<4D6963726F736F667420576F7264202D20B0BEB74CA4C0A4E8B57B5FA440BAFBBC75A9CAAA692E646F63>
xdx
0u = xu
x
1θ x
x dx+


1θ 2θ
1 1 2 2sin tan ; sin tan x x dx
u u x x
θ θ θ θ +
u u uT dx x x x x
uT dx x
ρ∂ ∂ =
∂ ∂ (6)

= ∂ ∂


Gρ
( ),x tσ
2
2
uA dx A qAdx Adx x t σσ σ ρ∂ ∂ − + + + = ∂ ∂
(9)
dx
∂ ∂ (10)
ε ∂ = ∂
ρ∂ ∂ ∂ + = ∂ ∂ ∂ ⇒
ρ∂ ∂ + =
∂ ∂ (11)

= ∂ ∂
( ),x tτ
2 2
= ∂ ∂
= ∂ ∂
ξ η ξ η ξ η
∂ ∂ ∂ ∂ ∂ ∂ ∂ = + = +
2 2 2 2
2 2 22u u u u u x x x ξ ξ η η ∂ ∂ ∂ ∂ ∂ ∂ = = + + ∂ ∂ ∂ ∂ ∂ ∂ ∂
(16)
o o u u u u uc c t t t
ξ η ξ η ξ η
∂ ∂ ∂ ∂ ∂ ∂ ∂ = + = − +
t t t ξ ξ η η ∂ ∂ ∂ ∂ ∂ ∂ = = − + ∂ ∂ ∂ ∂ ∂ ∂ ∂
(18)

2 2 2 2 2 2
+ + = − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
( ) ( ) ( ),u f gξ η ξ η= + (22)
( ) ( ) ( )o o,u x t f x c t g x c t∴ = − + + (23)
( ) ( ),0u x U x= ( ) ( ),0u x V x=
6
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( )
( ) ( )
o o o o
, f x c t g x c tu x t t t t
f x c t x c t g x c t x c t x c t t x c t t
∂ − ∂ +∂ = +
∂ ∂ ∂ ∂ − ∂ − ∂ + ∂ +
= + ∂ − ∂ ∂ + ∂
′ ′= − − + +
( ) ( ) ( )o oc f x c g x V x′ ′− + = (26)
( ) ( ) ( ) o
(24)(27)
( ) ( ) ( ) o
c ζ ζ= − ∫ (28)
c ζ ζ= + ∫ (29)
x c t
b f x c t U x c t V d
c ζ ζ
x c t
b g x c t U x c t V d
c ζ ζ
, 1 1 2 2
u x t f x c t g x c t
= − + +
= − + + − − + +
x c t x c t x c t
− + +
o o o
x c t
+
x a − < <

(32)
( ) ( ) ( )o o 1, 2
u x t U x c t U x c t = − + + (35)

8
d’Alembert
( ) ( )1 o,u x t f x c t= − 0t = ( ) ( )1 ,0u x f x= ( )1 ,u x t
0t = oc
( ) ( )2 o,u x t g x c t= + ( )2 ,u x t 0t = oc

-3 -2 -1 0 1 2 3 4 5 6 -0.5
0
0.5
1
1.5
oc ox ot ( )U x ( )V x
( ),x t ( )o o,x t
o o o ox c t x c t− = − (36)
o o o ox c t x c t+ = + (37)
d’Alembert
( ) ( )1 1 o 1,u x t f x c t= −
( )2 ,u x t
1x x
2x 1 o 1x c t+ 2 o 1x c t+
0t = 1t t=
oc
9
( ) ( ) ( ) ( )o o o
o o o o o o o o o o o
o
x c t
x c t u x t U x c t U x c t V d
c ζ ζ
− = − + + + ∫ (38)
( ),u x t ( )o o,x t ( )U x ( )V x [ ]o o o o o o,x c t x c t− +

( )U x ( )V x ot ox
[ ]o o o o o o,x c t x c t− + ( )o o,x t
( )o o,x t
[ ]o o o o o o,x c t x c t− +
[ ]o o o o o o,x c t x c t− +
[ ]o o o o o o,x c t x c t− +
-1 0 1 2 3 4 5 6 -0.5
0
0.5
1
1.5
2
2.5
3
o o ox c t−
( )o o,x t
o o ox c t+

