16
!"#$%&'()*+ -./0#*+% 123 !"#$%&" ()$"&*)+, -$$.,//0001)2$&34*3151%6$35731)#18./*()(%&)/+"#$%&"/ !"#$%&'()*+ 405(% ! 9":"&)$"; *: ) %:2$)4+" 4)#5<&3%:; )$(32.-"&" ! =): $&):2.3&$ (3(":$%( ):; ":"&<7 3>"& +3:< ;*2$):#"2 ! =): *:;%#" (*?*:< 3@ $-" )$(32.-"&" !=): .+)7 #&%#*)+ &3+"2 *: $-" ;">"+3.(":$ 3@ .+):"$)&762#)+" )$(32.-"&*# 2$&%#$%&"

0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

  • Upload
    others

  • View
    18

  • Download
    0

Embed Size (px)

Citation preview

Page 1: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

!"#$%&'()*+,-./0#*+%,123

!"#$%&"'()$"&*)+, -$$.,//0001)2$&34*3151%6$35731)#18./*()(%&)/+"#$%&"/

!"#$%&'()*+,405(%

! 9":"&)$";'*:')'%:2$)4+"'4)#5<&3%:;')$(32.-"&"! =): $&):2.3&$ (3(":$%( ):; ":"&<7 3>"& +3:< ;*2$):#"2! =):'*:;%#"'(*?*:<'3@'$-"')$(32.-"&"!=):'.+)7'#&%#*)+'&3+"2'*:'$-"';">"+3.(":$'3@'.+):"$)&762#)+"')$(32.-"&*#'2$&%#$%&"'

Page 2: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

!"#$%&'()*+,405(%

A)+47'BCDDEF

66$$%%%%77..,,440055(( 11''$$))**88$$//""0099,,$$%%++**999900""**$$// ::))0055**"".. 440055(( 55(())""**++0099 $$%%++**999900""**$$//

;)(%%<)(,+$$)-*/0"(!)&<"62#)+"')$(32.-"&*#'(3$*3:2'2)$*2@7'-7;&32$)$*#'"G%*+*4&*%(1'

H:'$-*2'#)2"'$-"'.&"22%&"'! #):'4"'%2";')2'$-"'>"&$*#)+'#33&;*:)$",

,'>"&$*#)+'>"+3#*$7'*:'.&"22%&"'#33&;*:)$"

,'<"3.3$":$*)+'

" # dp dt!

d" = gdz

(1978)

I:')..&3?*()$"'&"+)$*3:2-*.'4"$0"":'! ):;'",

" # dp dt ~ w /$g

! " ! # $ # " #%!& ! ,'.&"22%&"J'# , )+$*$%;"'

Page 3: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

;)(%%<)(,+$$)-*/0"(

;)(%%<)(,+$$)-*/0"(KL&*(*$*>"'"G%)$*3:2M

"! v "t

= #! v $ %! v #& "

! v "p

# fk ' ! v #%(

"#"p

= $RTp

"#! v + $%

$p= 0

"#"t

= $! v % &# $'

"#"p

+1cp

pS

p(

) *

+

, -

R /C p

˙ H

-3&*N3:$)+'(3(":$%('"G1

#3:$*:%*$7'"G1

2$)$"'"G1 O -7;&32$)$*#'"G%*+*4&*%(

$-"&(3;7:)(*#2'"G1

f = 2"sin#

˙ H

,'=3&*3+*2'.)&)("$"&

,';*)4)$*#'-")$*:<'&)$"'BP/(Q/2F

' ,'%:*$'>"#$3&'*:'$-"'>"&$*#)+';*&"#$*3:

!!! ,'>"&$*#)+'>"+3#*$7

,'-3&*N3:$)+'>"+3#*$7

Page 4: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

"! v "t

= #! v $ %! v #& "

! v "p

# fk ' ! v #%(

:($%")$&'*+,=9$4

R-" &)$*3 4"$0"":'$-"')##"+"&)$*3:'BST/!F'):;'$-"'=3&*3+*2'@3&#"'B@SF

=3&*3+*2'@3&#"

