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Part 4
RAYLEIGH AND LAMB
WAVES
Rayleigh Surface Wave
x1x2
x3
surface wave
x3
x1
Partial Wave Decomposition Displacement potential: = ∇ϕ + ∇×u ψ Wave equations:
2 2
2 2 2 2sd2 2 2 2
sdandk k
c t c t1 ∂ ϕ 1 ∂
∇ ϕ = = − ϕ ∇ = = −∂ ∂
ψψ ψ
Wave velocities:
sd2 andc cλ+ μ μ
= =ρ ρ
Wave propagates in the 1x direction:
2 1 3 22
0 0, 0,ux∂
= ⇒ = ψ = ψ = ψ = ψ∂
Simplified wave equation:
2 2
2d2 2
1 30k
x x∂ ϕ ∂ ϕ
+ + ϕ =∂ ∂
2 2
2s2 2
1 30k
x x∂ ψ ∂ ψ
+ + ψ =∂ ∂
Axial displacement (tangential to the surface)
11 3
ux x∂ϕ ∂ψ
= −∂ ∂
Transverse displacement (normal to the surface):
33 1
ux x∂ϕ ∂ψ
= +∂ ∂
Normal stress in the propagation direction:
2 2 2 2
11 2 2 2 1 31 3 1( ) 2 ( )
x xx x x∂ ϕ ∂ ϕ ∂ ϕ ∂ ψ
τ = λ + + μ −∂ ∂∂ ∂ ∂
Normal stress parallel to the surface:
2 2 2 2
33 2 2 2 1 31 3 3( ) 2 ( )
x xx x x∂ ϕ ∂ ϕ ∂ ϕ ∂ ψ
τ = λ + + μ +∂ ∂∂ ∂ ∂
Shear stress:
2 2 2
31 13 2 21 3 1 3(2 )
x x x x∂ ϕ ∂ ψ ∂ ψ
τ = τ = μ + −∂ ∂ ∂ ∂
We seek solution in the following form
1 1( )3( ) i k x tF x e − ωϕ =
1 1( )3( ) i k x tG x e − ωψ =
1 RR
k kcω
→ =
From the wave equation,
2 2 2
2 2 2Rd d2 2 2
1 3 30 ( )Fk k k F
x x x∂ ϕ ∂ ϕ ∂
+ + ϕ = ⇒ = −∂ ∂ ∂
2 2 2
2 2 2s sR2 2 2
1 3 30 ( )Gk k k G
x x x∂ ψ ∂ ψ ∂
+ + ϕ = ⇒ = −∂ ∂ ∂
Solutions:
2 2
3 R d3( ) x k kF x Ae− −= and 2 2
3 sR3( ) x k kG x Be− −= The potential functions can be rewritten as follows
d 3 R 1( )x i k x tAe e−κ − ωϕ =
s 3 R 1( )x i k x tBe e−κ − ωψ =
2 2d R dk kκ = − and 2 2
s sRk kκ = −
The surface wave solution is a combination of coupled partial (longitudinal and shear) waves.
The partial wave amplitude ratio (B /A) is determined
by the condition that the surface is traction free.
Both partial waves are evanescent, i. e., R s dk k k> > or R s dc c c< <
Boundary conditions:
Both normal and tangential stresses vanish on the surface ( 3 0x = )
2 2 2 2
33 2 2 2 1 31 3 3( ) 2 ( )
x xx x x∂ ϕ ∂ ϕ ∂ ϕ ∂ ψ
τ = λ + + μ +∂ ∂∂ ∂ ∂
2 2s RRd( 2 ) 2k i k= λ+ μ κ ϕ − λ ϕ − μ κ ψ
2 2 2
31 13 2 21 3 1 3(2 )
x x x x∂ ϕ ∂ ψ ∂ ψ
τ = τ = μ + −∂ ∂ ∂ ∂
2 2d R s R2 ( )i k k= − μ κ ϕ − μ κ + ψ
At the surface, without the common term of R 1( )i k x te − ω :
2 2
s RRd2 2
d R s R
( 2 ) 200 2 ( )
k i k ABi k k
⎡ ⎤λ+ μ κ − λ − μ κ⎡ ⎤ ⎡ ⎤⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦− μ κ −μ κ +⎣ ⎦
2 2 2 2 2 2 2 2
s sR R R Rd d( 2 ) 2 ( 2 ) 2 ( )k k k k k kλ+ μ κ − λ = μ − λ+ μ = μ −μ = μ κ +
2 2s s RR
2 2d R s R
( ) 200 2 ( )
k i k ABi k k
⎡ ⎤κ + − κ⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥
− κ − κ +⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
Characteristic equation:
2 2 2 2s s dR R( ) 4 0k kκ + − κ κ =
