=GVAaP=C9C=GVEHgfAtEHGIC9@BMOMOR=gWJ*LDGICGk3GIJVPSdDTIGICuz%gfbebZGoU]NHC=JVdOP=ToMOR=TIJITIUOGVbebZGVAaPu\NQE@BMOM1@BR*@µºHPUOGZP=gfdOP=GICYwy@wIgfADCcWdDG>E@9UOTVM3GVADUD@BADJIGxUOGFE@(PSR*@BADCSNQPSNHgfAcad1@fU]R*@BPSNHcWdDG → [±fX9UD@BADCE_^`@BbeMOEQNQPSdDUOGxUOTVbZgWU]dOEHTIG5GIC=P¯PSR=ICwy@BNQOEHGVbZGVAWPC*GVADCSNQOEHG(nE@ewgfRSbZGYUOGYE@ZU]NHCSPSRSNQOdOPSNHgfAiUOG(UOTkwygfRSbo@BPSNHgfADCK@Bd]UOGVEnoUOG

µ = 1z0b»gfdORKEHGYJu@BEHJVdOE:MOR=TIC*GVAaP=T9CSdOR>E@

¦D|sdOR=G aDzQ®u[]\n(µ) ∝

1 µ < 1

e−1,4(µ−1)3/2µ ≥ 1

©aDzQ®`« :@UOTVMlGVADUD@BADJIG#UOGiE_^`@BbeMOEQNQPSdDUOGtU]d½CSNQ|sA1@BE5UOTVbZg-U]dOEHT#GVAwygfADJVPSNHgfAUOGiE_^`@BbeMOEQNQPSdDUOGtUOGiMlgfbeM1@B|GªGICSP

cWd1@fU]R*@BPSNHcWdDG9UOICKEHGICKPSR=IC/wy@BNQOEHGIC5@BbeMOEQNQPSdDUOGICYUOG7M3gfbeM1@B|Gp© −40 dB« ¥ dDC=cWd^`noE@ZPSR*@BADCSNQPSNHgfA·©~NQADU]NHcWdDTIG9M1@BR

: 5] j `f 8

−40 −30 −20 −10 0 10−100

−80

−60

−40

−20

0

20

40

n

(µ)

OP= ')J^ ' "!J 2 !$ OP= ')JU ' " 6JP J !$

+A,.021,3H4H02,/7"8:96;=<%780>"<%0.

"

+.,."

SUT2VEW !X2$/!Y &ê tj b_ g` "f j 5] e j ] _ Ec 5_ j($ fe tuj 5_ 5_ t 6e t f!"5_ t k_ N` a 5_5]dc egf [` 5_ f% c Jel_ _ 5_`` b`5_ j fle 5_ _ε0

00 dB 2

E@5¿1IJ*LDG5CSdORE@5¦D|sdOR=G aDzQ®u[a«nYdOADGxUOTVM3GVADUD@BADJIGxGVAMOdONHC*C*@BADJIGK[±fX@BdOP=gfdORUOG0 dB

U^qGkj]JVNQP*@BPSNHgfAzO¹ E3GICSPONHGVAGVAaP=GVADU]dcadDGoEHGw_@BNQP7U^qGkjOJVNQP=GVR7EHGobeNQEQNHGVd|sR*@BAadOE@BNQR=GnE_^`@BbeMOEQNQPSdDUOG

0 dBgfdndOADG@BbeMOEQNQPSdDUOGpCSdOMlTVRSNHGVdOR=G

R=GVbZGVPYGVA#Ju@BdDC=GEHG9bZgWUOVEHGdOPSNQEQNHC=T9NHJVN_zAtGkGVPu\nJIGICY@BbeMOEQNQPSdDUOGICu\E@CSPSRSdDJVPSdOR=GeU]d#beNQEQNHGVd#|sR*@BAadOE@BNQR=Gs\EHGICJ=L1@µºHADGIC>UOGKwygfR=JIGs\]C=gfAaP>C=dDC=JIGVMOPSNQOEHGIC>U^q¢VPSR=G5bZg-U]N¦1TIGICM1@BRE_^`@fJVPSNHgfA²UOGKE_^qgfADUOG(@fJIgfdDCSPSNHcWdDGs\OGVP>UD@BADCJIG5Ju@fCu\-E@U]NHCSPSRSNQOdOPSNHgfAeUOGJIgfAaP*@fJVP=C

n(µ)GIC=P%UOTVM3GVADUD@BAaP=GUOGE_^qGkjOJVNQP*@BPSNHgfAZ@fJIgfdDCSPSNHcadDGszs :GIC»R=TIC=dOEQP*@BP=CcadON]JIgfRSR=GICSM3gfADUOGVAaP

@Bd]j@BbeMOEQNQPSdDUOGIC/U^qGkj]JVNQP*@BPSNHgfAiCSdOM3TVRSNHGVdOR=GIC/n0 dB

C=gfAWPFUOgfADJYneJIgfADCSNHUOTVR=GVR/@urGIJYMORSdDUOGVADJIGszAeJIgfbeM1@BR*@BAaP¡JIGVPSP=GC=NQb(dOE@BPSNHgfA²© ¦D|sdOR=G aDzQ®u[a«»@IrGIJEHGICR=TICSdOEQP*@BP=CGkj]MlTVRSNQbZGVAWP*@Bd]j©cWdON-C*gfAaP%PSR*@fJITICCSdOREHGIC

¦D|sdOR=GIC aDzQ®uÀGVP aDzQ®s®xM1@BR>GkjOGVbeMOEHG«k\]NQE:@BMOM1@BR*@µºHPFcadDGKE@9PSR*@BADCSNQPSNHgfAhcad1@fU]R*@BPSNHcWdDG → [±fXUD@BADCE@UOTVMlGVADUD@BADJIGGVAi@BbeMOEQNQPSdDUOGUOG7MlgfbeM1@B|G9U]diC=NQ|sA1@BEUOTVbZg-U]dOEHTs\1PSR*@BADC=NQPSNHgfA#@fC*C=gWJVNHTIG9@BdiJVE@BMOM3GVbZGVAaP5UOGICKJIgfAaP*@fJVP=Cu\@oEQNHGVd15 − 20 dB

@urf@BAWPE_^`@BbeMOEQNQPSdDUOGªUOGhMlgfbeM1@B|Gibo@µj]NQbo@BEHGªGkj-M3TVRSNQbZGVAaP*@BEHG0 dBz GhADgfb(OR=G#UOGiJIgfAaP*@fJVP=C

JVE@BMOM1@BAaP=C©JIgfAWP*@fJVP=C(w_@BNQOEHGIC«YA^qGICSPUOgfADJoM1@fC9CSd]oC*@BAWPUD@BADC7E@hU]NHCSPSRSNQOdOPSNHgfAn(µ) = cste

EHgfR=C*cadDGµ < 1MlgfdORw_@BNQR=GY@BMOM1@BR*@µºHPSR=GYE@ZUOTVMlGVADUD@BADJIG7GVAMOdONHC=C*@BADJIG([±fXONHGVA²@urf@BAWPFdOADG(@BbeMOEQNQPSdDUOG7U^qGkj]JVNQP*@BPSNHgfAhUOG0 dBz

G7w_@BNQPKcWdDG(E_^`@BbeMOEQNQPSdDUOGeUOTVbZg-U]dOEHTIGeJVR=gfNHC=C=G9UOG7wy@wIgfA#cWd1@fU]R*@BPSNHcadDG9bo@BEQ|sR=T9EHGJVE@BMOMlGVbZGVAWPYUOG7JIGVRSP*@BNQADCJIgfAaP*@fJVP=Ctwy@BNQOEHGIC·©JIGVd]jM3gfdORtcadON

µ < εa/ε0«hMOR=gur-NHGVAaPUOGE_^`@BdO|sbZGVAWP*@BPSNHgfA MOR=gf|sR=GIC*CSNQrGÜOGICtJIgfAaP*@fJVP=C

JVE@BMOM1@BAaP=C#@IrGIJE_^`@BbeMOEQNQPSdDUOGUOGMlgfbeM1@B|Gεa/ε0

zF :GCSNQ|sA1@BE9UOTVbZgWU]dOEHTR=TICSdOEQP*@BAaPu\/bZ¢VbZGtMOR=grGVA1@BAaP#UOGJIgfAaP*@fJVP=C>JVE@BMOM1@BAWP=Cu\OJVR=gBºHP>UOgfADJxMOEQdDCr-NQP=GKcWdDG

ε3/2

aGVPR=GICSP=GKMOR*@BPSNHcadDGVbZGVAWPFcWd1@fU]R*@BPSNHcadDG ¥ dDC=cWd^`n

εa/ε0 = 1©0 dB«kz% M1@BRSPSNQRUOG/JIGVPSP=G/@BbeMOEQNQPSdDUOGKUOG>M3gfbeM1@B|Gs\aGVAWrWNQR=gfAE@xbZgfNQPSNHTKUOGIC¯JIgfAaP*@fJVP=CMOR=TIC=GVAWP=CUD@BADCEHGbeNQEQNHGVd

@BdR=GVMlgsCFJVE@BMOMlGYGVPEHG(CSNQ|sA1@BErs@BRSNHG5JIgfbebZGε

3/2az

F¦DAZUOG>JIgfbeMOR=GVADU]R=G>E_^ NQA]¿DdDGVADJIG/UOGICw_@BNQOEHGIC¡JIgfAaP*@fJVP=CCSdORE_^`@BbeMOEQNQPSdDUOGKU]dZCSNQ|sA1@BE1UOTVbZg-U]dOEHTs\dOADGM3gfRSPSNHgfAUOG7JIGICxJIgfAWP*@fJVP=CYGICSP5@ ¥ gfdOP=TIG7wygfRSbZGVEQEHGVbZGVAaP(nmE@oU]NHC=PSRSNQOdOPSNHgfA

n(µ)zlA#Gk3GVPu\lM3gfdORKgfOP=GVAONQRxdOADG9PSR*@BADCSNQPSNHgfA

2 → 3/2M3gfdOR(UOGoMOEQdDCYw_@BNQOEHGIC7@BbeMOEQNQPSdDUOGICUOGZM3gfbeM1@B|Gs\»E@U]NHCSPSRSNQOdOPSNHgfA

n(µ)UOgfNQP7JIgfbeM3gfRSP=GVR7dOADGoMOEQdDC

|sR*@BADUOGMOR=gfMlgfRSPSNHgfAÜOGJIgfAaP*@fJVP=Cw_@BNQOEHGICu\GVP9CSd]oC@BbebZGVAaPeEHg-Ju@BEQNHC=TIGorGVR=C9EHGIC7w_@BNQOEHGIC7rs@BEHGVdOR=CUOGµz¾WdOR9E@

¦D|sdOR=G aDzQ®PaD\WdOADG>M3gfRSPSNHgfApUOG>w_@BNQOEHGICJIgfAWP*@fJVP=C>@YTVP=Tx@ ¥ gfdOP=TIGKGVAmUOGIC*C=gfdDCUOGµ = 0.1

zW :@7U]NHCSPSRSNQOdOPSNHgfApdOPSNQEQNHC=TIGMlgfdOREHG(Ju@BEHJVdOE»@9E@7wgfRSbZGo§

n(µ) ∝

10 µ < 0.11 0.1 < µ < 1

e−1,4(µ−1)3/2µ ≥ 1

©aDzQ®¬s«

!"#%$&('*) &('+&,-)./0!&!$('211 .%

−40 −30 −20 −10 0 10−100

−80

−60

−40

−20

0

20

40

n

(µ)

O/P= %' U %' ! !$ O2P= %')JE = ' 6JP !$

+-,."

"

+-,/.021,354602,/7"8F96;=<%780>"<%0.

SUTVEW X !Y &ê tuj 5_ l` "uf j 5]Je j ] _ Ec 5_ j$ f6e tj b_ 5_ t e t f!"b_ t _ ` a b_ b]dc egf [`5_ f% 3 t %]`_ _ k_ t ` tHtuj ]Jel_ f %_ 5_ f d` cfa j _` t f 5_µ = 0.1

?5@BADC9JIGoJu@fCu\»EHGZADgfb7OR=GpUOGoJIgfAaP*@fJVP=CeJIgfbeMORSNHCGVAWPSR=G0 < µ < 0.1

GICSP9C=GVADCSNQOEHGVbZGVAWPTV|a@BE@BdADgfb7OR=GpUOGJIgfAWP*@fJVPmJIgfbeMORSNHCGVAaPSR=G

0.1 < µ < 1GVPoGVAaPSR=G

1 < µ < +∞ zGIJVN/NQADU]NHcadDGhcadDGM3gfdORodOADGi@BbeMOEQNQPSdDUOGU^qGkjOJVNQP*@BPSNHgfAtUOG eεaε0

' 0.1GVAWrWNQR=gfA#dOA#PSNHGVR=C5UOGIC5JIgfAWP*@fJVP=CYJVE@BMOMlGs\GVP5cWdDG9MlgfdOR eεa

ε0' 0.5 ' −14 dB

\3Js^qGICSPE@mbZgfNQPSNHTUOGIC5JIgfAaP*@fJVP=C(cadONJVE@BMOM3GVAaPuz ¡^`@BbeMOEQNQPSdDUOGoUOG7M3gfbeM1@B|G eεa

ε0

noE@fcWdDGVEQEHGeC=G9MOR=g-U]dONQPKE@mPSR*@BADCSNQPSNHgfAX→ [±fXpCSdOR5E@m¦D|sdOR=G*aDzQ®PaC=GeCSNQPSdDGeGVAar-NQR=gfAn −12 dB

\3JIGecadON¡JIgfRSR=GIC=MlgfADUtndOADGMOR=gfM3gfRSPSNHgfAUOGeJIgfAaP*@fJVP=CJVE@BMOM1@BAWP=C¡C=GCSNQPSd1@BAWP¯@BdOP=gfdOR¡UOGE@KbZgfNQPSNHTU]dADgfb7OR=GP=gfP*@BEDUOGJIgfAaP*@fJVP=Cuza¾WNOE@/M3gfRSPSNHgfAZUOGJIgfAWP*@fJVP=C¡JIgfbeMORSNHC=GGVAWPSR=G

