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A theory of elastic waves In isotropic media Usually solution of this equation is represented as a sum is a scalar potential is a vector potential however why not to do differently?
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A model of the Earthquake surface waves
V.K.Ignatovich. FLNP JINR
STI2011 June 8
This report is along the papersV.K. Ignatovich and L.T.N. Phan.
Those wonderful elastic waves. Am.J.Phys.
v. 77, n. 12, pp. 1093-I17, (2009)
A.N. Nikitin, T.I. Ivankina, and V.K. Ignatovich The Wave Field Patterns of the Propagation
of Longitudinal and Transverse Elastic Waves in Grain-Oriented Rocks
Physics of the Solid Earth, 2009, v. 45, n. 5, pp. 424-436
And a little bit more
A theory of elastic waves In isotropic media
ljlj tu 22 )()( uu jjj u
Usually solution of this equation is represented as a sum
][ φu is a scalar potential
φ is a vector potential
however why not to do differently?)exp(),( tiit rkAru
22)(2 ijuF
u
i
j
j
iij x
uxuu
21
ijijijij uuF 2)( u
)()(22 uuuu t
)exp(),( 0 tiiut rkAru 10 Au
)()( 22 AkkAAkkA k
)()(22 AκκAAκκA k
2: k
kkκ κttA 32211
κtt ,, 21
02,122 k
02 322 k
22222,1 tck 2tc
2
2223 2 lck
22 2tl cc
)()(22 uuuu t
2/ lt ccAll this is trivial. Reflection from interfaces is less trivial
Reflection from a free surface
2A 2A3A
)exp()exp()exp()( 3332222222 zikrzikrzikz rri AAAu
ln
)()exp(),( |||| ztiit urkru
2||
22 kck nn tl cccc 23
03 kcAt such a critical angle A Longitudinal Surface wave appears
Calculations of reflection amplitudes0 jiji n ijijij u 2)( u
0)()( unununΣ tiizik
rzik
rzik
i eereret ||||3223322222),( rkAAAru
03322222 rri rr ΣΣΣΣ
22222 )( iiii k AAnkΣ 22222 )( rrrr k AAnkΣ
333333 )( rrrr kk AAnknΣ
2||2 kki nlk 2||22 kkki nlA
:
2
||22||
22
2
2k
kkkki
nlΣ
2
||22||
22
2
2k
kkkkr
nlΣ
3
2||
22||3
3
2k
kkkkr
nlΣ
03322222 rri rr ΣΣΣΣ
2
||22||
22
2
2k
kkkki
nlΣ
2||22
||3
3
23222
21
kkkk
kkrr
||2
2||
22
3
23222 2
1kkkk
kkrr
2
||3222
||22
2||
22||2
2
332
4
22
kkkkk
kkkkkkr
2
||3222
||22
22||
22
2||32
224
4
kkkkk
kkkkkr
nl,lc
k 3 2
||233 kkk
2
||3222
||22
2||
22||2
2
332
4
4
kkkkk
kkkkkkr
22
||22
2||32
22||
22
2||32
224
4
kkkkk
kkkkkr
22
2
3 sinsin ck
k
22232
sinsin)sin()2sin(2)2(cos
)2cos()2sin(sin2
c
cr
-- angle of incidence
sin2|| kk
cos22 kk
222sin ltc cc
)2tan(sin2)(32 cccr
1)( 222 cr
2||
2222
2||
233 kcckkkk lt
1)(22 cr
sincos2 nlA r lA 3r
cccr 2tansin2)(32
sincos2 nlA i
llA 8.612tantancos2 ccc
65.0sin ltc cc
462 A
1)(22 cr
lAAAA cccrri rr cos22tansin23322222
71)(| 22 Atliss ccEEQ
c
l12tantancos2 ccc
ctl rrccQ sin)()( 232
232
2
222
232
sinsin)sin()2sin(2)2(cos
)2cos()2sin(sin2
c
cr
Tomas Lokajicek, Vladimir Rudajev
V.K. Ignatovich. A proposal of a UCN experiment to check an earthquake waves model.Europhys. Lett. 92 (69002-p1-4) 2010.