.&"22%&"'<&);*":$'@3&#"

(m s-2)

U3& 27:3.$*# 2#)+" BVCWWW 5(F (3$*3:2 *: X)&$-Y2 )$(32.-"&",-3&*N3:$)+'2#)+"'!'Z'CWWW'5(.&"22%&"'2#)+"'L'Z'CWWW'-L)>"+3#*$7'2#)+"'S'Z'CW'(/2$*("'2#)+"'!/S'Z'CW[ 2=3&*3+*2'.)&)("$"&'@'Z'CW6\ ]N

Ro "UfL

=3&*3+*2'@3&#"

.&"22%&"'<&);*":$'@3&#"

.&"22%&"'#3:$3%&20*:;

:($%")$&'*+,=9$49"32$&3.-*# @+30 4"#3("2 ) <33;)..&3?*()$*3: @3& 2()++ ^32247:%(4"&2

fk " ! v g = #$%

_"@*:*$*3: 3@ <"32$&3.-*# @+30

! v g =1f! k " #$

ug = "1f#$#y, vg =

1f#$#x

3&

"#! v g = 0

$ = 0

9"32$&3.-*# @+30 *2 $036;*(":2*3:)+

9"32$&3.-*# @+30 *2 :3$ ) <33; )..&3?*()$*3: :")& $-" "G%)$3& B$ *2 2()++F 3&@3& 2#)+"2 ` aBCWW5(F B% *2 2()++F 0-"&" ^3 Bb S/@!F *2 +)&<"1

()

Page 5: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

>'()#09,4*/-

_*@@"&":$*)$*:<''''''''''''''''''''''''''''''''0*$-'&"2."#$'$3'! ):;'%2*:<''''''''''''''''''''''J'K$-"&()+'0*:;K'&"+)$*3:'*2'34$)*:";,'

3&''

#3+; 0)&(

.&"22%&"'<&);*":$'@3&#"

=3&*3+*2'@3&#"

]3+$3:'BTWW\F

fk " ! v g = #$%

"#"p

= $RTp

"! v g"p

= #Rfp! k $ %T

"ug"p

=Rfp"T"y

"vg"p

= #Rfp"T"x

?0"*"<-(@09"*"<-(,+)$%%,%(+"*$/,$=,A0)%B%,0"#$%&'()(

R"(."&)$%&"

X)2$0)&;'0*:;

Page 6: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

C$)"*+*".R-"'&'(&)*+,'-.J'/J')43%$')'#+32";'#3:$3%&'*2';"@*:";')2'$-"'+*:"'*:$"<&)+')+3:<'$-"'#3:$3%&'3@'$-"'#3(.3:":$'3@'$-"'>"+3#*$7'>"#$3&'$-)$'*2'+3#)++7'$):<":$'$3'$-"'#3:$3%&,'

C =! v " d" l

l# =

! v cos$dll# = (% &

! v )" " n dSS##

%:*$'>"#$3&':3&()+'$3'$-"'2%&@)#"

BA$35"2Y'$-"3&"('0)2'%2";1F

_*>*;*:<'$-"'#*&#%+)$*3:'"/ ":#*&#+*:<')'2()++')&")'"0J'):;'#3:2*;"&*:<'$-"'+*(*$'"0!WJ'0"'<"$'$-"'1-(,'&',2,'

" =#C#S

= ($ %! v )&" k

='v'x

('u'y

]3+$3:'BCDDTF

D<0%*@E($%")$&'*+,0&&)$F*#0"*$/X>":'*:')'2()++'^32247':%(4"&'&"<*("J':3:6<"32$&3.-*#'B)<"32$&3.-*#F'#3(.3:":$2'.+)7'#&%#*)+'&3+"2'*:';&*>*:<'>"&$*#)+'0*:;2'):;'$"(.3&)+'#-):<"21