Rayleigh Velocity
s
d
1 2( )2(1 )
cc
− νξ = =
−ν
Rs
cc
η =
Exact Rayleigh equation:
6 4 2 2 28 8(3 2 ) 16(1 ) 0η − η + − ξ η − − ξ = Approximate expression:
0.87 1.12
1+ ν
η ≈+ν
Shear Velocity / Longitudinal Velocity
Ray
leig
h V
eloc
ity /
Shea
r Vel
ocity
0.88
0.89
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
approximationexact
Partial Wave Composition The partial wave amplitude ratio:
2 2s d RR
2 2s R s R
22
k i kBA i k k
κ + κ= = −
κ κ +
By substituting the Rayleigh wave number Rk from the characteristic equation:
d
s
B i iA
κ= − = − ζ
κ
( ) 1.5 1.6vζ = ζ ≈ −
d 3xA e−κϕ = and s 3xi A e−κψ = − ζ
(without the common R 1( )i k x te − ω term)
Displacements Axial displacement (tangential to the surface):
d 3 s 31 R s1 3
( )x xu i A k e ex x
−κ −κ∂φ ∂ψ= − = − ζκ∂ ∂
Transverse displacement (normal to the surface):
d 3 s 33 d R3 1
( )x xu A e k ex x
−κ −κ∂φ ∂ψ= + = − κ − ζ∂ ∂
3
3 d R
1 R s0x
u ki iu k=
κ − ζ= = − ζ
− ζκ
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2Depth / Rayleigh Wavelength
Nor
mal
ized
Par
ticle
Dis
plac
emen
t.
normal (u3)
tangential (u1)
ν = 0.3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 20 0.5 1 1.5 2Depth / Rayleigh Wavelength
Nor
mal
ized
Par
ticle
Dis
plac
emen
t.
normal (u3)
tangential (u1)
ν = 0.3
Stresses
d 3 s 313 R d2 ( )x xi A k e e−κ −κτ = − μ κ −
d 3 s 32 233 sR( )( )x xA k e e−κ −κτ = μ + κ −
13 R d2 233 sR
2i k ik
τ κ= − = − ζ
τ + κ
2
s d R2 211 d 3 s 3R d 2 2
s R
4( 2 ) exp[ ] exp[ ]
kA k x x
k
⎧ ⎫μκ κ⎪ ⎪⎡ ⎤τ = − λ+ μ −λκ −κ − −κ⎨ ⎬⎣ ⎦ κ +⎪ ⎪⎩ ⎭
Depth / Rayleigh Wavelength
Nor
mal
ized
Stre
ss
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
τ
13τ
τ33
11
ν = 0.3
LAMB PLATE WAVES
Lowest-order Lamb waves in a plate
a) symmetric
b) asymmetric
x1
x2
x3
d
wave direction
2
Potential Method 1 1 1 3 3 2 1 1 3 3exp[ ( )] exp[ ( )]d d d dA i k x k x A i k x k xϕ = + + −
2 23 1d dd dk k k i= − = κ or 2 2
1d d dk kκ = −
1 1 1 3 3 2 1 1 3 3exp[ ( )] exp[ ( )]s s s sB i k x k x B i k x k xψ = + + −
2 23 1s s ssk k k i= − = κ or 2 2
1s ssk kκ = −
ks1 = kd1 = k, 2 2d dk kκ = − , 2 2
s sk kκ = −
3 1 3 1cosh( )exp( ) sinh( )exp( )s d a dA x i k x A x i k xϕ = κ + κ 3 1 3 1sinh( )exp( ) cosh( )exp( )s s a sB x i k x B x i k xψ = κ + κ
2 1sA A A= + , 2 1aA A A= − , 2 1sB B B= − and 2 1aB B B= +
Boundary Conditions
33 3 33 3 31 3 31 3( ) ( ) ( ) ( ) 0x d x d x d x dτ = = τ = − = τ = = τ = − =
normal stress:
2 2 2 2
33 2 2 2 1 31 3 3( ) 2 ( )
x xx x x∂ ϕ ∂ ϕ ∂ ϕ ∂ ψ
τ = λ + + μ +∂ ∂∂ ∂ ∂
shear stress:
2 2 2
31 13 2 21 3 1 3(2 )
x x x x∂ ϕ ∂ ψ ∂ ψ
τ = τ = μ + −∂ ∂ ∂ ∂
Example of Boundary Conditions Normal stress on the upper surface of the plate: 33 3( ) 0x dτ = =