Experiments byLokajicek Tomas, Rudajev Vladimir
4E-005 6E-005 8E-005 0.0001time of flight [s]
-0.0 8
-0.0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
90 deg., 30 dB
80 deg., 30 dB
70 deg., 30 dB
65 deg., 30 dB
60 deg., 30 dB
55 deg., 30 dB
50 deg., 30 dB
40 deg., 36 dB
30 deg., 36 dB
20 deg., 36 dB
10 deg., 36 dB
5 deg., 36 dB
S5_S5_signal
4E-005 6E-005 8E-005 0.0001tim e of flight [s]
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
90 deg., 30 dB
80 deg., 30 dB
70 deg., 30 dB
65 deg., 30 dB
60 deg., 30 dB
55 deg., 30 dB
50 deg., 30 dB
40 deg., 36 dB
30 deg., 36 dB
20 deg., 36 dB
10 deg., 36 dB
5 deg., 36 dB
S5_S5_reference
90 deg.
0 deg.
113,5 mm
90 deg.
recieverS-wave transducer5 MHz resonant frequencydiameter 5 mm
transmitterS-wave transducer5 MHz resonant frequencydiameter 5 mm
material:
in 90 deg. P-wave time propagation: 41,8 s
perpsexthickness: 20 mmP-wave velocity: 2,72 km/sS-wave velocity: 1,37 km/s
[ ]S-wave time propagation: s82.9 [ ]
reference transducerP-wave transducer1 MHz resonant frequencydiameter 10 mm
012tantancos2 lA ccc
5.0sin l
tc cc
0)30cos()60cos()60sin()30sin(12tantancos
ccc
62tansin4 2 cc
232rQ
02 A
57.0sin l
tc cc
llA 4.112tantancos2 ccc
steel
So, to observe an effect we need a material with
ct/cl>0.6
Anisotropic media
jlljut
2
2
ijklc -- a set of phenomenologocal constants
klijklij uc
j
l
l
jjl x
uxu
u21
In general 21 constants
222 )(2 jljljll uauuF
lmjmmlmjjllljljl
jl auaauauuuF
22
)()(22 AkkAA k
)]())[(()])(()([ 2 AakAakakAkakAaa k
But anisotropy means a vector and an additional constant. So we can define
)exp(),( tiit rkAru
kkκ cbκA |][|][ κaκac
κa
aκaκκcb
][
))(())(()( 222 AaakAkkAak k
))(()(2 AkakAaa k
0)( 222 akk
κaab
0)(2 22 kakka
0)(2)(42 222 akkaakk
All we need is a linear vector algebra
κbc ,,
0)(2 22 kakka
0)(2)(42 222 akkaakk
)cos4)(1(2sin 2222 czz
22sin4))cos41(1()cos41(1
12222222
cc
z
22 kz 222 2 tl ccc
2cos1)( vt
)(cos κa
22sin4))cos41(1()cos41(1
1)(2222222
cc
vql
22sin4))cos41(1()cos41(1
1)(2222222
cc
vqt
1tc
58.1 tl ccc
5.0
)()( vqlcc tql
)()( vqtcc tqt
It is important to saythat we cannot exclude
by averaging of values over alldirections of propagation,
because all the values depend on
22 )()(cos aκ
Polarization of waves
babaκκabaκ
babaκκA ))((2
)()(41
))((2122222
2
EV
V
ql
qlql
κabaκbabaκ
babaκκA ))((2
)()(41
))((2122222
2
EV
V
qt
qtqt
122 qtql AA 0qtqlAA
))((212 abaκ VbκA
2
)()(16)(411)(411 22222222 abaκaκaκ
cc
κ a
b
22
c
12,qtqlV
In an anisotropic medium propagate plane waves of only 3 modes
• transverse with Аt~[kxa] and ct2=ct0(1+)
• quasi transverse with Аqt in the plane [k,a]
• quasi longitudinal with Аql in the plane [k,a]
quasi longitudinalquasi transverse
)exp(),( tiit krAru
a akk
2Atransverse
a k1A 3A
Reflection of a quasi transverse wave from a free surface
0,,, rqlltrqtttiqt rr ΣΣΣΣ
)()()( AknAnkAknΣ
)()()())(())(( AakAkaanAnkaAakna
One can find an analytical solution
of two reflected waves
2.0)cos( a)(
sin)(
sin)(
sin
qtrqlrql
rql
rqtrqt
rqt
VVV
5.222
c
5.0
nl a
a rqt
rql
quasi longitudinal wave becomes surface one at 6.0
It seems possible to find such a direction of vector a
that for given elastic parameters the amplitude of the
surface longitudinal wave becomes maximal.
2)2()2(
2
2
ql
qt
VV
For instance
Summary• Reflection of elastic waves from free surfaces is
accompanied by beam splitting.• At some critical angle of the incident shear
wave polarized in the incidence plane a longitudinal surface wave is created.
• Its amplitude and energy can be large, and its polarization along the surface is alike to devastating earthquake waves.
• For observation of such waves the materials with ratio ct/cl>0.6 are needed.
Thanks