R-"'&")+'0*:;'*2';*>*;";'*:$3'<"32$&3.-*#'0*:;'):;')<"32$&3.-*#'0*:;1

R-"'&)$*3'3@'$-"'()<:*$%;"2'3@'$-"')<"32$&3.-*#'):;'<"32$&3.-*#'0*:;2'*2'$-"'2)("'3&;"&'3@'()<:*$%;"')2'$-"'^32247':%(4"&'B)&3%:; W1C'*:'X)&$-Y2')$(32.-"&"F

!)<&):<*): ;"&*>)$*>",'

! v = ! v g +! v a

! v g "1f0

! k # $%

ddt

=""t

+! v # $+%

""p

dg

dt"##t

+! v g $ %

R-"');>"#$*3:'$"&('*2')..&3?*()$";'47');>"#$*3:'47'<"32$&3.-*#'0*:;,

Page 7: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

G("0@&90/(,0&&)$F*#0"*$/

R-"'@*&2$63&;"&'R)7+3&'2"&*"2')..&3?*()$*3:'3@'$-"'=3&*3+*2'.)&)("$"&'$ ,

0-"&"'

c*;6+)$*$%;"'4"$)6.+):"',XG%)$3&*)+ 4"$)6.+):"','

f = 2"sin#~ 2" [sin#0 + (# $ #0)cos#0]= f0 + % y

f0 = 2"sin#0

$ =2"acos#0 ~

dfdy

f = f0 + " y

f = " y

?

7

W#W

! v = ! v g +! v a

! v g "1f0

! k # $%

D<0%*@E($%")$&'*+,5$)"*+*".,(H<0"*$/H:$&3;%#*:<'4"$)6.+):"')..&3?*()$*3:'$3'$-"'G%)2*6<"32$&3.-*#'!)<&):<*):;"&*>)$*>"'):;'&"$)*:*:<'2()++ G%):$*$*"2 $3 $-" @*&2$'3&;"&J'$-"'&)$"'3@'#-):<"'3@'$-"'<"32$&3.-*#'0*:;'*2'<*>":'47'

""t

+! v g # $

%

& '

(

) * ! v g = +( f0 + ,y)

! k - (! v g +

! v a ) +$.

~ + f0! k - ! v a + ,y

! k - ! v g

B'''''''''''''''''''''''''''''''0)2'%2";F

A*:#"'<"32$&3.-*#'0*:;'*2':3:;*>"&<":$'B'''''''''''''''''FJ'$-"'#3:$*:%*$7'"G1'*2

"#! v g = 0

$ = 0

"#! v a +

$%$p

= 0

a."&)$*:<'&3$)$*3:'B(*F'$3'BCF'):;'%2*:<'BTFJ'0"'-)>"'$-"'G%)2*6<"32$&3.-*#'>3&$*#*$7'"G%)$*3:1''

BCF

BTF

Page 8: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

D<0%*@E($%")$&'*+,5$)"*+*".,(H<0"*$/

d3&$*#*$7'#-):<"2'0*$-'$*("'$-&3%<-6 );>"#$*3:'3@ )423+%$" >3&$*#*$7'B'''''''''''''F'47'<"32$&3.-*#'0*:;'B F'6 >"&$*#)+';*>"&<":#" B-3&*N3:$)+ ;*>"&<":#"F

"#g

"t= $! v g % &(#g + f ) + f0

"'"p

"g #$vg$x

%$ug$y

=&2 ' ( f0

,'<"32$&3.-*#'>3&$*#*$7

+, - . /0,

]3+$3:'BTWW\F

6$%%7.,405(

!"$'%2'#3:2*;"&')'$036;*(":2*3:)+'(3$*3:'B! b'WF'

""t

+! v g # $

%

& '

(

) * (+g + f ) = 0

! I423+%$"'>3&$*#*$7'B''''''''''''F'*2'#3:2"&>";')+3:<'$-"'<"32$&3.-*#'0*:;' 1

I 4)2*# 2$)$"'0-"&"')'-3(3<":"3%2'N3:)+'@+30'"?*2$2,

+, - . /0,

fu = "#$ #y

" g = 012

?