0 < µ < 0.1GICSP>@BdO|sbZGVAaP=TIGs\1E@9PSR*@BADCSNQPSNHgfA²X → [±fX9rf@e@urgfNQREQNHGVdMlgfdOR>UOGIC/@BbeMOEQNQPSdDUOGIC>UOGxMlgfbeM1@B|GMOEQdDC5wy@BNQOEHGIC7cadDG −12 dB

z:K^qGICSPYEHGZJu@fC7CSdORYE@¦D|sdOR=G aDzQ®ugfvE@pPSR*@BADCSNQPSNHgfA2 → 3/2

@EQNHGVdUOIC −17 dBU^`@BbeMOEQNQPSdDUOGUOGM3gfbeM1@B|Gsz E_^ NQAWrGVR=C=Gs\sCSN]E@/MlgfRSPSNHgfAeUOGJIgfAaP*@fJVP=C¯JIgfbeMORSNHC=GGVAaPSR=G0 < µ < 0.1

GICSPU]NQbeNQAadDTIGs\JIgfbebZGxJs^qGICSPEHGKJu@fCUD@BADCE@7CSNQb(dOE@BPSNHgfAPSR*@fJITIGxCSdORE@(¦D|sdOR=G aDzQ®Ò\WE@(PSR*@BADCSNQPSNHgfA@YEQNHGVdpM3gfdORUOGIC@BbeMOEQNQPSdDUOGICUOGxM3gfbeM1@B|GYCSdOM3TVRSNHGVdOR=GIC7©~P °WMONHcWdDGVbZGVAaP −5 dB

«kz GIC»UOTVP*@BNQEHC»¦DADCUOG¡E@/U]NHCSPSRSNQOdOPSNHgfA

n(µ)MlgfdOR

µ 1C=gfAWPU]NoJVNQEHGICn>R=GIJIgfADCSPSRSdONQR=GM1@BR=JIGcWdDG¡EHGICJIgfAaP*@fJVP=C

w_@BNQOEHGICFC*GYbo@BAONwyGICSP=GVAaP5nePSR*@urGVR=CFdOADG7NQAaP=TV|sR*@BEHG9UOGYE@oU]NHCSPSRSNQOdOPSNHgfAiUOG7JIgfAaP*@fJVPe©~rgfNQR/E_^qTIcWd1@BPSNHgfA¨©aDzQ®Pa-«S«kz1 GMOR=gfOEHVbZGmNQAarGVR=C*GmcWdON¯JIgfADCSNHCSP=GnPSR=gfdOrGVR9E@pwygfRSbZGoUOGoE@U]NHCSPSRSNQOdOPSNHgfAUOGoJIgfAWP*@fJVP=Cp©~ADgfP*@BbebZGVAWPrGVR=C(EHGICw_@BNQOEHGICKJIgfAaP*@fJVP=C«KneM1@BRSPSNQRxUOG7E@ZU]°-A1@BbeNHcadDG9UOG7JVR=gfNHC*C*@BADJIG7U]diCSNQ|sA1@BE%UOTVbZgWU]dOEHT9GICSP/UOgfADJ@oMORSNHgfRSNU]NoJVNQEHGn9R=Tu@BEQNHC*GVRFUOGKwy@wIgfA²MOR=TIJVNHC=G(@IrGIJYEHGICR=TICSdOEQP*@BP=CFGkj]M3TVRSNQbZGVAaP*@Bd]j²gfOP=GVAadDCFNHJVN_z?9^`@BdOPSR=GICewgfRSbZGICMOEQdDCR=Tu@BEQNHCSP=GICoUOGmE@ªU]NHC=PSRSNQOdOPSNHgfA·UOGpJIgfAaP*@fJVP=CZM3GVdOrGVAaPZ¢VPSR=GpNQbeMOEHTVbZGVAWP=TIGICoUD@BADCZJIG

Ju@BEHJVdOE_z]AM1@BRSPSNHJVdOEQNHGVRu\OJIGVRSP*@BNQADGICU]NHCSPSRSNQOdOPSNHgfADCADG/wygfAaP@BMOM1@BR*@µºHPSR=GY@BdDJVdOADGKTIJ*LDGVEQEHGxM1@BRSPSNHJVdOEQNHVR=GY@fC=C=g-JVNHTIGK@BdbeNQEQNHGVd\lJIgfAaPSR*@BNQR=GVbZGVAWP7noE@ewygfRSbZG7UOG9U]NHCSPSRSNQOdOPSNHgfA#UOT ¥ nZdOPSNQEQNHC=TIG7NHJVN_\lgfvhdOADG7MlgfRSPSNHgfAªUOG9JIgfAaP*@fJVP=CKw_@BNQOEHGICx@TVP=T@ ¥ gfdOP=TIG9GVA#UOGIC=C*gfdDCKUOG

µ = 0.1z:K^qGICSP/EHGJu@fCxUOGIC/wgfADJVPSNHgfADCxMOdONHC*C*@BADJIGs\3UOG7E@ZwygfRSbZG

n(µ) = µκ ©~MlgfdORµ < 1

«gfvκM3GVdOPMOR=GVADU]R=G5UOGICrf@BEHGVdOR=CM3gsCSNQPSNQrGICY© wygfADJVPSNHgfAJVR=gfNHC=C*@BAaP=G«gfdpUOGICrs@BEHGVdOR=CADTV|a@BPSNQrGIC(© wygfADJVPSNHgfA

UOTIJVR=gfNHC*C*@BAaP=Gs\U]NQrGVRS|GVAWP=GeGVAµ = 0

«kzgfbebZGEHGJu@fCκ = 0

@mTVP=T7PSR*@fJITCSdORxE@Z¦D|sdOR=G aDzQ®u[O\3GVPKADG9MOR=TIC=GVAaP=GM1@fCxdOADGZEHgWJu@BEQNHC*@BPSNHgfACSd]oC*@BAaP=GZrGVR=C5EHGICxw_@BNQOEHGICYJIgfAaP*@fJVP=C(MlgfdOR5M3GVRSbZGVPSPSR=GedOADGePSR*@BADCSNQPSNHgfAX → [±fX²@Irs@BAaPdOADG5@BbeMOEQNQPSdDUOGYUOGKM3gfbeM1@B|G5UOG

0 dB\-C=GVdOEHGICEHGICrs@BEHGVdOR=CADTV|a@BPSNQrGIC>UOG

κ\]cWdON3bZVADGVAaP/n7dOADGxU]NQrGVRS|GVADJIGYUOG

n(µ)EHgfR=C=cWdDG

µ → 0\DC=gfAaPFR=Tu@BEQNHCSP=GICM3gfdORF¢VPSR=GYGVA²@fJIJIgfR=Uh@urGIJ5EHGIC>Gkj]MlTVRSNHGVADJIGICFR=Tu@BEQNHC=TIGICuzQz

: 5] j `f (/

−40 −30 −20 −10 0 10−100

−80

−60

−40

−20

0

20

40

n

(µ)

OP= ')J^ ' "!J 2 !$ OP= ')JU ' " 6JP J !$"

+-,$"

+A,.012,/3H460,78:96;=<%780>"<%0.

SUT2VEW !X2 Y &ê tj b_ g` "f j 5] e j ] _ Ec 5_ j($ fe tuj 5_ 5_ t 6e t f!"5_ t k_ N` a 5_ 5]dc Je:f% ` b_ f% 3 t %]` _ _ _ t ` t5tuj ] eg_ f?_ 5_ f%d` cfa j _` t f b_µ = 0.1

−40 −30 −20 −10 0 10−100

−80

−60

−40

−20

0

20

40

n

(µ)

OP= ')J^ ' "!J 2 !$ OP= ')JU ' " 6JP J !$

+-,."

"

+-,.012,/3H460,78F9H; <7"80><?0.

SUT2VEW !X2 !Y &ê tj b_ g` "f j 5] e j ] _ Ec 5_ j($ fe tuj 5_ 5_ t 6e t f!"5_ t k_ N` a 5_ 5]dc Je:f% ` b_ f% 3 t %]` _ _ _ t ` t5tuj ] eg_ f?_ 5_ f%d` cfa j _` t f b_µ = 0.1

!"#%$&('*) &('+&,-)./0!&!$('211 .%

−40 −30 −20 −10 0 10−100

−80

−60

−40

−20

0

20

40

n

(µ)

O/P= %' U %' ! !$ O2P= %')JE = ' 6JP !$

+-,$"

"

+A,.012,/3H460,78:96;=<%780>"<%0.

SUTVEW !X2 Y Z N` a 5_ 5] c Jegf [` 5_ f%d` 5_ t _ t `` f_ n(µ) ∝ µκ t µ < 1fq_

κ = −0.5_ ufe 53 _ _ fe tuj 5_ ` "uf j 5] e j ] 3 j _J` -f`` ]_

¾WdORE@9¦D|sdOR=G aDzQ®¬]\]E@U]NHCSPSRSNQOdOPSNHgfAiCSdONQrs@BAaP=G(@eTVP=TxdOPSNQEQNHC*TIGZ§

n(µ) ∝

µ−0.5 µ < 1

e−1,4(µ−1)1/2µ ≥ 1

©aDzQ®u´«

gfbebZG#E@wgfR=JIGGICSPR=GVEQNHTIGnE@ÜOTkwygfRSbo@BPSNHgfA U]d JIgfAWP*@fJVP²M1@BRE@R=GVE@BPSNHgfA UOG£/GVRSPedf ∝ µ3/2 \E@U]NHC=PSRSNQOdOPSNHgfApUOGFwygfR=JIGICUOGKJIgfAaP*@fJVP=CJIgfRSR=GICSM3gfADUD@BAaPFn7JIGVPSP=GKU]NHCSPSRSNQOdOPSNHgfAUOG/UOTkwygfRSbo@BPSNHgfAUOG/JIgfAWP*@fJVP=CGICSPUOG

E@ZwygfRSbZGfκ−1/3 MlgfdOR µ < 1

zb»gfdORκ = −1/2

\3E@mU]NHCSPSRSNQOdOPSNHgfAtUOG(wgfR=JIGGICSPxUOgfADJGVAf−1/3−1/2 = f−5/6 z¡EQEHG²R=GICSP=GiNQAWP=TV|sR*@BOEHGs\JIGicWdONxGICSPmADTIJIGIC*C*@BNQR=GiMlgfdOR@fC=CSdOR=GVRdOAADgfb(OR=G#UOGiJIgfAaP*@fJVPp¦DAONxUD@BADCpEHG²beNQEQNHGVdz

¯^`@BbeMOEQNQPSdDUOG7UOG5M3gfbeM1@B|G7neE@fcadDGVEQEHGYC*G5MOR=gWU]dONQP/E@PSR*@BADCSNQPSNHgfAiGICSPFGVAar-NQR=gfAhUOG −12 dB\OJIG(cWdONJIgfRSR=GICSM3gfADU

@Bd]jGkj]MlTVRSNHGVADJIGICbZGVADTIGICFADgfP*@BbebZGVAaPKMlgfdOR8 kPa

GVP24 kPa

© ¦D|sdOR=GIC aDzQ®uÀeGVP aDzQ®s®µ«kz?x@BADCmEHGiJu@fCmUOT ¥ ntTVPSdDU]NHT

n(µ) = csteM3gfdOR

µ < 1©κ = 0

«k\E@U]NHCSPSRSNQOdOPSNHgfA½UOGwygfR=JIGi@tE@#wygfRSbZGn(f) ∝ f−1/3 z3GVPSP=G(U]NHCSPSRSNQOdOPSNHgfA\3U]NQrGVRS|GVAWP=G7EHgfR=C*cadDG f → 0

ADGYMOR=TIC=GVAWP=G7JIGVM3GVADUD@BAaPKM1@fCKUOGYEHg-Ju@BEQNHC*@BPSNHgfAC=d]oC*@BAaP=G9rGVR=CKEHGIC/wy@BNQOEHGIC5JIgfAWP*@fJVP=CxMlgfdOR5cWdDG(EHG9JVE@BMOMlGVbZGVAWP5NQADU]dONHC=G9dOADG9PSR*@BADCSNQPSNHgfAtX → [±fX@Irs@BAaP 0dBU^`@BbeMOEQNQPSdDUOG7U^qgfADUOGYUOG5M3gfbeM1@B|Go© ¦D|sdOR=G aDzQ®u[a«kz

E_^ NQAarGVR=C=Gs\3CSN»E@oU]NHCSPSRSNQOdOPSNHgfA#UOG9JIgfAaP*@fJVP=C5GICSP/PSR=gfMhONHGVAªEHg-Ju@BEQNHC=TIG7rGVR=Cµ = 0

\1E@oPSR*@BADCSNQPSNHgfAtX → [±fX@BMOM1@BR*@µºHP5M3gfdORYUOGICY@BbeMOEQNQPSdDUOGICYUOGM3gfbeM1@B|GPSR=ICKwy@BNQOEHGICz:K^qGICSPxEHGeJu@fCxEHgfR=C=cadDGκ = −1

M1@BRYGkjOGVbeMOEHGs\CSdORE@¦D|sdOR=G aDzQ®uÓ\gfvE@²PSR*@BADCSNQPSNHgfA·@BMOM1@BR*@µºHPrGVR=C −35 dB

z?xGoP=gfdOP=GIC9w_@wIgfADCu\JIGVPSP=GowygfRSbZGUOGU]NHC=PSRSNQOdOPSNHgfAGVAWPSR*@µºHADGdOADGU]NQrGVRS|GVADJIGiU]d ADgfb7OR=GhUOGJIgfAWP*@fJVP=CmUD@BADCoEHGbeNQEQNHGVd·MOdONHC*cadDGE@tU]NHCSPSRSNQOdOPSNHgfAUOGpwygfR=JIGICoUOGJIgfAWP*@fJVP=C

n(f)GICSPMOR=gfM3gfRSPSNHgfAOADGVEQEHG7n

f−4/3 z¼FADGU]NHCSPSRSNQOdOPSNHgfAtMOEQdDCYR=Tu@BEQNHCSP=G©κ = −1/3

«k\cWdONUOGMOEQdDCxwgfdORSAONQPYUOGICYR=TICSdOEQP*@BP=CYGVAt@fJIJIgfR=UtcWd1@BAaPSNQP*@BPSNw@urGIJEHGICoGkj]M3TVRSNHGVADJIGICZbZGVADTIGICu\GICSPedOPSNQEQNHC=TIGM3gfdORZEHGJu@BEHJVdOE/GVPZPSR*@fJITIG²CSdORoE@i¦D|sdOR=GRaDzQ®uOz¡ »@ªPSR*@BADCSNQPSNHgfA X→ [±fXGICSP@BEHgfR=CCSNQPSdDTIGn −7 dB