2 2 2 2 2
33 2 2 2 21 31 3 1( ) 2( )d
s
cx xc x x x
∂ ϕ ∂ ϕ ∂ ψ ∂ ϕτ μ = + + −
∂ ∂∂ ∂ ∂/
Partial derivatives (without the common 1exp( )i k x :
2
23 32
1[ cosh( ) sinh( )]s d a dk A x A x
x∂ ϕ
= − κ + κ∂
2
23 32
3[ cosh( ) sinh( )]s d a dd A x A x
x∂ ϕ
= κ κ + κ∂
2
3 31 3
[ cosh( ) sinh( )]s s s a si k B x B xx x∂ ψ
= κ κ + κ∂ ∂
Normal stress on the upper surface of the plate:
2 2 2 233 ( )cosh( ) ( )sinh( )
2 cosh( ) 2 sinh( ) 0
s d s s d a
s s s s s a
k d A k d A
i k d B i k d B
τ μ = +κ κ + +κ κ
+ κ κ + κ κ =
/
Full 4×4 Boundary Condition Matrix
0000
s
a
s
a
AABB
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦
a
2 2 2 2
2 2 2 2
2 2 2 2
( )cosh( ) ( )sinh( ) 2 cosh( ) 2 sinh( )
( )cosh( ) ( )sinh( ) 2 cosh( ) 2 sinh( )
2 sinh( ) 2 cosh( ) ( )sinh( ) ( )cosh( )
2 si
s d s d s s s s
s d s d s s s s
d d d d s s s s
d
k d k d i k d i k d
k d k d i k d i k d
i k d i k d k d k d
i k
+ κ κ +κ κ κ κ κ κ
+ κ κ − +κ κ κ κ − κ κ=
κ κ κ κ − + κ κ − + κ κ
− κ
a
2 2 2 2nh( ) 2 cosh( ) ( )sinh( ) ( )cosh( )d d d s s s sd i k d k d k d
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥κ κ κ + κ κ − + κ κ⎢ ⎥⎣ ⎦
by summing and subtracting the 1st and 2nd rows as well as the 3rd and 4th rows
2 2
2 2
2 2
2 2
( )cosh( ) 0 2 cosh( ) 0
0 ( )sinh( ) 0 2 sinh( )
0 2 cosh( ) 0 ( )cosh( )
2 sinh( ) 0 ( )sinh( ) 0
s d s s
s d s s
d d s s
d d s s
k d i k d
k d i k d
i k d k d
i k d k d
⎡ ⎤+ κ κ κ κ⎢ ⎥
+κ κ κ κ⎢ ⎥= ⎢ ⎥
κ κ − + κ κ⎢ ⎥⎢ ⎥κ κ − + κ κ⎢ ⎥⎣ ⎦
a
Symmetric subsystem:
2 2
2 2( )cosh( ) 2 cosh( ) 0
02 sinh( ) ( )sinh( )s d s s s
sd d s s
k d i k d ABi k d k d
⎡ ⎤+κ κ κ κ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥
κ κ − +κ κ ⎣ ⎦⎢ ⎥ ⎣ ⎦⎣ ⎦
Asymmetric subsystem:
2 2
2 2( )sinh( ) 2 sinh( ) 0
02 cosh( ) ( )cosh( )s d s s a
ad d s s
k d i k d ABi k d k d
⎡ ⎤+κ κ κ κ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥
κ κ − +κ κ ⎣ ⎦⎢ ⎥ ⎣ ⎦⎣ ⎦
Characteristic Equations Symmetric subsystem:
2 2 2 2( ) tanh( ) 4 tanh( ) 0s s d s dk d k d+ κ κ − κ κ κ = Asymmetric subsystem:
2 2 2 2( ) tanh( ) 4 tanh( ) 0s d d s sk d k d+ κ κ − κ κ κ = Group velocity:
gck
∂ω=
∂
pg p
cc c k
k∂
= +∂
1
pg
p
p
cc c
c
=∂ω−∂ω
Potential Amplitude Distributions #1 Potential amplitude ratios:
ands s a aA B A B/ / from the boundary conditions Symmetric waves: 3 1cosh( )exp( )s dA x i k xϕ = κ
3 12 22 sinh( ) sinh( )exp( )
( )sinh( )d d
s ss s
i k dA x i k xk d
κ κψ = κ
+κ κ
Asymmetric waves: 3 1sinh( )exp( )a dA x i k xϕ = κ
3 12 22 cosh( ) cosh( )exp( )
( )cosh( )d d
a ss s
i k dA x i k xk d
κ κψ = κ
+κ κ
Stress and Displacement Distributions #2 displacement components:
11 3
ux x∂ϕ ∂ψ
= −∂ ∂
33 1
ux x∂ϕ ∂ψ
= +∂ ∂