7 3) 45678

3) 9:544

Page 9: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

6$%%7.,405(

R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"'<"3.3$":$*)+'."&$%&4)$*3:

" # g =$2 " %

f0, " u g = &

1f0

' " % 'y, " v g =

1f0

' " % 'x

A%42$*$%$*:<'$-"2"'*:$3'BCF'):;'&"$)*:*:<'@*&2$'3&;"&'$"&(2'3:+7J'0"'<"$'

""t#2 $ % + u

""x#2 $ % + &

" $ % "x

= 0

R-"';">*)$*3:'@&3('$-"'4)2*#'2$)$"'*2';":3$";'47'BFY',

""t # $ g + (u + # u g )

" # $ g"x

+ # v g" # $ g"y

+ # v g% = 0 BCF

6$%%7.,405(

A)+47 BCDDEF

c " u = " #k 2 + l2

3 N3:)+'0)>":%(4"&* ("&*;*3:)+'0)>":%(4"&& N3:)+'.-)2"'>"+3#*$7

! L&3.)<)$*3:'3..32*$"'$3'$-"'.+):"$)&7'&3$)$*3:

! $ "@@"#$'B+)$*$%;"'>)&*)$*3:'3@'$-"'=3&*3+*2'.)&)("$"&'$F'*2':"";";1

! !3:<"& 0)>"2 B2()++"& 3F'.&3.)<)$"'@)2$"&1'

L322"22*:<'):<%+)&'(3(":$%('*:'$-"';*&"#$*3:'3..32*$"'$3'$-"'.+):"$)&7'&3$)$*3:'

I22%(*:<')'0)>"'23+%$*3:'''''''''''''''''''''''''''''''''''''''''''''''''''J'$-"'.-)2"'>"+3#*$7'3@'^32247'0)>"'*2'34$)*:";,'

" # = ˆ # exp[i(kx + ly $ kct)]

Page 10: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

6$%%7.,405(

A)+47 BCDDEF

""t

+! v g # $

%

& '

(

) * (+g + f ) = 0

X)&$-

A)$%&:

P%.*$"&

Page 11: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

:)05*".,405(H:'=)&$"2*):'#33&;*:)$"2J'0*$-3%$'$-"')22%(.$*3:'3@'-7;&32$)$*#'"G%*+*4&*%(J'$-"'<3>"&:*:<'"G%)$*3:2')&",

"u"t

+ u"u"x

+ w"u"z

=1#"p"x

"w"t

+ u"w"x

+ w"w"z

= $1#"p"z

"#"t

= $"(#u)"x

$"(#w)"z

"%"t

+ u"%"x

+ w"%"z

= 0

% =p#R

pSp

&

' (

)

* +

R /C p

R

-3&*N3:$)+'(3(":$%('"G1'B?6)?*2'3:+7F

#3:$*:%*$7'"G1

>"&$*#)+'(3(":$%('"G1

$-"&(3;7:)(*#2'"G1

:)05*".,405(XG%)$*3:2'@3&';*2$%&4):#"2,

-3&*N3:$)+'(3(":$%('"G1'B?6)?*2'3:+7F

#3:$*:%*$7'"G1

>"&$*#)+'(3(":$%('"G1

$-"&(3;7:)(*#2'"G1

" # $ u #t

= %# $ p #x

" # $ w #t

= %# $ p #z

% $ " g

# $ " #t

= %" # $ u #x

%#(" $ w )#z

1& # $ & #t

+ $ w N 2

g= 0

$ " =$ p

cs2 % " $

& &

cs2 =

cpcvRT

N 2 = g d ln" dz

2 23%:;'2."";

4%37):#7'@&"G%":#7

Page 12: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

:)05*".,405(H23$-"&()+')$(32.-"&",

N 2 = g cpT

" (z) = "s exp(#z /H)