U^`@BbeMOEQNQPSdDUOGUOGmM3gfbeM1@B|Gsz%K^qGICSP9E_^`@BbeMOEQNQPSdDUOGpUOGZM3gfbeM1@B|Gbo@µj-NQb7dOb@fC*C=gWJVNHTIGn/dOADGPSR*@BADCSNQPSNHgfAoX → [±fX]\sgfDC=GVRSrTIG>MlgfdOR%dOADGJIgfAaPSR*@BNQAWP=G/C=P*@BPSNHcadDGTVEHGVrTIG>UOG 64 kPa

© ¦D|sdOR=G aDzQ®uXa«kzGVE@7CSNQ|sAON¦1GKcadDG/E@7U]NHCSPSRSNQOdOPSNHgfAUOG/JIgfAWP*@fJVPMlGVdOPU]NoJVNQEHGVbZGVAaPu\DUD@BADCADgsCJIgfADU]NQPSNHgfADCGkj-M3TVRSNQbZGVAaP*@BEHGICu\O¢VPSR=GbZgfNQADCFONHGVAEHg-Ju@BEQNHC=TIG5rGVR=C>EHGIC>wy@BNQOEHGIC/JIgfAaP*@fJVP=Cuz< :@MOdONHC=C@BADJIG

κ = −1/3R=GVMOR=TIC=GVAWP=GYdOADGYU]NHCSPSRSNQOdOPSNHgfA

n(f)MOR=gfM3gfRSPSNHgfAOADGVEQEHG7nf−2/3 z?x@BADCE@9C=GIJVPSNHgfACSdONQrs@BAaP=Gs\]EHGICR=TIC=dOEQP*@BP=CGkj-M3TVRSNQbZGVAaP*@Bd]jC=gfAaP>U]NHC*JVdOP=TICU]dM3gfNQAaP>UOGKr-dDGxUOGICTVrGVAWPSdDGVEQEHGIC

: 5] j `f

−40 −30 −20 −10 0 10−100

−80

−60

−40

−20

0

20

40

n

(µ)

OP= ')J^ ' "!J 2 !$ OP= ')JU ' " 6JP J !$

"

+.,."

+-,/.021,354602,/7"8:96;I<%780>"<%0.

SUT2VEW !X2 Y Z ` a b_ 5]dc egf [` b_ f%d` 5_ t _ t N`` f _ n(µ) ∝ µκ t µ < 1fq_

κ = −1_ fe 43 _F_ fe tuj 5_ ` "uf j 5]Je j ] 3 j _` f`` ]%_

−40 −30 −20 −10 0 10−100

−80

−60

−40

−20

0

20

40

n

(µ)

OP= ')J^ ' "!J 2 !$ OP= ')JU ' " 6JP J !$"

+-,$"

+-,/.021,/3H4602,/7"8:96;I<7"80><?0.

SUT2VEW !X2.Y Z ` a b_ 5]dc egf [` b_ f%d` 5_ t _ t N`` f _ n(µ) ∝ µκ t µ < 1fq_

κ = −0.3_ fe 43 _F_ fe tuj 5_ ` "uf j 5]Je j ] 3 j _` f`` ]%_

!"#%$&('*) &('+&,-)./0!&!$('211 .%

@BMOMOEQNHJu@BPSNHgfADC\DUOGIC>MOR=gfOEHVbZGICFR=GVADJIgfAaPSR=TICKgfdUOGIC/@BbZTVEQNHgfR*@BPSNHgfADCxne@BMOMlgfRSP=GVRuz

Z ` b``

&('_& ¸;186 1;*2>, :@·bZTVPSLDgWUOGUOGMOR=TVM1@BR*@BPSNHgfA¸U]d beNQEQNHGVd|sR*@BAWdOE@BNQR=GADgfA]JIgfADC=gfEQNHUOT ¥ gfdDGdOA R[ZfEHGNQbeM3gfRSP*@BAaP#CSdORiE@

R=TVMlTVP*@BONQEQNQP=TpUOGIC9R=TICSdOEQP*@BP=CeGkj]MlTVRSNQbZGVAWP*@Bd]j3z%¼/AUOGIC9NQADU]NHJu@BP=GVdOR=CZUOGmE@iCSP*@BONQEQNQP=TpU]dbeNQEQNHGVd|sR*@BAWdOE@BNQR=GGICSPE@MOR=GIC=CSNHgfAC=P*@BPSNHcadDGZJu@BMOP=TIGZM1@BRYE_^ NQAaP=GVRSbZTIU]N@BNQR=GpU]dJu@BMOP=GVdOR(UOGewgfR=JIGsz »@UOTVRSNQrGoUOGE@pMOR=GIC*CSNHgfACSP*@BPSNHcWdDGbZGICSdOR=TIG GVA¶wygfADJVPSNHgfA U]d P=GVbeMDCCSdONQPtdOADG EHgfNZEHgf|a@BRSNQPSLObeNHcadDGsz9 gfR=C=cWdDGË@MOR=GIC*CSNHgfA CSP*@BPSNHcWdDGÄDG¨rs@BRSNHGMOR*@BPSNHcadDGVbZGVAWPZMOEQdDCGVAwgfADJVPSNHgfA U]dP=GVbeMDC©y@BdlgfdOPeUOGMOEQdDCSNHGVdOR=CLDGVdOR=GICP °WMONHcWdDGVbZGVAaP«k\¯EHGobeNQEQNHGVd¨M3GVdOP¢VPSR=G²JIgfADCSNHUOTVR=TiJIgfbebZGiCSP*@BONQEQNHC=Tsz \i@BEQ|sR=TiJIGVE@]\¯dOADG²MlGVPSNQP=GhM3GVRSPSdORS1@BPSNHgfA Gkj]P=TVRSNHGVdOR=Gs\JIgfbebZG²dOANQbeM1@fJVPbZTIJu@BAONHcadDG9CSdOR/EHGYR=TIJVNQMONHGVAaPxM1@BRKGkj]GVbeMOEHGs\<R=GVE@BADJIG(EHG(beNQEQNHGVdiUD@BADC/dOADG7MOL1@fC=G(UOG(R=GVE@µjO@BPSNHgfAiEHgf|a@BRSNQPSLObeNHcadDGrWNHC=NQOEHGC=dORE@KMOR=GIC*CSNHgfAZCSP*@BPSNHcWdDGJu@BMOP=TIGsz FMOR=IC%dOAeADgfb7OR=G>UOgfAOADT>U^ NQbeM1@fJVP=Cu\sJIGVd]j-JVNOA^qgfAaP¡MOEQdDCU^ NQA]¿DdDGVADJIGCSdOR9E_^qTVP*@BPU]dbeNQEQNHGVd\GVPJs^qGICSPniJIGmbZgfbZGVAaPZcWdDGoEHGICGkj]MlTVRSNHGVADJIGICMOR=TIC*GVAaP=TIGICeUD@BADCeJIGUOgWJVdObZGVAWPZgfAaPeTVP=TbZGVADTIGICuz ¡^ LONHCSP=gfNQR=GiU]dJ=L1@BRS|GVbZGVAaPU]d beNQEQNHGVd½@#dOADGhNQA]¿DdDGVADJIGiC=dORmC*@CSPSRSdDJVPSdOR=GyxagsC*ÀsÀO\ DKgFs´D\xAONsB\JIG

cadONM3GVdOPYC=GPSR*@fU]dONQR=GeM1@BR7UOGICYJIgfAaPSR*@BNQAWP=GIC7R=TIC=NHU]dDGVEQEHGIC5LDgfRSN dugfAaP*@BEHGIC9NQbeM3gfRSP*@BAaP=GIC(UD@BADCYEHGZJu@fCYU^ dOADGw_@BNQOEHGJIgfAaPSR*@BNQAWP=GFCSP*@BPSNHcWdDGMOR=TIJITIUOTIGM1@BRdOADGwygfRSP=GJIgfAaPSR*@BNQAWP=G/C=P*@BPSNHcadDGrGVRSPSNHJu@BEHGszW?x@BADC¯JIGP°-MlGUOGJIgfA]¦D|sdOR*@BPSNHgfADCu\gfvtdOADGoU]NQbeNQAWdOPSNHgfAUOGoMOR=GIC=CSNHgfAC=P*@BPSNHcadDG@BMOMOEQNHcadDTIGmGICSPYR=Tu@BEQNHC=TIG ¥ dDCSP=Gm@Irs@BAaP9E_^qGkj-M3TVRSNHGVADJIGoUOGmU]°WA1@BbeNHcWdDGGVAp@BbeMOEQNQPSdDUOG5U]dpCSNQ|sA1@BEUOTVbZg-U]dOEHTs\WEHGICR=TICSdOEQP*@BP=CgfOP=GVAWdDCC=gfAaPU]NTVR=GVAaP=CUOGxJIGVd]joMOR=TIC=GVAWP=TICuzb@BRGkjOGVbeMOEHGs\E@¨PSR*@BADCSNQPSNHgfA

2 → 3/2MlGVdOPª@IrgfNQRªEQNHGVdU^`@BlgfR=U MlgfdORiE@¨M3gfE@BRSNHC*@BPSNHgfArGVRSPSNHJu@BEHGU^qgfADUOGUOGtM3gfbeM1@B|G

UOGªJVNHC*@BNQEQEHGVbZGVAaP²GVPGVADCSdONQP=GªM3gfdORpE@MlgfE@BRSNHC*@BPSNHgfA½LDgfRSN dugfAWP*@BEHGszKgfbebZGt@BdDJVdOA @BdOPSR=GªbZg°GVAUOGªbZGICSdOR=GGkj-M3TVRSNQbZGVAaP*@BE9UOGICJIgfAaPSR*@BNQAWP=GICªUD@BADCEHG#beNQEQNHGVd|sR*@BAWdOE@BNQR=GA^qGIC=P²U]NHCSM3gfAONQOEHGs\NQEYA^`@M1@fChTVP=T#MlgsC*CSNQOEHG#UOGJIgfA]w~R=gfAWP=GVR%EHGIC»R=TIC=dOEQP*@BP=CgfOP=GVAWdDC¯@IrGIJdOADGCSPSRSdDJVPSdOR=GJIgfAOAadDGU]dbeNQEQNHGVd\fGVP%@BNQADCSN_\BUOGrf@BEQNHUOGVR%EHGIC%gfDC=GVRSrs@BPSNHgfADCGk3GIJVPSdDTIGICz»GVMlGVADUD@BAWPu\:E@R=gfOdDCSP=GIC=C=GZUOGoJIGVPSP=GZP=GIJ=LOAONHcWdDGZM3GVRSbZGVP(UOGoMlGVADC=GVRYcWdDGZUD@BADC7JIGZJu@fCYMOR=TIJVNHCu\EHGICJIgfAaPSR*@BNQAWP=GIC9C=P*@BPSNHcadDGIC5R=TIC=NHU]dDGVEQEHGICm©GVP(GVAtM1@BRSPSNHJVdOEQNHGVR9EHGICYJ*L1@µºHADGIC7UOGwygfR=JIGeLDgfRSN dugfAWP*@BEHGIC«YC*gfAaP(CSdDC*JIGVMOPSNQOEHGICU^q¢VPSR=GªEHGICCSdOMOM3gfRSP=CUOGªE@bZTVbZgfNQR=G#U]d½beNQEQNHGVd cad1@BAWP²nE_^ LONHCSP=gfNQR=GtUOGªC=gfAJ=L1@BRS|GVbZGVAWPhGVPUOgfNQrGVAWP²¢VPSR=GMOR=TIC=GVAaP=GIC/UD@BADCEHG5beNQEQNHGVdz ¡^ NQA]¿DdDGVADJIGUOGE@FMOR=TVM1@BR*@BPSNHgfAZU]d9beNQEQNHGVdeCSdORE@>wgfRSbZGUOGE@/U]NHC=PSRSNQOdOPSNHgfAZUOGJIgfAWP*@fJVP=CA^qGICSPM1@fC»PSRSNQr-N@BEHGn

UOTIJVRSNQR=GGVP»A^`@>M1@fCTVP=T¯TVPSdDU]NHTGVA7UOTVP*@BNQEOnADgfPSR=GJIgfAOA1@BNHC=C*@BADJIGszf¹ EaGICSP%nFCSdOMOM3gsC=GVR:cWd^`nFMOR=GIC=CSNHgfA7C=P*@BPSNHcadDGTV|a@BEHGs\dOAGVbeMONQEHGVbZGVAaP|sR*@BAWdOE@BNQR=GJIgfbeM1@fJVPCSP*@BONQEQNHC=TmJIgfbeMlgfRSP=GoMOEQdDC7UOGmJIgfAaP*@fJVP=C9cad^ dOAGVbeMONQEHGVbZGVAaP|sR*@BAadOE@BNQR=GADgfA]JIgfbeM1@fJVPuz¡GVMlGVADUD@BAWPu\»NQEGICSP9U]NoJVNQEHGU^qGVAUOTIU]dONQR=GmE@MOR=gfM3gfRSPSNHgfAR=GVE@BPSNQrGpUOGmJIgfAaP*@fJVP=C9w_@BNQOEHGIC7GVPUOGJIgfAaP*@fJVP=C>wgfRSP=CM1@BR>GkjOGVbeMOEHGs\DgfdGVADJIgfR=G5EHG(JIgfbeMlgfRSP=GVbZGVAWP/UOGxE@eU]NHCSPSRSNQOdOPSNHgfA²rGVR=CEHGICw_@BNQOEHGIC>UOTkwgfRSbo@BPSNHgfADCUOG>JIgfAaP*@fJVP=Cz-GPSR*@urf@BNQE<ADTIJIGIC=CSNQP=GdOADG|sR*@BADUOGFRSNQ|sdDGVdORUD@BADC¯EHGMOR=gfP=gWJIgfEHG/Gkj-M3TVRSNQbZGVAaP*@BE3nxbZGVPSPSR=GFGVAmgWGVdOr-R=Gsz GMOR=gfOEHVbZGNQAWrGVR=C=GcadONJIgfADCSNHCSP=Gn UOTIU]dONQR=G¨n¨M1@BRSPSNQRªUOGE@·U]°WA1@BbeNHcWdDGUOGJVR=gfNHC=C*@BADJIGU]dCSNQ|sA1@BE