10
[ ]1 2,x x t
( ),x t ( )1,0x ( )2 ,0x
1 ox x c t= − (39)
2 ox x c t= + (40)
[ ]1 2,x x
[ ]1 2,x x
( ),u x t [ ]1 2,x x ( ),u x t
[ ]1 2,x x [ ]1 2,x x

-1 0 1 2 3 4 5 6 7 8 -0.5
0
0.5
1
1.5
2
2.5
3

x
t
1 ox x c t= − 2 ox x c t= +
11

-1 0 1 2 3 4 5 6 7 8 -0.5
0
0.5
1
1.5
2
2.5
3

o ox x c t= +o ox x c t= −
t
12
= ∂ ∂
( ) ( ) ( ),u x t X x T t= 2
2

2 2 o 0T k c T+ = (43)

( ) ( ) ( )1 2sin cosX x A kx A kx= + (44)
( ) ( ) ( )3 o 2 osin cosT t A kc t A kc t= + (45)
(wave number) k (circular frequency)ωλ
(wave length)
o o
π ω ω π λ λ
= = ⇒ = = (46)

13
A kx A kx A t A t
A A kx t A A kx t
A A kx t A A kx t
ω ω
ω ω
ω ω
(47)
o 1c = 1λ = 2k π= 2ω π= 1 0.1t = 2 0.8t =
3 2.3t =
-1
0
1
-1
0
1
-1
0
1
-1
0
1
2 u=sin(kx)*sin(ωt)
( )sin sin cos cos sinα β α β α β+ = +
(47)
( ) ( ) ( ) ( ) ( )1 2 3 4, sin sin cos cosu x t B kx t B kx t B kx t B kx tω ω ω ω= + + − + + + −
(48)
( ) ( )1 , sinu x t kx tω= + okcω =
( ) ( )1 sin cosu kx tω= ( ) ( )2 cos sinu kx tω=
( ) ( )3 cos cosu kx tω= ( ) ( )4 sin sinu kx tω=
1t t=
2t t=
3t t=
14
( ) ( )1 o, sinu x t k x c t= + λ x
2k π λ
= ( )okx t k x c tφ ω= + = + t tΔ x
oc tΔ φ oc
o 1c = 1λ = 2k π= 2ω π= 1 0.1t = 2 0.8t =
3 2.3t =
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
= ∂ ∂
t ∂
1, sin cos 3
u x t x c t x t c t= + +
o 1.0c =
= ∂ ∂
( ) ( ) 0
t =
(14)
( ) ( ) ( )o o,u x t f x c t g x c t= − + + (52)
(49)(50)
( ) ( ) ( )o
x c t
b f x c t U x c t V d
c ζ ζ
x c t
b g x c t U x c t V d
c ζ ζ
( ) ( )o o 0f c t g c t− + = 0t ≥ (55)

( ) ( )f x g x= − − 0x ≤ (56)
( )of x c t− ox c t−
( )of x c t−
17
xt c
≤
x c t
x c t u x t U x c t U x c t V d
c ζ ζ
xt c
≥
x c t
b g x c t U x c t V d
c ζ ζ
x c t
b f x c t g x c t U x c t V d
c ζ ζ
x c t
x c t u x t U x c t U x c t V d
c ζ ζ
x c t
x c t
x c t
x c t
xU x c t U x c t V d t c c
u x t xU x c t U x c t V d t
c c
ζ ζ
ζ ζ
o
xt c
≥ x
( ) ( ) ( )( )o o 1, 2
u x t U x c t U x c t = + − − − (61)
( )1 2
( )( )o 1 2
U x− − oc

0
= ∂ ∂
( ) ( ) 0
t =
1 1 ; 2 2
x c t x c t
xU x c t U x c t V d t c c
u x t
ζ ζ
= ∂ ∂
( ) ( ) 0
t =

( ) ( ) ( )1 2sin cosX x A kx A kx= + (44)
( ) ( ) ( )3 o 2 osin cosT t A kc t A kc t= + (45)
(68) ( ) ( )0 0X X= = (44)
2 0A = ; ( )1 sin 0A k = (69)