A%42$*$%$*:<'$-"'0)>"'23+%$*3:'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''Be @&"G%":#7F'*:$3'$-"'<3>"&:*:<'"G%)$*3:2'4"@3&"J'):'"G%)$*3:'@3&'$-"'>"&$*#)+'>"+3#*$7'" *2'34$)*:";')2'

" w (x,z,t) = ˆ w (z,t)exp[i(kx +#t)]

d2(" ˆ w )dz2 +

1H

d(" ˆ w )dz

+# 2

cs2 $ k 2 +

N 2k 2

# 2

%

& '

(

) * " ˆ w = 0

ˆ w (z) = W (z)exp(z /2H) " # ˆ w 2(z)$W 2(z)

=3:2*;"&*:<'$-"')(.+*$%;"'<&30$-'0*$-'-"*<-$'*:')'2$&)$*@*";')$(32.-"&"J'0'*2')22%(";'$3'-)>"'$-"'@3&(

:)05*".,405(R-":'$-"'"G%)$*3:'4"#3("2

d2Wdz2

+"2

cs2 # k

2 +N 2k 2

"2#

14H 2

$

% &

'

( ) W = 0

I22%(*:<')'0)>"'23+%$*3:''''''''''''''''''''''''''''B(,'>"&$*#)+'0)>":%(4"&FJ'$-"';*2."&2*3:'&"+)$*3:'*2'34$)*:";,

W " exp(imz)

m2 ="2

cs2 # k

2 +N 2k 2

"2#

14H 2

A3+%$*3:2'@3&')#3%2$*#6<&)>*$7'0)>"'B#3::"#$";'$3'23%:;'0)>"F'):;'*:$"&:)+'<&)>*$7'0)>"'"?*2$1

I..&3?*()$"'23+%$*3:'@3&'*:$"&:)+'<&)>*$7'0)>"'*2

;< " =<'<

'< - >< - ?@A<

Page 13: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

:)05*".,405(U3&')'+)&<"6-3&*N3:$)+'2#)+"'0)>"2'B$7.*#)+'*:'.+):"$)&7')$(32.-"&"2FJ

6 !3:<'."&*3;'0)>"2'-)>"':")&6-3&*N3:$)+'.-)2"'2%&@)#"2'6 A-3&$'."&*3;'0)>"2'-)>"'$*+$";'.-)2"'2%&@)#"2

'<

>< " ;<

=<

]3+$3:'TWW\

BCDfCF

Page 14: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

A)+47 BCDDEF

I(.+*$%;"'<&30$-'0*$-'-"*<-$'):;'0)>"'4&")5*:<! $%&4%+":#" <":"&)$*3:J'("):60*:;')##"+"&)$*3:

c3%:$)*:'0)>"2'3:'X)&$-

c3%:$)*:'0)>"2'3:'c)&2

3:'d":%2

Page 15: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

>'()#09,"*-(L+):"$)&762#)+"'<&)>*$7'0)>"'<":"&)$";'47'$-"'(3>"(":$'3@'$-"'23+)&'-")$*:<'&"<*3:'*:'$-"';*%&:)+'#7#+"

A3+%$*3:2'3@'!).+)#"Y2'$*;)+'"G%)$*3:

BCDfCF

>'()#09,"*-(d"&$*#)+'2$&%#$%&"

]3+$3:'BCDDTF

X?#*$)$*3:'("#-):*2(,6 X)&$-', 23+)&'-")$*:<'3@'2$&)$32.-"&*#'3N3:"'+)7"&6 d":%2','23+)&'-")$*:<'3@'#+3%;'+)7"&6 c)&2','23+)&'-")$*:<'3@')$(32.-"&*#';%2$'

Page 16: 0#*+%,123...6$%%7.,405(R-"'>"+3#*$7'):;'>3&$*#*$7')&"'&"+)$";'$3'$-"

>'()#09,"*-(,*/,C(/<%B%,0"#$%&'()(

_*&%:)+ A"(*6;*%&:)+