UOTVbZgWU]dOEHTs\9E@U]NHCSPSRSNQOdOPSNHgfA UOG JIgfAaP*@fJVP=C UD@BADCEHGbeNQEQNHGVd GICSPTVrGVAaPSdDGVEQEHGVbZGVAWPGVAar-NHC*@B|Gu@BOEHGsze :@ wygfRSbZGcad1@BEQNQP*@BPSNQrGpUOGmE@iU]NHCSPSRSNQOdOPSNHgfAMlGVdOP¢VPSR=GpUOTIU]dONQP=GUOGmE_^`@BbeMOEQNQPSdDUOGUOGoM3gfbeM1@B|GnE@hPSR*@BADCSNQPSNHgfA

2 → 3/2GVP#E@ rf@BEHGVdORbZgu°GVAOADG·UOGÜOTkwygfRSbo@BPSNHgfA C=P*@BPSNHcadDGÜD@BADC#EHG beNQEQNHGVd UOTIU]dONQP=G UOGE_^`@BbeMOEQNQPSdDUOG@BDC*gfEQdDG U]dCSNQ|sA1@BE(UOTVbZg-U]dOEHTszKGVM3GVADUD@BAaPiE@¨CSPSRSdDJVPSdOR=Gt¦DADGtUOG#E@Ü]NHCSPSRSNQOdOPSNHgfAGICSPU]NoJVNQEHGnR=GVPSR=gfdOrGVRiGVA R*@BNHC=gfAUOGE_^ NQAaP=TV|sR*@BPSNHgfA C=M1@BPSN@BEHGC=dORiEHGbeNQEQNHGVd¸GVP#UOgfADJ¨CSdORªE@ U]NHCSPSRSNQOdOPSNHgfA\YGkGIJVPSdDTIGEHgfR=C#U]dMOR=gWJIGIC*CSdDC#UOGUOTVbZgWU]dOE@BPSNHgfAzK^qGICSP¡@BEHgfR=CE_^ NQAaP=TV|sR*@BEHGUOGE@KU]NHCSPSRSNQOdOPSNHgfAeCSdOREHGICUOTkwygfRSbo@BPSNHgfADCCSP*@BPSNHcWdDGIC%cadON]GICSP:NQbeM3gfRSP*@BAaP=GnpdOADGp@BbeMOEQNQPSdDUOGmUOGoMlgfbeM1@B|GUOgfAOADTIGs\»JIGmcadON@MORSNHgfRSNADGZM3GVRSbZGVP(M1@fC@IrGIJoJIGVPSP=GC=GVdOEHGoNQA]wgfRSbo@BPSNHgfAUOGUOTVP=GVRSbeNQADGVRpEHGICmU]NTVR=GVAaP=GICpCSPSRSdDJVPSdOR=GICEHg-Ju@BEHGICmUOGiJIGVPSP=GhU]NHC=PSRSNQOdOPSNHgfAz »@#bZGICSdOR=GiJIgfbeMOEHTVbZGVAaP*@BNQR=GtUOG²E@MOL1@fC=GoU]dCSNQ|sA1@BEUOTVbZgWU]dOEHTGICSPYdOADGmNQA]wgfRSbo@BPSNHgfANQbeM3gfRSP*@BAaP=GpcadONGICSP(M3GVdOP¢VPSR=GpnbZ¢VbZGU^`@BMOMlgfRSP=GVR9dOADGC=gfEQdOPSNHgfAhnJIG5MOR=gfOEHVbZG(NQAarGVR=C=GYM1@BR>E_^ NQAaP=GVRSbZTIU]N@BNQR=GU]dwygfRSbo@BEQNHCSbZGYUOG5¤DgfdORSNHGVRuz<?xGICwygfRSb(dOEHGIC/@BA1@BEQ°-PSNHcWdDGICU^ NQAarGVR=CSNHgfAigfAaPFTVP=T(gfOP=GVAadDGIC>bo@BNHCFADG5C=gfAaPFM1@fCMOR=TIC=GVAaP=TIGICKUD@BADC>JIGYUOg-JVdObZGVAaPuz ¡^ NQA]¿DdDGVADJIG5UOGKE@9U]NHCSP*@BADJIGxJu@BR*@fJVP=TVRSNHCSPSNHcWdDG5U^`@BPSP=TVAad1@BPSNHgfAhUOGKE_^ NQAWP=GVADCSNQP=T5UOGKMlgfbeM1@B|G

àCSdORE_^qGVbeMOE@fJIGk

bZGVAaP5UOG7E@oPSR*@BADC=NQPSNHgfA2 → 3/2

@mTVP=T9TVrs@BEQdDTIG7AadObZTVRSNHcWdDGVbZGVAaP7GVAidOPSNQEQNHC*@BAWPKE@owygfRSb(dOEHG©2aDzQ®ua«>U]dªbZg-UOVEHGUOTVrGVEHgfMOMlTsz-¾WdORE@K¦D|sdOR=G aDz`XsÀO\fPSR=gfNHC¡UOTVM3GVADUD@BADJIGIC¯UOGE_^`@BbeMOEQNQPSdDUOGFU]doCSNQ|sA1@BEDUOTVbZg-U]dOEHT>GVAwgfADJVPSNHgfAmUOGE_^`@BbMOEQNQPSdDUOGeUOG9MlgfbeM1@B|GC=gfAWP5PSR*@fJITIGICxMlgfdORxPSR=gfNHCxrs@BEHGVdOR=C5UOG7E@mEHgfAO|sdDGVdOR

àz3 ¡^ NQA]¿DdDGVADJIGeUOGE_^`@BPSP=TVAWd1@BPSNHgfAUOG

!"#%$&('*) &('+&,-)./0!&!$('211 .%

−40 −30 −20 −10 0 10 20−100

−80

−60

−40

−20

0

20

n

(µ)

O/P= %' U %' ! !$ O2P= %')JE = ' 6JP !$

à = 0.1m

à = 0.01m

"

+-,."

)

SUTVEW !X!Y 6 _ _ 5_ j f j *" _ %f?f%] N` 53 _ $f%] [f

àb_ j$ _ `] 5_ t e t f!"5_ `

j _ ` " uf j 5]Je j ] ) + _`Fqf j _ ` à = 0.01, 0.05, 0.1` j ` ]%_` Hf[` j f c e j _ 0 2 t _

` a 5_ f%d` 0 % ]_E` _ _ #" %_ 2 _ uf _ t b_ f%d`-c fa j _` _ 5_`` b`b_

µ = 0.1

E_^qgfADUOG5UOGKM3gfbeM1@B|G5GICSPUOgfADJxPSR=ICwy@BNQOEHG5C=dORE_^qGVbeMOE@fJIGVbZGVAWP/UOGxE@7PSR*@BADCSNQPSNHgfA2 → 3/2

\OdOADGKrs@BRSN@BPSNHgfA²U^ dOAw_@fJVP=GVdORF®uÀZ©GVAWPSR=G

0.01mGVP

0.1m«UOG

àGVAaPSR*@µºHA1@BAWPdOAoUOTVMOE@fJIGVbZGVAWPUOGbZgfNQADC¯UOG

1 dBCSdOR¡E_^qGVbeMOE@fJIGVbZGVAaP

UOGZE@PSR*@BADCSNQPSNHgfA2 → 3/2

z%GVM3GVADUD@BAaPu\E_^ NQA]¿DdDGVADJIGUOGZE_^`@BPSP=TVAWd1@BPSNHgfA£K¤CSdOR(EHGZAONQrGu@Bd¨@BDC=gfEQdU]dCSNQ|sA1@BEUOTVbZg-U]dOEHT5A^qGICSPM1@fCFADTV|sEQNQ|Gu@BOEHGs\DEHGxMOLDTVADgfbZVADG7UOGYUOTVbZgWU]dOE@BPSNHgfAiTVP*@BAaPFJVdOb7dOE@BPSNwz GICGkj]M3TVRSNHGVADJIGIC·MOR=TIC=GVAWP=TIGICgfAaPTVP=TGkGIJVPSdDTIGIC·M3gfdORUOGIC·w~R=TIcWdDGVADJIGIC·UOG M3gfbeM1@B|G JIGVAaPSR=TIGICCSdOR

80.192 kHz\3JIG9cadONJIgfRSR=GICSM3gfADU#nZdOADG9EHgfAO|sdDGVdOR5Ju@BR*@fJVP=TVRSNHCSPSNHcWdDGeU^`@BPSP=TVAad1@BPSNHgfAtM3gfdOR/E_^ NQAWP=GVADCSNQP=TeUOG9Mlgfb

M1@B|GhGICSPSNQbZTIGhn2 − 4 cm

zGICmJIgfADU]NQPSNHgfADC\GVA·JIgfA ¥ gfADJVPSNHgfA½@IrGIJhEHGU]N@BbZVPSR=GhGkGIJVPSNw(U]d·PSR*@BADC*U]dDJVP=GVdOR@fC*CSdOR=GVAaP7dOAtwygfADJVPSNHgfAOADGVbZGVAaPeUOGZE_^`@BAWP=GVAOADGmM1@BR*@BbZTVPSRSNHcWdDGUD@BADC(dOAR=TV|sNQbZGmUOG BtGICSP=GVRSrGVEQPuz%GoR=TV|sNQbZGmUOGwygfADJVPSNHgfAOADGVbZGVAaPu\-M1@BRgfMOMlgsCSNQPSNHgfAp@BdZR=TV|sNQbZGKUOG cGVRe^WP*@u°²©~rgfNQR¯EHG>UOTVOdOPU]dmJ*L1@BMONQPSR=GKX«¡GICSP¡ONHGVAp@fUD@BMOP=T/n5E@UOTVP=GIJVPSNHgfA<±BcWd1@BAaPSN¦1Ju@BPSNHgfAiUOGFE_^`@BAONHC*gfPSR=gfMONHGKU]dbeNQEQNHGVd\-EHG>w_@BNHC=JIGu@BddOEQPSR*@fC=gfADgfR=G5UOG/MlgfbeM1@B|GxTVP*@BAaPJIgfEQEQNQbo@BP=T©U]NQrGVRS|GVADJIG/w_@BNQOEHGFU]dmwy@BNHC*JIGu@Bd<«kz%b%@BRJIgfADC*TIcadDGVAWPu\WEHGICADgfb(OR=GICU^qgfADUOGKUOGFM3gfbeM1@B|G©GVPM1@BRE@YbZ¢VbZGKgWJIJu@µC=NHgfAeE@xMlgfE@BRSNHC@BPSNHgfAmUOGIC¡gfADUOGICUOG>JVNHC*@BNQEQEHGVbZGVAaP«¯gfAaPdOADG>U]NQR=GIJVPSNHgfAoONHGVAmUOTk¦DAONHGsz E_^ NQAarGVR=C=Gs\WUD@BADC¡EHGR=TV|sNQbZGUOG cGVRe^-P*@I°gfvEHGxCSNQ|sA1@BEUOTVbZgWU]dOEHT(GICSP|TVADTVR=T5UD@BADCdOAJ=L1@BbeMhU]NlR*@fJVP=TYU^qgfADUOG5UOGxM3gfbeM1@B|Gs\OE@U]NQR=GIJVPSNHgfAUOGKM3gfE@BRSNHC*@BPSNHgfAUOGICgfADUOGICUOGxM3gfbeM1@B|GKUOG5JVNHC@BNQEQEHGVbZGVAaPFGICSPbo@BEUOTk¦DAONHGs\OJIG5cWdON3GIC=PbZgfNQADCw_@IrgfR*@BOEHG7n(dOADGUOTVP=GIJVPSNHgfAhUOG5E_^`@BAONHC=gfPSR=gfMONHG(UD@BADC>EHG5beNQEQNHGVdz

j b` (

&(' 2>,78 =6 1;=2F, GICR=TICSdOEQP*@BP=CgfOP=GVAWdDCpbZgfAaPSR=GVAaPhcadDGhE_^qGk3GVPpU^`@BdOP=gBUOTVbZg-U]dOE@BPSNHgfAADgfAEQNQADTu@BNQR=GiGIC=PJu@BM1@BOEHGªUOGªJu@µ

R*@fJVP=TVRSNHC=GVR5C=TVEHGIJVPSNQrGVbZGVAaP(EHGICFw_@BNQOEHGICxJIgfAaP*@fJVP=C5MOR=TIC*GVAaP=C5UD@BADCxEHGICKbeNQEQNHGVd]jh|sR*@BAWdOE@BNQR=GIC5ADgfA]JIgfADC=gfEQNHUOTICuz F¦DAU^qGkj-MOEQNHcWdDGVR5EHGICYU]NTVR=GVADJIGICYU^`@BbeMOEQNQPSdDUOGIC9U]dtCSNQ|sA1@BE¡UOTVbZgWU]dOEHTM3gfdORYU]NTVR=GVAaP=GICYM3gfE@BRSNHC*@BPSNHgfADC7U^qgfADUOGICYUOGMlgfbeM1@B|Gs\lGVPFE@ZPSR*@BADCSNQPSNHgfAªJVE@BNQR=G