(72)
22
( ) o ocos sinn n n c n t c n tT t c dπ π = +
(73)

u x t X x T t
=
= +
(74)
n n nc c D= n n nd d D=
( ),nu x t
( ) ( )
u x t u x t
∞
=
∞
=
= =
= +
∑
= =
= =
u x t c n n xd V x t
π π∞
∫ (78)
( ) 0
o
π = ∫ (79)
23
n c n tV d c n
π πζ ζ ζ π
π πζ ζ ζ π
∞
=
= +
∫ ∑
∫ (80)
2 2 n n nA c d= + 1tan n
n n
c d

( ) ( ), sin sinn n n n n xu x t A tπ ω θ = +
(81)

nω
nθ nθ x nω ( )U x ( )V x
o 1
c Tπ πω ρ

x n n n n
− = 0


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
0
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
0
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
0
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
0
1

= ∂ ∂
( ) ( ) 0
t =
( ), 0
x
2 2 2
n n n
c n t c n t n x c d
π π π
( ) ( )
2 2 2
c n t c n t n x c d
π π π
n d V d
26
= ∂ ∂
( ) ( ),0u x V x
( ) ( ) ( )
u x t X x T t
=
= +
( ) o o 0 0
n
=
= + + + ∑
1d V dζ ζ= ∫ (94)
( ) 0
∫ (97)
( ) 0
o
π = ∫ (98)
27


= ∂ ∂
( ) ( ) 0
t =
n n
c t t c n t n xu x t c n
π π π π π
∞
=
− − = +
∑
= ∂ ∂
x =
( ) 0
, 0
t
c n t n x u x t
n π π
∂ ∂ =
ω 1 1 2 2k c k cω = =
1 1
( )1i k x t ru A e ω− +′= (102)
( )2i k x t tu Be ω−= (103)
A A′B
0x =
( )1i k x tAe ω− ( )2i k x tBe ω−
( )1i k x tA e ω+′
0x =
A A B′+ = (106)
( )1 1 1 1 2 2E k A E k A E k B′+ − = (107)

−′ = +
= +


33

α →∞
0 1 2 3 4 5 6 7 8 9 10 -1
-0.5
0
0.5
1
1.5
2
α
34

(1) 2 1E E= 2 1ρ ρ= 2 1c c= 1α = 0A′ = B A=
-10 -8 -6 -4 -2 0 2 4 6 8 10 1
1
1
1
1
1
-10 -8 -6 -4 -2 0 2 4 6 8 10 1
1
1
1
1
1
(2) 2 10.8E E= 2 10.5ρ ρ= 2 11.265c c= 0.6325α = 0.2251A A′ =
1.2251B A=
-10 -8 -6 -4 -2 0 2 4 6 8 10
0.8
1
1.2
1.4
-10 -8 -6 -4 -2 0 2 4 6 8 10
0.8
1
1.2
1.4
(3) 2 10.01E E= 2 10.01ρ ρ= 2 1c c= 0.01α = 0.9802A A′ =
1.9802B A=
-10 -8 -6 -4 -2 0 2 4 6 8 10 0
0.5
1
1.5
2
-10 -8 -6 -4 -2 0 2 4 6 8 10 0
0.5
1
1.5
2
x
x
tσ
35
(4) 2 11.5E E= 2 12ρ ρ= 2 10.866c c= 1.732α = 0.2679A A′ = −
0.7321B A=
-10 -8 -6 -4 -2 0 2 4 6 8 10
0.8
1
1.2
1.4
-10 -8 -6 -4 -2 0 2 4 6 8 10
0.8
1
1.2
1.4
(5) 2 12E E= 2 10.5ρ ρ= 2 12c c= 1α = 0A′ = B A=
-10 -8 -6 -4 -2 0 2 4 6 8 10 1
1
1
1
1
1
-10 -8 -6 -4 -2 0 2 4 6 8 10 1
1
1
1
1
1
(6) 2 140E E= 2 140ρ ρ= 2 12c c= 40α = 0.9512A A′ = −
0.0488B A=
-10 -8 -6 -4 -2 0 2 4 6 8 10 0
0.5
1
1.5
2
-10 -8 -6 -4 -2 0 2 4 6 8 10 0
0.5
1
1.5
2
x
x
36

1. Graff, K. F.(1975), Wave Motion in Elastic Solids, Ohio State Univ. Press. 2. Achenbach, J. D.(1984), Wave Propagating in Elastic Solids,
North–Holland, Amsterdam.
Amsterdam.
7. (2003)()