#3 stress components:
2 2 2 2
11 2 2 2 1 31 3 1( ) 2 ( )
x xx x x∂ ϕ ∂ ϕ ∂ ϕ ∂ ψ
τ = λ + + μ −∂ ∂∂ ∂ ∂
2 2 2 2
33 2 2 2 1 31 3 3( ) 2 ( )
x xx x x∂ ϕ ∂ ϕ ∂ ϕ ∂ ψ
τ = λ + + μ +∂ ∂∂ ∂ ∂
2 2 2
31 13 2 21 3 1 3(2 )
x x x x∂ ϕ ∂ ψ ∂ ψ
τ = τ = μ + −∂ ∂ ∂ ∂
Dispersion Curves
normalized velocity: s
cc
η =
normalized frequency: ss
2 f dk dcπ
Ω = =
0
1
2
3
4
5
0 1 2 3 4 5 6Normalized Frequency
Nor
mal
ized
Pha
se V
eloc
ity
symmetric modes asymmetric modes
A0
A1 A2
S0
S1 S2
0
1
2
0 1 2 3 4 5 6Normalized Frequency
Nor
mal
ized
Gro
up V
eloc
ity
symmetric modes asymmetric modes
A0 A1 A2
S0
S1 S2
(steel, cd = 5,295 m/s, cs = 3,141 m/s)
Asymptotic Behavior Low-Frequency Asymptotes Lowest-Order Symmetric Mode (s0):
20lim and
(1 )p g pd
Ec c cω →
= =ρ − ν
2 2 2 2( ) tanh( ) 4 tanh( ) 0s s d s dk d k d+ κ κ − κ κ κ =
2 2 2 2( ) 4 0s s d s dk d k d+ κ κ − κ κ κ =
2 2 2 2 2( ) 4 0s dk k+ κ − κ =
2 2 2d dk kκ = − and 2 2 2
s sk kκ = −
2 2 2 2 2 2(2 ) 4 ( ) 0s dk k k k k− − − =
4 2 2 4 4 2 24 4 4 4 0s s dk k k k k k k− + − + =
2 2 4 2 24 4 0s s dk k k k k− + + =
42
2 24( )s
s d
kkk k
=−
2
22 1 ss
d
cc cc
= −
1 22 2
sd
cc
− ν=
− ν
21 2 2 2 2(1 )2 12 2 1 (1 ) (1 )s sc c c− ν μ + ν μ
= − = = =− ν − ν − ν ρ − ν ρ
plate2(1 )Ec c= =
− ν ρ
Lowest-Order Asymmetric Mode (a0):
4 20lim and 2
3 (1 )p g pd
Ec d c cω →
= ω =ρ − ν
2 2 2 2( ) tanh( ) 4 tanh( ) 0s d d s sk d k d+ κ κ − κ κ κ = 1st-order approximation
0lim tanh( ) ...θ→
θ = θ +
2 2 2 2( ) 4 0s d d s sk d k d+κ κ − κ κ κ =
2 2 2 2 2( ) 4 0s sk k+κ − κ =
2 2 2( ) 0sk −κ = or 0sk = (only at 0ω = )
3rd-order approximation
3
0lim tanh( )
3θ→
θθ = θ −
3 3 3 32 2 2 2( ) ( ) 4 ( ) 0
3 3d s
s d d s sd dk d k d
κ κ+κ κ − − κ κ κ − =
2 2 2 22 2 2 2 2 2(2 ) (1 ) 4 ( )(1 ) 0
3 3d s
s sd dk k k k k
κ κ− − − − − =
4 2 2 2 2 2 2 2 2 2 2 23 (2 ) ( ) 4 ( ) 0s s sdk k k k k d k k k d− − − + − =
after further algebraic manipulation:
4 4 2 2 4 2 2 2 2 4 2 2 4 23 (4 4 )( ) 4 ( 2 ) 0s s s s sdk k k k k k k d k k k k k d− − + − + − + =
46 4 2 2 4 4 2 2 2 2 4 2 6 4 2 2 4
23 4 4 4 4 4 8 4 0s
s s s s s sd d dk k k k k k k k k k k k k k k k k k
d− + − + − + + − + =
42 4 4 2 2 2 2 4 2 4 2 2 4
23 4 4 4 4 0s
s s s s sd d dk k k k k k k k k k k k k k
d− + − + − + =
after neglecting all but the highest power terms in k (since k is large)
44 2 4 2
23 4 4 0s
s dk k k k k
d− + =
24 2 2 2
24 (1 )3
ss
d
cc d cc
= ω −
4 2 2 2 2 2 24 1 2 2 1(1 )3 2(1 ) 3 1s sc d c d c− ν
= ω − = ω− ν − ν
2 24
23 (1 )E dc ω
=ρ − ν
platef3
c dc c
ω= =
Low-Frequency Asymptotes of the Fundamental Lamb Modes
steel plate (cd = 5,900 m/s, cs = 3,200 m/s)
0
1
2
3
4
5
6
0 1 2 3 4 5Frequency × Thickness [MHz mm]
Phas
e V
eloc
ity [k
m/s]
.