2 → 3/2\3gfDC=GVRSrTIG(UD@BADC/E@mUOTVMlGVADUD@BADJIG9GVAi@BbeMOEQNQPSdDUOGU]dhC=NQ|sA1@BE»UOTVbZgB

U]dOEHTeGVAªwgfADJVPSNHgfAUOGE@pUOTkwgfRSbo@BPSNHgfAUOGM3gfbeM1@B|Gs\GVPYcWdON%NQAaP=GVRSr-NHGVAaP(ONHGVA@Irs@BAaPYE@pUOTkwygfRSbo@BPSNHgfACSP*@BPSNHcWdDGbZgu°GVAOADGs\-EHG>beNQEQNHGVdpUOgfNQPJIgfbeMlgfRSP=GVRdOAm|sR*@BADUmADgfb(OR=GxUOG/JIgfAaP*@fJVP=Cw_@BNQOEHGICK©y@BdOP=gfdORUOG `sÀ $¸UOGICJIgfAaP*@fJVP=Ck«kz?xGYMOEQdDCu\<JIGIC/JIgfAWP*@fJVP=CFw_@BNQOEHGICKUOgfNQrGVAaP5¢VPSR=G7C=d]oC*@BbebZGVAaPKEHgWJu@BEQNHC*TICFrGVR=C/EHGICFPSR=ICFw_@BNQOEHGIC/UOTkwygfRSbo@BPSNHgfADCxCSP*@µPSNHcadDGICz1AiM1@BRSPSNHJVdOEQNHGVRu\3UD@BADCFADgsCKJIgfADU]NQPSNHgfADC5Gkj-M3TVRSNQbZGVAaP*@BEHGIC\<dOADG(EHgfN»dOAONwygfRSbZG9GVAiUOTkwygfRSbo@BPSNHgfADCKMlgfdOR/EHGICwy@BNQOEHGICFJIgfAaP*@fJVP=C©JIgfRSR=GICSMlgfADUD@BAWPKndOADG(U]NQrGVRS|GVADJIG(GVAMOdONHC=C@BADJIG −1/3

MlgfdORFE@eU]NHCSPSRSNQOdOPSNHgfAhUOGKwygfR=JIGIC«ADGMlGVRSbZGVP¯M1@fC¯UOG>UOTIJVRSNQR=GFEHGIC¡JIgfbeMlgfRSP=GVbZGVAWP=CgfDC=GVRSrTICuz?5@BADC¡EHGIC¡beNQEQNHGVd]jZ|sR*@BAadOE@BNQR=GICbZgfNQADC¯R=TV|sdOEQNHGVR=CJIgfbebZGEHGC@BOEHGA1@BPSdOR=GVEDgfddOA9beNQEQNHGVde|sR*@BAadOE@BNQR=GM3gfEQ°WU]NHCSMlGVR=C*Gs\EHGADgfb(OR=GUOGJIgfAaP*@fJVP=Cwy@BNQOEHGICGICSP%GVADJIgfR=GCSdOM3TVRSNHGVdORGVPE@EHg-Ju@BEQNHC*@BPSNHgfAMOEQdDCbo@BR=cWdDTIGs\OEHGYJIgfbeM3gfRSP=GVbZGVAaPKcad1@fU]R*@BPSNHcWdDGYA^`@u°a@BAaPFMOdh¢VPSR=G5gfDC=GVRSrTZ3@BNsBDz :@hUOTVP=GIJVPSNHgfA GVP7E@hcWd1@BAaPSN¦1Ju@BPSNHgfA UOGmE_^`@BAONHC=gfPSR=gfMONHGpADgfAEQNQADTu@BNQR=GMOR=TIC=GVAWP=GpUD@BADC7EHGbeNQEQNHGVdGICSP9R=GVADU]dDG

MlgsC*CSNQOEHG¯|sR*fJIG>nKUOGVd]jgfDC=GVRSrf@BPSNHgfADCU]NHC=PSNQADJVP=GICuzs :@/MOR=GVbeNHVR=GGICSP%E@KU]N3TVR=GVADJIGGVAWPSR=GEHGIC%AONQrGu@Bd]j9UOGICCSNQ|sA1@Bd]jUOTVbZgWU]dOEHTICmNHC=CSdDCUOG²UOGVd]j M3gfE@BRSNHC*@BPSNHgfADCpU^qgfADUOGICpUOGM3gfbeM1@B|GhUOGhJVNHC*@BNQEQEHGVbZGVAaPU]NTVR=GVAaP=GIC#©~M1@BR*@BEQEHVEHGªGVPgfRSPSLDgf|gfA1@BEHG9@Bd]jJ*L1@µºHADGIC/UOGYwgfR=JIG«kzD »@ZC=GIJIgfADUOG7GICSP>E@eU]NTVR=GVADJIG7GVAWPSR=GYEHGIC/GVbeMOE@fJIGVbZGVAaP=CxUOGIC>PSR*@BADCSNQPSNHgfADCMOdONHC=C*@BADJIG

2 → 3/2UD@BADCiEHGJIgfbeM3gfRSP=GVbZGVAaPtUOGE_^`@BbeMOEQNQPSdDUOGÜOGIChC=NQ|sA1@Bd]j UOTVbZgWU]dOEHTIC#GVA wygfADJVPSNHgfAUOG

E_^`@BbeMOEQNQPSdDUOGUOG MlgfbeM1@B|GszGICUOGVd]j¸NQADU]NHJu@BP=GVdOR=Cu\7JIgfLDTVR=GVAaP=CGVAaPSR=GGVd]j¶UD@BADCEHGICtbZGICSdOR=GICGkGIJVPSdDTIGICu\MlGVRSbZGVPSP=GVAWPiU^`@fJIJITIUOGVRiR=GVE@BPSNQrGVbZGVAaPªMOR=TIJVNHC=TVbZGVAWPt@BdAONQrGu@BdU^`@BAONHC*gfPSR=gfMONHGU]d M1@BR*@BbZVPSR=GADgfA EQNQADTu@BNQR=GGk3GIJVPSNw¯U]dhbeNQEQNHGVd\<GVPxUOgfADJYNQADU]NQR=GIJVP=GVbZGVAWP(@BdhAONQrGu@BdªU^`@BAONHC=gfPSR=gfMONHG7EQNQADTu@BNQR=Gsz3 GICFrs@BEHGVdOR=CKUOGICK@BAONHC=gfPSR=gfMONHGICgfDC=GVRSrTIGIC>Gkj]MlTVRSNQbZGVAWP*@BEHGVbZGVAaP5C=gfAWPFJIgfEQEHGIJVP=TIGICFUD@BADCEHGYP*@BOEHGu@Bd,aDzQ®sz

bR=GIC=CSNHgfAhCSP*@BPSNHcadDG8 kPa 24 kPa 64 kPa?KN3TVR=GVADJIGZUOGAONQrGu@Bd]jnE_^`@BMOM1@BRSNQPSNHgfAUOGE@PSR*@BA]

CSNQPSNHgfAhX → [±fX©~R*@BMOM3gfRSP ε (h)a /ε

(v)a«

éUc ∼ ¬9Uc ®uUc?KN3TVR=GVADJIGGVAaPSR=G¡EHGICAONQrGu@Bd]j(UOGIC:CSNQ|sA1@Bd]jYUOTVbZg-U]d]EHTICNHC=CSdDCUOGICUOGVd]jmM3gfE@BRSNHC*@BPSNHgfADC>U^qgfADUOGICUOGKM3gfbM1@B|GªUD@BADCpE@R=TV|sNHgfAUOGiJVR=gfNHC=C*@BADJIG#cWd1@fU]R*@BPSNHcWdDG©C*@BADC>JVE@BMOM3GVbZGVAaP«

®uXµ*®PaoUc ∼ Uc ®u´µ*®ueUc

?KN3TVR=GVADJIGGVAaPSR=G¡EHGICAONQrGu@Bd]j(UOGIC:CSNQ|sA1@Bd]jYUOTVbZg-U]d]EHTICNHC=CSdDCUOGICUOGVd]jmM3gfE@BRSNHC*@BPSNHgfADC>U^qgfADUOGICUOGKM3gfbM1@B|GYn9E_^`@BbeMOEQNQPSdDUOG(UOGxM3gfbeM1@B|Gxbo@µj-NQb7dOb ©y@IrGIJJVE@BMOMlGVbZGVAWP«

éUc èUc ®s®5Uc

DFNQrGu@BdÜOGpUOTkwygfRSbo@BPSNHgfA UOGoM3gfbeM1@B|GnE_^`@BMOM1@BRSNPSNHgfAiUOG7E@oPSR*@BADCSNQPSNHgfAªX → [±fXoM3gfdOR/EHG7CSNQ|sA1@BEUOTkbZgWU]dOEHTNHC=C=d7U^qgfADUOGICUOGMlgfbeM1@B|GMlgfE@BRSNHC*TIGICC*GVEHgfAEHGIC>J=L1@µºHADGICKUOGKwygfR=JIG

*®uUc ∼ *®ueUc éUc

W XuY ]f t j f 5_J` \^] %_ _` _ %_ j f "5] k] f% ` " uf j b]Je j ] t f 5_` 5_` b_t e t f!"b_ 5_ N` f jdj _ eg_ t j fN`]_`p` _ j j _J` Jr f_` 5_Uc _ _ dr " uf j _Jel_ +

$_ [` _Jela j _ 5_ _`

?]J` j f d` t _ el_ 5_ eg_J` ?_ iIt f tuj 5` _ ` ba`_ Jqf [` Hf_` i j$ f N` t _ t f?fe 8 ?_ b_ j ]?f ] 3 [f%?f 43[_ Hf[` j _ e j _ i _ $ 5a _ ?_ _Jel_ j fpqf j _ b_ j($ f N` t _ j ]%f %_+ _` qf j _ ` 5a _ [_J` t _ t ?_J`` v` f 53 [_ 5_

24 kPa` Qe [` t %] N`_J`C3[_ t j _` f ?_J`

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Europhysics Letters PREPRINT

Acoustic second harmonic generation with shear to longi-

tudinal mode conversion in granular media

V. Tournat1, V.E. Gusev1, V.Yu. Zaitsev2 and B. Castagnede1

1 Universite du Maine, Av. Olivier Messiaen, 72085 Le Mans Cedex 9, France2 Institute of Applied Physics, 46 Uljanova Street, Nizhny Novgorod, 603950 Russia

PACS. 83.80.Fg – Granular solids.PACS. 81.05.Rm – Porous materials; granular materials.PACS. 43.25.+y – Nonlinear acoustics.

Abstract. – Excitation of longitudinal acoustic wave at the 2nd harmonic by sinusoidalshear acoustic wave in a granular material is reported. The amplitude of the 2nd harmonicexhibits beatings (typical to nonlinear processes in dispersive media), which is observed not asa function of the distance from emitter, but with increasing amplitude of the primary (pump)shear wave. The effect is attributed to varying contribution of clapping intergrain contactsto the total nonlinearity of the medium with increasing pump amplitude, which modifies theeffective length of the nonlinear interaction. This is consistent with the idea that in granularassemblages there is an important amount of contacts loaded much weaker than in average.

Introduction. – Second harmonic excitation is a classical nonlinear effect which is useful,for example, for frequency-up conversion in optics [1,2] and material characterization in acous-tics [3–5]. The efficiency of the transformation of the pump wave at fundamental frequencyω into its 2nd harmonic at frequency 2ω is determined not only by the local nonlinearity ofa medium, but also depends on the possibility to synchronously accumulate locally excited2ω-waves. The synchronism conditions are closely related to the wave velocity dispersion inthe medium. In the case of collinear wave interaction, the nonlinear sources at 2ω (producedby the pump wave) propagate with the phase velocity of the pump wave c(ω) = ω/k(ω), whilethe excited second harmonic propagates with velocity c(2ω) = 2ω/k(2ω). Here k denotes thewavenumber. If c(ω) 6= c(2ω) (i.e. ∆k ≡ k(2ω)−2k(ω) 6= 0) the beating of the forced and freewaves occurs. Such beating is a hallmark of dispersive nonlinear interactions [1, 2, 6, 7]. Thisleads to periodic spatial modulation of the 2ω amplitude along the interaction length, and thebeating spatial period limits the accumulation (coherence) length of the second harmonic byLc = π/∆k. In optics, where the attenuation of the pump wave in negligible, the length ofthe nonlinear crystal should be optimized correspondingly.

In acoustics, the dispersion is usually much less than in optics, and the process of the2ω excitation in a homogeneous medium is commonly considered as a quasi-synchronous onewith 2ω-wave accumulation limited by the pump wave absorption or diffraction. Dispersion ischaracteristic of acoustic modes in waveguides, where the periodic spatial modulation of sum-and difference-frequency sound amplitude was experimentally observed [6]. A similar effect

c© EDP Sciences

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2 EUROPHYSICS LETTERS

was reported for Rayleigh surface waves for which the dispersion was introduced by loadingthe substrate with a thin film [7, 8]. It was demonstrated experimentally that dispersionsuppresses higher harmonics generation in surface acoustic waves [9].

In the present letter we describe observations of beatings in the amplitude of the secondharmonic excited in a granular medium. There are two important differences between thereferences cited above and our experiments.

First, the asynchronism between the nonlinear force and the second harmonic wave inour experiments is not due to velocity dispersion for a particular acoustic mode, but dueto the mode conversion in the nonlinear process. It is well known that the excitation of thelongitudinal acoustic wave at 2ω by the shear acoustic pump wave at ω is theoretically allowedin solids [3]. However, to the best of our knowledge, this process has never been experimentallyobserved because of weak nonlinearity of homogeneous solids and large difference betweenthe velocities cL and cS of longitudinal (L) and shear (S) waves [3]. The first of these twofactors (limiting 2ω-wave amplitude) was overcome in our experiment by choosing the granularmaterial in which the nonlinear parameter exceeds those typical to consolidated homogeneousmaterials by two to four orders in magnitude (depending on the level of external static pre-loading) [10, 11].

Second, the beatings in 2ω amplitude are found not when the observation distance isvarying, but when the amplitude of the pump shear wave is increased. Theoretical estimatespresented below attribute this effect to the nonlinear transformation of the wave interactionregion, which is due to the change in the mechanism of the 2ω excitation with the increasingpump amplitude.