fundamental symmetric mode (S0) thin-plate dilatational wave approximation fundamental asymmetric mode (A0) thin-plate flexural wave approximation
High-Frequency Asymptotes of the Fundamental Modes
Symmetric Mode:
2 2 2 2( ) tanh( ) 4 tanh( ) 0s s d s dk d k d+ κ κ − κ κ κ =
Asymmetric Mode:
2 2 2 2( ) tanh( ) 4 tanh( ) 0s d d s sk d k d+ κ κ − κ κ κ =
2 2d dk kκ = − is pure real since dc c<
2 2
s sk kκ = − is pure real since sc c<
lim tanh( ) 1θ→∞
θ =
2 2 2 2( ) 4 0s d sk k+κ − κ κ =
both fundamental modes approach the Rayleigh velocity
High-Frequency Asymptotes of the Higher-Order Modes
Symmetric Modes:
2 2 2 2( ) tanh( ) 4 tanh( ) 0s s d s dk d k d+ κ κ − κ κ κ =
Asymmetric Modes:
2 2 2 2( ) tanh( ) 4 tanh( ) 0s d d s sk d k d+ κ κ − κ κ κ =
2 2d dk kκ = − is pure real since dc c<
2 2
s sk kκ = − is pure imaginary since sc c>
tanh( ) tan( )i iθ = θ
0sκ =
sc c→
all higher-order modes approach the shear velocity
Cut-Off Frequencies
kcω
= , ss
kcω
= , dd
kcω
= , 2 2s sk kκ = − , 2 2
d dk kκ = −
c→∞ , 0k→
s si kκ = is pure imaginary, d di kκ = is pure imaginary
tanh( ) tan( )i iθ = θ
Symmetric Modes:
2 2 2 2( ) tanh( ) 4 tanh( ) 0s s d s dk d k d+ κ κ − κ κ κ =
2 2 2 2( ) tan( ) 4 tan( ) 0s s d s dk k i k d k k k i k d− + =
4 sin( )cos( ) 0s s dk k d k d =
cos( ) 0dk d = or sin( ) 0sk d =
(2 1)2dk d n π
= − or sk d n= π
2 (2 1)2dd n λ
= − or 2 sd n= λ
(2 1)4
dc
cf nd
= − or 2
sc
cf nd
=
where n = 1, 2, ...
1 3 1 3( ) ( )u x u x= − and 3 3 3 3( ) ( )u x u x= − −
Asymmetric Modes:
2 2 2 2( ) tanh( ) 4 tanh( ) 0s d d s sk d k d+κ κ − κ κ κ =
4 cos( )sin( ) 0s s dk k d k d =
cos( ) 0sk d = or sin( ) 0dk d =
(2 1)2sk d n π
= − or dk d n= π
2 (2 1)2sd n λ
= − or 2 dd n= λ
(2 1)4
sc
cf nd
= − or 2
dc
cf nd
=
1 3 1 3( ) ( )u x u x= − − and 3 3 3 3( ) ( )u x u x= −
Cut-Off Frequencies
symmetric asymmetric
shear 2s
ccf nd
= (2 1)4
sc
cf nd
= −
dilatational (2 1)4
dc
cf nd
= − 2
dc
cf nd
=
general symmetric Lamb mode
pure transverse shear resonancepure transverse longitudinal resonance
general asymmetric Lamb mode
pure transverse shear resonancepure transverse longitudinal resonance