Experiment. – The scheme of the experiment is shown in fig. 1. The emitter excited shearwaves at frequency f = 5.12kHz with vertical (V) or horizontal (H) polarization. For thetransducer of a radius a ' 2cm and the estimated length of the shear pump wave λω ' 4.4cm,the diffraction length Ld ∼ πa2/λω ∼ 3cm is comparable to the transducer dimensions andis much less than 2ω -signal observation distance R ' 16cm (see fig. 1). Consequently, theexcitation of the second harmonic takes place in the essentially spherically-diverging pump-wave rather than in the plane-wave geometry. Variation of the external stress by an order ofmagnitude 7.2 kPa ≤ P0 ≤ 72 kPa changed the average static strain from ε0 ' 7 10−5 toε0 ' 3.3 10−4. Here and in the following, the compression corresponds to positive strain. Theglass beads were of 2mm diameter. The receiver was created for the longitudinal polarization,so that its residual sensitivity to laterally polarized waves was significantly reduced. It wasverified that the received signal at 2ω corresponded to L-wave, for which the signal propagationtime and its polarization were checked. The observed mode conversion between the pump S-wave and the longitudinal 2nd harmonic is allowed by the symmetry considerations, whereasexcitation of the shear 2nd harmonic in isotropic homogeneous materials is forbidden [3].

In fig. 1 parts (a) and (b), the dependence of the 2ω amplitude on the amplitude of thevertically polarized shear pump is presented. The beatings are clearly visible and reproducible.On the x-axis, 0dB corresponds to the maximal used pump strain amplitude about ε max

a '

1.4 10−5. It is more than an order of magnitude less than the average static strains in themedium (2.3 10−4 and 2.9 10−4 for the applied static stresses 41kPa and 60kPa respectively).Thus on the x-axis the average static strain of the medium is located around ∼ 20dB andhigher. In fig. 1(b) the measurements correspond to larger static preloading than in fig. 1(a).Comparison of the figures indicates the shift of the first local minimum (marked by verticalarrows) to higher pump amplitude for higher static strain.

In fig. 1(c) the dependence of 2ω longitudinal amplitude on the amplitude of the hori-zontally polarized shear pump wave is shown for the same static loading as in the case of

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V. Tournat et al.: Acoustic harmonic generation in granular media 3

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L-receiver

V-shear emitter

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Fig. 1 – Scheme of the experiment and longitudinal second harmonic level as a function of the shearpump strain level. Vertically polarized pump (V-shear), plots (a) and (b). Horizontally polarizedpump (H-shear), plot (c). Insert in plot (b) is for longitudinal pump.

vertical polarization in fig. 1(a). Comparison of the figures demonstrates that the first mini-mum appears for the horizontally polarized pump at smaller amplitudes than for the verticallypolarized.

Discussion. – Our theoretical analysis attributes the beating in the 2nd harmonic ampli-tude to the unproportional modification of the spatial distribution of the nonlinear 2ω sourceswith increasing pump amplitude. Both nonlinear acoustic experiments in granular media andtheir theoretical interpretations indicate that, at sufficiently strong excitation, the weak clap-ping inter-grain contacts can provide competitive [12, 13] or even dominant [11] contributionto the nonlinear part of the stress-strain relation σ(ε) if compared with the contribution fromcontacts which remain closed during the whole period of acoustic loading. Indeed, in realgranular materials, there is a significant portion of contacts with loading much weaker thanin average [11, 12], which essentially contribute to the resultant non-linearity of σ(ε). Let usmodel for simplicity the distribution of weak contacts by a single fraction statically pre-loadedby the strain µε0 (which for µ 1 is significantly lower than the average loading ε0). Sep-arating out explicitly the static (σ0, ε0) and oscillatory (σ, ε) parts of the stress and strain,the stress-strain relation may be rewritten as:

σ0 + σ = bn0(ε0 + ε)3/2H(ε0 + ε) + bn1(µε0 + ε)3/2H(µε0 + ε) (1)

Here H(. . .) is the Heaviside function; n0 and n1 are the mean numbers of average loaded andweak contacts per grain, and factor b depends on elastic moduli of individual grains and theporosity of the packing. Estimates based on wave velocities data in grainy materials indicate

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that n0,1 may be comparable (n0 ∼ n1) [13]. Nondimensional coefficient |µ| 1 characterizesthe extent of unloading of the softer fraction. The power 3/2 in eq. 1 corresponds to theclassical Hertzian nonlinearity. Below, we consider only positive 0 < µ 1 corresponding toinitially weakly compressed contacts. Then for |ε| µε0 the first and higher derivatives ofeq. (1) with respect to ε characterize the linear and nonlinear elastic moduli of the material,respectively:

∂mσ(ε0)

∂εm∼ bn0(1 +

n1

n0µ3/2−m)ε

3/2−m0 (2)

Thus for the linear modulus (m = 1) the relative contribution of the weak contacts is ∼µ1/2 1 and may be negligible. In contrast, for the quadratic (m = 2) nonlinear modulusresponsible for 2ω excitation, the contribution of the weak fraction is ∼ µ−1/2 1 and thusmay strongly dominate over the nonlinearity of the average-loaded contacts.

For a propagating wave beam containing initially one or several sinusoidal waves, the afore-mentioned material nonlinearity results in the creation of running virtual nonlinear sources.These sources radiate combination frequencies, for example, higher harmonics nω of a sinu-soidal pump. In underwater acoustics, this effect is known as parametric sound radiation, andis mostly used for the frequency down-conversion via difference frequency ω1 − ω2 generation(|ω1 − ω2| ω1, ω2, when ω1 ∼ ω2). In liquids, where shear waves do not propagate, thenonlinearity of the state equation is almost perfectly quadratic in the acoustic strain, and theeffect allows for a relatively simple but quite rigorous description. Using the perturbationapproach, the stress in the nonlinearly radiated wave may be expressed in the integral form:

σnl(~r) = <e1

4π

∫Q(~r′)

eikrad|~r−~r′|

|~r − ~r′|d3~r′ (3)

where the integration is performed over the volume of the nonlinear sources Q(~r′) ≡ Q(x′, y′, z′)created in the medium by the primary (pump) beam; ~r is the position of the observation point,krad is the wave number of the nonlinearly radiated signal (for example of the second har-monic). For liquids, the nonlinear source Q is proportional to the square of the primary wave

amplitude: Q ∼[εae−iωt+ikr′

]2

. In liquids both the primary and the nonlinearly radiated

waves are of L-type and due to the absence of dispersion the co-propagating interacting wavesare synchronous. This formally means that the oscillating exponential factors in the nonlinearsource and in the radiated wave can perfectly compensate each other.

For the experiments is granular solids, the situation is more complex because of the non-linear conversion between L- and S-waves. However, for the qualitative understanding of theobserved beating effect we may omit the detailed discussion of coupling between S- and L-modes (having different polarizations) and will focus on the phase synchronism between therunning nonlinear sources and the radiated wave. The integral solution in form of eq. (3) issufficient for our purpose to discuss the phase properties of the integrand and its functionalamplitude behavior in order to explain the beating effect. Concerning the absolute level of theradiated harmonic, certainly the coupling between S- and L-modes with different polarizationsmay additionally affect the numeric factors compared with those in solution (3).

In the considered case the nonlinear sources propagate with the phase velocity of the S-wave, so that the nonlinear source for the 2nd harmonic Q ∼ e−i2ωt+i2kS(ω)r′

, where thewavenumber kS = ω/cS(ω). The radiated second harmonic of the L-wave propagates withanother phase velocity cL and wavenumber krad = kL(2ω) = 2ω/cL(2ω). The oscillatingfactors of the source and the radiated harmonic thus do not compensate each other andproduce beating ∼ ei∆kr′

= ei[kL(2ω)−2kS(ω)]r′

under the integral. The half spatial period of

,


the beatings limits the distance over which the contributions of the nonlinear sources can besynchronously accumulated. Another important difference from liquids is that, in the granularmaterial, the nonlinear source remains quadratic only for small enough amplitudes εa of thepump-strain, εa/ε0 µ. In this case the expansion of eq. 1 leads to the nonlinear source of theform Q = Q2 ∼ (3/16)(µε0)

−1/2ε 2a . For stronger εa > µε0 the nonlinearity is progressively

becoming of the clapping type (see the Heaviside function in eq. (1)). Thus, after singlingout the 2ω Fourier harmonic, the nonlinear source corresponding to the essentially clapping

regime of the weak contacts takes the form: Q = Q3/2 ∼ (3/4π)ε3/2

a . The magnitudes of thenonlinear source in the two regimes coincide at ε cr

a ' 16µε0/π2, which we shall consider asthe characteristic transition amplitude between the quadratic and clapping regimes. In thisapproximation, the interaction region in eq. (3) is subdivided into two parts corresponding tothe quadratic/clapping regions of the nonlinear sources:

ε 2ωa (~r) ' <e

const.

4π

∫

r′<Lcr

Q3/2(~r′)

eikrad|~r−~r′|

|~r − ~r′| d3~r′ +

∫

r′>Lcr

Q2(~r′)

eikrad|~r−~r′|

|~r − ~r′| d3~r′

(4)

Schematically these regions are shown in fig. 2. At small-amplitude excitation, the clappingregion does not exist at all. With increase of the pump amplitude the clapping region appearsnear the radiator and extends in the bulk, the transition distance Lcr being determined bycondition εa(r = Lcr) = 16µε0/π2. The pump wave (see the previous section) is sphericallydiverging, so that εa(r) ' εa(r = Ld)Ld/r. Note that εa(r = Ld) is approximately equal tothe pump wave amplitude at the radiator εa(0). Performing the integration over the crosssection of the pump beam and approximating |~r − ~r′| ' r in the denominators in eq. (4), thelatter is reduced to:

ε2ωa (R) ' const.<e

ε

3/2a (0)

π√

Ld

Lcr∫

Ld

ei∆kz′

(z′)1/2dz′ +

ε 2a (0)

4(µε0)1/2

R∫

Lcr

ei∆kz′

z′dz′

(5)

where Lcr = π2Ldεa(0)/(16µε0). Thus at small amplitudes for which there is no clapping yet(and formally Lcr < Ld) the first integral vanishes and only the second quadratic in εa termremains. Integral (5) is readily expressed via the Fresnel, Sine and Cosine integrals. In fig.2,expression (5) is plotted for three values of the unloading parameter µ and other parametersclose to those under the experimental conditions (observation length R = 16cm, ∆k = 95 m−1

corresponding to pump at 5.12 kHz, and cS = 225 m/s, cL = 335 m/s). The pump strainmaximum ε max

a (0) (0 dB) is chosen an order of magnitude smaller than the static pre-strainε0. The resultant behavior of the harmonic in fig. 2 essentially depends on µ. For example,for µ = 10−2 (in fig. 2) the initially quadratic and then oscillating dependence for the 2ndharmonic at higher pump amplitudes is qualitatively similar to the experimental curves shownin fig. 1 for the same pump-amplitude range.

In accordance with the proposed physical model, the nonlinear force (exciting 2ω) growsproportionally to ε 2

a everywhere in space unless εa near the emitting transducer exceeds ε cra .

After that, with increasing amplitude of pump wave in a larger and larger space region near the

emitting transducer “clapping” nonlinearity dominates, and the force will grow ∼ ε3/2

a (whilethe dependence ∼ ε 2

a will persist outside of this region). Due to slowing of the force growthnear the emitter, the effective length Leff of the nonlinear antenna emitting 2ω increaseswith pump wave amplitude. In fig 3, the distribution of the nonlinear force is presented fordifferent ratios of εa(0)/ε cr

a in the case of spherical pump wave. Clearly, there is an increase

! % /


−40 −30 −20 −10 0−80

−60

−40

−20

0

1 2

(a) (b)PSfrag replacements

Q3/2Q2

z

Pump emitterµ = 10−1

µ = 10−2

µ = 10−3

Lcr

Second

harm

onic

level,

dB


Fig. 2 – Insert (a), plots of result (5) for three values of the parameter µ; insert (b), qualitative schemeof the nonlinear antenna.

in antenna length with increasing pump amplitude L(1)eff < L

(2)eff < L

(3)eff . The characteristic

length of the force distribution plays here the role of the length of the frequency doublingcrystal in nonlinear optics. Due to asynchronism between the 2ω longitudinal wave and theforce created by shear waves the variation of this length leads to beatings in received secondharmonic amplitude. In accordance with fig. 3, in a granular medium, the variation of thelength of the nonlinear emitting antenna can be achieved by increasing pump wave amplitude.

Distance ∆z between neighboring positions of Lcr corresponding to the successive extremaof the harmonic can be roughly estimated from the phasing condition ∆z ∼ π/∆k ∼ 3.3cm.This means that, when the position Lcr of the clapping boundary gradually moves throughthe whole interaction length (following the increase in the pump amplitude), then, for theobservation distance R ∼ 16 cm, there could be not more than ∼ R/∆z ∼ 4− 5 intermediateextrema. Further increase in the pump amplitude does not produce additional extrema, butshould lead to the appearance of the smooth dependence with slope 3/2, which indicates thatthe nonlinear sources are in the essentially clapping regime over the whole interaction length.

1 1.5 2 2.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(1)

(2)

(3)

PSfrag replacements

Distance r/Ld

Am

plitu

de

ofth

enonlinea

rfo

rce

L(1)eff

L(2)eff

L(3)eff

L(2)cr L

(3)cr

Fig. 3 – Antenna body transformation for different levels of emitted pump strains: εa(0)/ε cra = 1

for curve (1), εa(0)/ε cra = 1.3 for curve (2) and εa(0)/ε cr

a = 1.7 for curve (3). Broadening of the

antenna at the level 1/2 is represented by the increasing of the effective lengths L(1)eff , L

(2)eff and L

(3)eff

corresponding to the three increasing pump strain levels, respectively.

0


In fig. 2, the plot for µ = 10−3 demonstrates such a curve with completely developed beatingsand slopes 2 and 3/2 before and after this region. When parameter µ is too large (insignificantunloading), the intermediate extrema may not appear at all for the amplitudes used (see fig.2, curve µ = 10−1 with the quadratic slope over the whole amplitude range).

These examples demonstrate that the nonlinear beatings are rather sensitive to presence ofweak contacts in granular packings. Thus the 2nd harmonic generation in shear waves providesa sensitive tool for evaluation of the weak inter-grain forces. The estimate µ = 10−2 obtainedfor the best agreement with the experimental observations, correlates well with data inferredfrom the experiments on the high-frequency pump demodulation in granular media [11].

The experimental observations that pump amplitude corresponding to the first minimumincreases with external loading and is higher for vertical polarization of the shear pump thanfor the horizontal one are consistent with the ideas that increased loading makes it moredifficult to initiate contact clapping (ε cr

aincreases because ε0 increases) and that due to

preferential direction of loading (forced chains [14]) εcr

afor vertically oriented contacts is

larger than for the horizontal because ε0 is larger for the former. The latter statement meansthat vertical loading induces anisotropy in the nonlinearity of the granular assemblage. Thenonlinearity depends on the direction of particle displacement in the acoustic wave.

Conclusion. – Experimentally observed beatings in the second harmonic amplitude withincreasing amplitude of the pump wave are attributed to asynchronous character of the processof 2ω excitation and the pump-induced variation of the 2ω effective excitation length. Thevariation of the effective length of the emitting antenna is attributed to increasing contributionof the clapping contacts to the nonlinear process.

∗ ∗ ∗

This work is supported by a DGA contract No 00.34.026.00.470.75.65. and RFBR grantNo 02-02-16237 (V.Z.).

REFERENCES

[1] Lauterborn W., Kurz T., and Wiesenfeldt M., Coherent Optics, Fundamental and Appli-

cations, edited by Springer (Berlin) 1993.[2] Santer E.G., Nonlinear optics, edited by John Wiley & Sons (New-York) 1996.[3] Zarembo L.K., and Krasilnikov V.A., Sov. Phys. Uspethi, 13 (1971) 778.[4] Yost W.T., and Cantrell J.H., in Rev. Prog. QNDE, edited by D.O. Thompson and D.E.

Chimenti, Vol. 9 (Plenum Press, New-York) 1990, p. 1669-1676.[5] Zheng Y., Maev R.Gr., and Solodov I.Yu., Can. J. Phys., 77 (1999) 927.[6] Hamilton M.F., Il’inskii Yu.A., and Zabolostkaya E.A., in Nonlinear Acoustics, edited by

M.F. Hamilton and D.T. Blackstock (Academic Press, San Diego) 1997, p. 151-175.[7] Mayer A.P., Phys. Rep., 256 (1995) 237.[8] Lean E.G., and Powell C.G., Appl. Phys. Lett., 19 (1971) 356.[9] Lee J., Singh M.P., and Zucker J., Appl. Phys. Lett., 36 (1980) 896.

[10] Belyaeva I.Yu., Ostrovsky L.A., and Timanin E.M., Acoust. Lett., 15 (1992) 221.[11] Tournat V. et al., Phys. Rev. Lett., (submitted) .[12] Belyaeva I.Yu., Zaitsev V.Yu., and Timanin E.M., Acoust. Phys., 40 (1994) 893.[13] Zaitsev V.Yu., Acoust. Phys., 41 (1995) 439.[14] Liu C.-h., Nagel S.R., Schecter D.A., Coopersmith S.N., Majumbdar S., Navayan O.,

and Witten T.A., Science, 269 (1995) 513.

/RSPSNHJVEHG9C*gfdObeNHCxnbLW°-Cuz3 GVPSPuz ©yXsÀsÀs[«KC=dOR/E@o|TVADTVR*@BPSNHgfAtUOGC=gfdDC_L1@BRSbZgfAONHcWdDGIC5GVPKUOG9ORSdONQP5UD@BADCKEHGICbeNQEQNHGVd]j|sR*@BAadOE@BNQR=GICuzGIC5Gkj]MlTVRSNHGVADJIGIC5GVPxEHGICxNQAaP=GVRSMOR=TVP*@BPSNHgfADC9UOGICxMOLDTVADgfbZVADGICYUOGeUOgfdOOE@B|GeUOG9M3TVRSNHgWUOGeGVP5UOG9R=gfdOP=GrGVR=CxEHG

J=L1@fgsCxC=gfAWPKGVA#@fJIJIgfR=Ui@urGIJ9E_^`@BA1@BEQ°-C=G9UOGIC/R=TICSdOEQP*@BP=CKMOR=TIC=GVAaP=TIC5UD@BADC/EHG7J*L1@BMONQPSR=G aDz< GICxJIgfAWP*@fJVP=CFw_@BNQOEHGICxU]dbeNQEQNHGVd|sR*@BAadOE@BNQR=G(ADgfA]JIgfADC=gfEQNHUOT(C*gfAaP>EHGICMORSNQADJVNQM1@Bd]jR=GIC=MlgfADC*@BOEHGICFUOGIC>Gk3GVP=C>ADgfA]_EQNQADTu@BNQR=GIC/gfDC=GVRSrTICz

®Paa

u

Subharmonics and noise excitation in

transmission of acoustic wave through

unconsolidated granular medium

V. Tournat , V. E. Gusev , B. Castagnede

Universite du Maine, Av. Olivier Messiaen, 72085 Le Mans Cedex 09, France.

Abstract

First laboratory-scale experimental observation of both subharmonics excitationand significant increase in noise level caused by propagation of the acoustic wave inunconsolidated granular material is reported. The bifurcation phenomenon, takingplace above a critical level of acoustic excitation (and opening the subharmonicsroute to chaos) is attributed to the interaction of acoustic wave with distributedsystem of highly nonlinear inter-grain contacts. The estimates demonstrated thatthese are weak contacts (loaded at least two orders of magnitude weaker than inaverage) that might be responsible for the observed nonlinear effects. The additionalintermittent contacts created by the acoustic wave (which are open in the absenceof acoustic loading) can also contribute. In the clapping (tapping) regime, each ofthese contacts individually is similar to an impact oscillator, for which the scenarioof period doubling cascade and the transition to chaotic behavior has been predictedtheoretically and observed experimentally earlier. The experiments confirm that thenonlinear interactions of acoustic waves in granular assemblages are highly sensitiveto the fraction of weakly loaded (and unloaded) contacts, information on which isdifficult to access by any other experimental methods.

Key words: Granular solids, Nonlinear acoustics, Nonlinear dynamics andnonlinear dynamical systemsPACS: 83.80.Fg, 43.25.+y, 05.45.-a.

1 Introduction

It had been already 25 years ago that theoretical discovery by Feigenbaum

of certain universal properties in period-doubling bifurcations of iterated one-

Email address: [email protected] (V. Tournat).

Preprint submitted to Elsevier Science 22 August 2003

%

dimensional maps had catalyzed an experimental search for analogous behav-iors in various nonlinear systems [1]. In acoustics, subharmonic sound emissionconnected with period doubling bifurcations has been found in some musi-cal instruments [2,3]. Bifurcations properties and routes to chaos in thermo-acoustic system has been studied [4,5]. A review on nonlinear dynamics inacoustics can be found in [6].

For the current communication, it is important that subharmonic route tochaos has been observed in acoustic wave propagation through a medium [7].Acoustic turbulence (acoustic cavitation noise) in insonified fluids has been at-tributed to complex nonlinear oscillations dynamics of individual gaz bubblesforced by the external sound field [7,8] (in superfluid helium-4 vortex line gen-eration leads to quantum turbulence [9]). Only later, the theories based on theanalysis of gas bubbles nonlinear oscillations have got experimental supportfrom the observations of period doubling and chaos in the sonoluminescencefrom a single bubble [10].

The sequence of advances in acoustic evaluation of nonlinear dynamics of un-consolidated granular medium (where an individual intergrain contact maybe expected to play the role of a bubble in fluid) is just the opposite one.First, the period doubling bifurcations [11,12] and transitions to chaos [13]have been observed experimentally in the dynamics of a single nonlinear con-tact. It should be noted that references [11,12] given here, have a tight relationwith a whole body of publications on bouncing ball phenomenon and impactoscillations, respectively. It is also interesting that the vibration of an individ-ual nanocontact in [11] has been forced by an incident acoustic wave. Earlier,sub-harmonics excitation and different regimes of transition to chaos has beenreported for acoustically induced vibrations of a contact interface betweensolids [14–16]. The quasi-chaotic behavior has been observed in the interac-tion of an ultrasonics welding horn with the surface of a sample [17]. Finally,quite recently the excitation of the subharmonics has been observed in theinteraction of both surface [18] and bulk [19] acoustic waves with individualcracks in the material. Thus, the period-doubling bifurcation and chaotic be-havior is a well documented phenomenon in the interaction of the acousticwaves with individual contacts.

Experimental results reported below are believed to be the first laboratory-scale experimental observation of the interaction of acoustic wave with thedistributed system of highly nonlinear contacts (which is a basic feature ofunconsolidated granular materials). Both excitation of the subharmonics andsignificant increase in noise level with increasing amplitude of acoustic wavepropagating through granular medium are observed. The influence of externalloading on the bifurcation phenomenon is documented. It is expected thatthe laboratory scale nonlinear acoustic experiments of the type presented hereand in [19] will be useful in the future for modeling of full-scale geophysical

2

experiments. For example, the excitation of subharmonics has been observedin full-scale geophysical experiments performed in natural environment (dy-namic surface loading of fissile rock in active earthquake zones and in largeunderground excavations at 20-80 Hz with signal detection at distances of 400m from loading source) [20]. In these experiments, the subsequent increaseof dynamic loading above the threshold of subharmonic excitation has led tofast increase of the wide-band noise in the up-expanding frequency range. Theobservations have been attributed [20] to nonlinear dynamics of cracks.

2 Experiments

(T&C Rochester NY)

Power amplifier +50 dB

Force sensor

Waveform generator(Agilent HP 33120A)

Analyzer

acquisition

Pre−amplifier +40 dB(Panametrics 5660S)

(Panametrics V3052)

(Panametrics V3052)

Glass beads (2 mm in diameter)

Screw to apply a static force

Ultrasonic emitter

Ultrasonic receiver(Stanford SR 785)

14 c

m

Fig. 1. Experimental setup.

The experiment is conducted in a cylindrical reservoir filled with glass beadsof 2 mm in diameter (fig 1). It is possible to load this granular assemblagevertically with static pressure P0 in the range 32kPa ≤ P0 ≤ 320kPa providingestimated average static pre-loading strain ε0 of Hertzian contacts in the range1.8 10−4 ≤ ε0 ≤ 8.5 10−4. Acoustic waves (longitudinal or shear) are emittedby transducers of 4 cm in diameter at fundamental (pump) frequencies f0 of12kHz or 10kHz. The transmitted signal is received at a distance of 14 cmby a transducer predominantly sensitive to longitudinal acoustic waves. Themaximum strain (for both longitudinal and shear) in the emitted acoustic waveis estimated to be about ε max

a' 1.4 10−5. For the typical values of velocities

of acoustic waves (propagating in the granular skeleton) of the order of c '

450m/s, this corresponds to maximum accelerations of 75m/s2 significantlyexceeding one related to gravity.

In Fig. 2, the spectrum of the received signal is presented at different levelsof acoustic excitation (acoustic wave amplitude is a control parameter). Atlow excitation level (ε

a' 4.4 10−9, Fig. 2(a)) only the higher harmonics of

the pump wave nω0 (n = 2, 4, . . .) are visible in addition to fundamental fre-quency (ω0 = 2πf0). At higher level of excitation (ε

a' 4.4 10−7, Fig. 2(b))

subharmonic ω0/2 appears in the spectrum. In addition, it is possible to iden-

3

%

0 20 40 60 80−100

−80

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0

0 20 40 60 80−100

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0

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0

0 20 40 60 80−100

−80

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0

(b)

(c) (d)

(a)

PSfrag replacements

εa ' 4.4 10−9 εa ' 4.4 10−7

εa ' 1.4 10−6 εa ' 1.4 10−5

Spec

tralam

plitu

de

(dB

)Spec

tralam

plitu

de

(dB

)

Spec

tralam

plitu

de

(dB

)Spec

tralam

plitu

de

(dB

)

Frequency (kHz)

Frequency (kHz)Frequency (kHz)

Frequency (kHz)

Fig. 2. Observed spectra for different strain amplitudes of fundamental pump waveat f0 = 12kHz. The static pressure applied on the granular assemblage is ' 300kPa.

tify 3ω0/4, 3ω0/2 and 5ω0/2. It seems that ω0/4 and higher order subhar-monics (ω0/8, ω0/16, etc...) are not observed because of low sensitivity ofreceiver below ≤ 3kHz and because these subharmonics peaks fall on a lowfrequency noise shoulder. From Fig. 2(c,d) it is clear that subsequent increasein pump amplitude leads to preferential rise of this noise shoulder in compari-son with the amplitudes of the subharmonics and only ω0/2 and (2n + 1)ω0/2(n = 1, 2, 3, . . . ), are clearly visible.

−40 −30 −20 −10 0−70

−60

−50

−40

−30

−20

−10

01/2

3/2PSfrag replacements

Subhar

mon

ics

leve

l(d

B)

Fundamental level (dB)

Fig. 3. Levels of received subharmonics 1/2 (in squares) and 3/2 (in circles) asa function of the fundamental (12 kHz) wave level. The fundamental level 0dB

corresponds to ε max

a' 1.4 10−5. The applied static pressure ' 200kPa corresponds

to average static pre-strain ε0 ' 6.2 10−4.

In Fig. 3, the measurements of subharmonic amplitude and that of 3ω0/2are presented as a function of pump wave amplitude. It is clear that the

4

0 20 40 60 80

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0

20

2

1/2 3/2 3

7/25/2

1

4

PSfrag replacements

Spec

tral

amplitu

de

(dB

)

Frequency (kHz)

Electronicnoise level

Acousticnoise

Fig. 4. Observed spectra for different applied static pressure. From the lower curve

to the upper one, P0 ' 72, 91, 102, 138, 170, 213, 281 and 313 kPa, respectively. The

strain amplitude is ε max

a' 1.4 10−5 and the pump wave frequency is f0 = 10kHz.

appearance of 3ω0/2 in the spectrum follows the period-doubling bifurcation.

In Fig. 4 the spectrum of the received signal is presented for the fixed ampli-tude of the insonification (ε max

a' 1.4 10−5) when another control parameter

(average static loading) is varied. The broadening of the noise spectrum withincreased static force is observed.

There are several other features in our data which show systematic behaviors.For example, monotonic increase in the amplitudes of harmonics and of thesubharmonics as well as of noise with increase of either dynamic or staticloading correlates with data presented in Fig. 2 and Fig. 4.

3 Discussion

It might be clear from the introduction that we attribute the observed period-doubling phenomena (in frequency response of the acoustic energy transportthrough granular medium) to nonlinear dynamics of inter-bead contacts. Sev-eral less or more plausible scenarios could be imagined at a qualitative level(Fig. 5). The first of them is possible for the individual beads, which are con-tacting with surrounding skeleton only due to gravity (Fig. 5(a), bead (1)).

This bead does not carry any load transmitted through the assemblage andmight be called the “spectator” [21]. The accelerations in the acoustic field(estimated to be significantly higher than acceleration of gravity at the max-imum level of excitation) will definitely cause bouncing of this “spectator”between the neighboring beads of the skeleton. Then the scenario of period-

5

8

doubling bifurcation may follow that of the bouncing ball phenomenon [11,22–24], through the restoring force on the bead flight might be provided not onlyby the gravitational acceleration g but also by the interaction of the flying“spectators” with the neighboring beads. More than one cycle of the acousticmotion may occur between the impacts of the “spectator” with the surround-ing beads, and so the motion of the “spectator” can exhibit subharmonics ofthe exciting frequency. However, this scenario is less probable in comparisonwith some others because the number of beads, which are completely free ofexternal loading is expected to be significantly lower than the number of beadsthat are carrying at least some load.

(a)

(b)

(c)

(1)

(1)

(1)

(2)

(1)

(2)(2)

(1)

(2)

(1)

Fig. 5. Illustration of plausible scenarios responsible for the period-doubling bifur-

cation in the medium.

In Fig 5(b), the bead marked by (1) misses a single contact with the sur-rounding skeleton (with the bead marked by (2)). The acoustic loading maycause closing of the gap between the beads (1) and (2) at least during apart of the acoustic period (Fig. 2(b)). This contact might be called “weaklyunloaded”. In this case, the scenario of period doubling cascade and the tran-sition to chaotic behavior may follow those predicted theoretically [24–29] andobserved experimentally [12,13] for the impact oscillations. The mathematicalformalism of the theory of grazing bifurcations [12,30,31] could be useful forthe understanding of the nonlinear dynamics at those levels of acoustic excita-tion when the tapping of one bead by another is just starting. It is importantthat the theory of the impact oscillations is sufficiently developed not onlyfor absolutely rigid constraints but also for the impacting Hertzian contacts[29]. The period-doubling bifurcation in the dynamics of the impact oscillatorsleads to such dynamics that the contacts (1) and (2) are colliding only once

6

in two periods of forcing acoustic wave. Both theoretical investigations andexperiments are usually limited to forcing frequencies that are of the orderof the resonance frequency ωres of the oscillators or higher [25–29]. The theo-retical prediction of subharmonic oscillations for ω0/ωres & 1 correlates withthe expected relation of period-doubling bifurcation with subharmonic reso-nances [25] (similar to the situation in nonlinear vibrations of gas bubbles [8]).From this point of view, for the subharmonic excitation, the granular systemshould contain local oscillators with resonance frequencies ωres of the orderof or less than the acoustic loading frequency ω0. Using the measured valuefor sound velocity (c ∼ 450m/s) and the value of bead diameter (a = 2mm)the resonance frequency of an element containing a contact and a bead sup-porting the average static deformation ε0 is estimated by a cut-off frequencyfcut for a chain of beads as fres = fcut = c/(πa) ∼ 70kHz. This value sig-nificantly exceeds the frequency of acoustic excitation f0 ∼ 12kHz. It can beconcluded that these are the beads with contacts that are weakly loaded (incomparison with average) who contributes to subharmonic cascade leading tochaos. In particular, the contacts with the neighbors of the bead numbered(1) in Fig. 5(b) should be weak in comparison with average contacts. To esti-mate the preloading of the weak contacts providing the resonance frequencyfµ′ ∼ µ′fres ≤ f0 (where µ′ ' 2 10−1) it should be taken into account that theresonance frequency is proportional to the square root of the rigidity. Thenthe rigidity of these contacts Kµ can be estimated as Kµ ∼ µ′2K0 ∼ 4 10−2K0,where K0 is the rigidity of contacts subjected to average loading. Because forthe nonlinear Hertzian contacts the rigidity is proportional to the square rootof the preloading strain, then the preloading εµ′ of these weak contacts canbe estimated as εµ′ ∼ µ′4ε0 . 1.6 10−3ε0. Consequently, the individual beadsinteracting with the neighbors through these weak contacts are the most plau-sible candidates for realization of the impact oscillations scenario for perioddoubling cascade in granular medium. However, it is worth mentioning thatthe resonance frequency of an oscillator is inverse proportional to the squareroot of its mass. Consequently, the diminishing of the resonance frequency incomparison with one of a single bead in a linear chain (subjected to averageloading) might be due not only to weakened contacts but also due to increasedmass of an oscillator if it is composed of several beads (of a cluster). This isanother opportunity to explain the existence in a granular assemblage of theoscillators with low resonance frequencies to be subjected to deeper theoreticalanalysis in the future.

Currently, it seems more plausible that low frequency resonances are duerather to weak contacts than to clustering of the beads. The recent experi-ments [32] on the demodulation of the acoustic waves in granular assemblagesdemonstrated that nearly half of the contacts are subjected to strains thatare less than 10% of the average strain ε0. The transition in the amplitude ofthe demodulated signal from ∼ ε 2

a to ∼ ε 3/2

a related to the initiation of con-tacts clapping provides the estimate (εa ∼ 0.1ε0) for the characteristic strain

7

(/

under which the weak contacts are localized. The experimental results on theexcitation of the subharmonic and the theoretical estimates presented abovedemonstrate that these new nonlinear acoustic experiments are sensitive evento significantly weaker loaded contacts (with εµ′ ∼ 10−3ε0).

An interesting point is that a significant part of the weakly loaded contactswith estimated preloading εµ′ . 1.6 10−3ε0 is expected to start clapping them-self before the experimentally observed threshold ε exp

a for the subharmonicexcitation (ε exp

a ∼ 1.3 10−3ε0). This observation leads to the hypothesis thatthe existence of the initial gap between the contacts (Fig. 5(b)) is maybenot necessary. Another initial configuration of beads different from that inFig. 5(b) is presented in Fig. 5(c). In comparison with the previous case, thereis initial contact between the beads (1) and (2) but it can be easily openedby acoustic loading during a part of the acoustic loading period. This contactmight be called “weakly loaded”. The difference between the clapping con-tacts in the configurations depictured in Fig. 5(b) and in Fig. 5(c) is that inthe former clapping starts with increasing acoustic wave via a short periodimpact (grazing impact) while in the latter via a short period opening of thegap. The experiment on the interaction of the acoustic wave with the planeinterface between two solids [14] indicate that this can lead to completelydifferent scenarios of nonlinear dynamics when the amplitude of loading isfurther increased. It is expected that the scenarios in Fig. 5(b) and Fig. 5(c)are more probable than Fig. 5(a).

All of the proposed mechanisms of pronounced nonlinear dynamics (Fig. 5)are expected to be suppressed when the static loading increases (because thelatter should induce diminishing in the number of weakly loaded contacts).That is why, the observed (Fig. 4) increase in the amplitude of the receivedsignal (at fundamental frequency, at its superharmonics and subharmonics,and of the noise level) is believed to be due to even faster diminishing of thesound absorption and scattering with increasing compaction (consolidation)of the granular assemblage.

Another question worth discussing is whether the processes of synchronousamplification of the subharmonic are effective in the system under considera-tion or not? In other words if the subharmonic signal emitted by the clappingcontacts might be significantly amplified in the field of the fundamental waveor not? It is well known that for the longitudinal waves the process of differencefrequency generation is allowed in collinear geometry of ω and ω/2 propagationωLA − (ω/2)LA → (ω/2)LA [33]. The amplitude of the scattered subharmonicwave (in the right-hand-side) is proportional to the product of the amplitudesof the interacting pump wave (ωLA-wave) and the initial subharmonic wave (inthe left-hand-side). Because of this, it should be admitted that the number ofsubharmonic phonons can increase due to the interaction with the pump waveand that actually the (ω/2)LA phonons can stimulate the following process of

8

spontaneous decay of the pump wave ωLA → (ω/2)LA + (ω/2)LA. The latterprocess results in a net stimulated amplification of the subharmonic if the am-plification in the nonlinear process over-compensates attenuation of (ω/2)LA.In the case of the shear pump wave, the process ωSA − (ω/2)LA → (ω/2)LA

(and correspondingly, ωSA → (ω/2)LA + (ω/2)LA) are forbidden in isotropicmaterials by the laws of momentum and energy conservation (not only incollinear wave interactions but also in non-collinear interactions as well [34]).Importantly, even if the considered processes are allowed, they do not seemto be relevant to our experiments. In fact, the shortest possible amplifica-tion length of the subharmonic can be estimated (by neglecting wave atten-uation) as the distance of shock formation in the initially sinusoidal planepump wave `nl ' λω/(2πΓ2εa) [34]. Here, εa is the strain amplitude in thepump wave and Γ2 is the parameter of quadratic acoustic nonlinearity. Thelatter can be estimated for our system from the experiments [32] on the self-demodulation of high-frequency acoustic bursts to be Γ2 . 2.5 103 for thestatic pre-strain ε0 ∼ 10−4. In the experiments presented above, the pre-strainis higher 6 10−4 . ε0 . 8.5 10−4 (corresponding to 200kPa . P0 . 300kPa).Taking into account that the parameter of quadratic nonlinearity for non-clapping Hertzian contacts varies inverse proportionally to the square root ofthe pre-strain, we estimate Γ2 . 10−3. Consequently, the amplification lengthof the subharmonic is estimated as `nl & 50cm for the maximum acousticdeformation ε max

a ' 1.4 10−5 achieved in our experiments.

This length significantly exceeds not only the effective interaction length (con-trolled by the diffraction length of the pump wave ∼ 4cm) but even the ob-servation distance (L ∼ 14cm). It is well known that the process of clapping,which is not taken into account here in the estimates of Γ2 and `nl, tendsto saturate the quadratic non-linearity [32], i.e. to produce a lower Γ2 and ahigher `nl compared to the non-clapping regime for the same applied dynamicstrain. Moreover, if the 3-D geometry of the pump beam would be taken intoaccount, i.e. the diminishing of the pump wave amplitude at the scale of thediffraction length, this would lead to even longer amplification lengths [34].Consequently, the process of subharmonic amplification is estimated to benegligible in our experiment.

To finish the discussion it is worth mentioning that the goal and the results ofthe experiment describe above are completely different from well-known exper-iments [35,36] where the acoustic loading of the granular media at frequencyf0 ∼ 10kHz has been used to study the slow dynamics of granular assemblagesthrough the measurements of low frequency (below 10Hz) fluctuations in thetransmitted acoustic signals. In our experiments this slow dynamics can alsomanifest itself as slow (at the time scale ≥ 2s) fluctuations in the amplitudesof harmonics, subharmonics and noise level. It might be caused by structuralrelaxation of the bead assemblage under the acoustic activation [35]. Looselypacked glass beads are metastable and acoustic action might play the role of

9

an effective temperature [37] (i.e. of a source of thermal fluctuations) initi-ating the system rearrangment towards a stable state. The other mechanismof slow dynamics might be thermoelastic expansion of contacts (which arenatural regions of strain concentration and acoustic energy absorption) [38].However, more detailed discussion of our preliminary experimental results andestimates of the characteristic slow time scales are beyond the scope of thepresent communication.

4 Conclusions

The first experimental observation of the excitation of subharmonics above thecritical amplitude of sinusoidal acoustic wave propagating in unconsolidatedassemblage of beads is reported. With the increasing amplitude of the pumpwave, the number of the detectable combination frequencies grows. The broad-ening of the frequency range, where combination frequencies are observed, isaccompanied by broadening of the noise spectrum as well. Significant increaseof the noise level with increasing pump amplitude is documented. The bifur-cation phenomenon, opening the subharmonic route to chaos, is attributed tohighly nonlinear forced oscillations of intermittent (clapping) contacts. Theseintermittent contacts appear or by opening (during part of the acoustic period)of the statically weakly loaded contacts (clapping) or by closing (during partof the acoustic period) of the tiny gaps between the neighbor grains (tapping).The bifurcation sequence for the tapping scenario is expected to qualitativelyfollow the one predicted theoretically and observed experimentally earlier foran individual impact oscillator. Both clapping and tapping has been observedbefore in the interaction of the acoustic wave with a single plane contactinterface between two solids. However, the observations described above arebelieved to be the first where the acoustic wave interaction with a distributed

system of nonlinear contacts leads to excitation of subharmonics and noise.

The conducted experiments confirm a predominant sensitivity of the nonlin-ear acoustic phenomenon to weakly loaded (and weakly “unloaded”) contacts,which might be forced to clapping (tapping) by the acoustic wave with anamplitude which is significantly less than the average loading. Further devel-opment of the nonlinear acoustic methods could provide a unique tool for thediagnostics of the contacts that are several orders of magnitude weaker thanthose carrying the average (static) load.

10

Acknowledgments

When conducting this research we greatly beneficiated from the fruitfull dis-cussions with V. Zaitsev and V. Nazarov (from the Institute of Applied Physics,Nyzhnii Novgorod, Russia). This work is supported by a contract with DGANo 00.34.026.00.470.75.65.

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Effet non linéaire d'auto-démodulation d'amplitude dans