Post on 31-Jul-2020
Part 3
REFLECTION AND TRANSMISSION
Normal Incidence
ρ , c1 1
ρ , c2 2
Incident Wave Reflection
Transmission
1( )ei x tki iu A − ω= 1( )1 1ei x tki i iu Z i A Z −ωτ = − = ω
1( )ei k x tr ru A − − ω= − 1( )
1 1ei k x tr r ru Z i A Z − − ωτ = = ω
2( )ei x tkt tu A − ω= 2( )2 2 ei x tkt t tu Z i A Z −ωτ = − = ω
(“physical” sign convention)
Boundary Conditions:
for any value of t at x = 0
i r tu u u+ =
i r tτ + τ = τ
i r tA A A− =
1 1 2i r tA Z A Z A Z+ =
Reflection/Transmission Coefficients
rd
i
ARA
= td
i
ATA
=
21
1 d dZR TZ
+ = 1 d dR T− =
displacement:
2 12 1
rd
i
Z ZARZ ZA
−= =
+
11 2
2td
i
ZATZ ZA
= =+
stress:
2 12 1
rs
i
Z ZRZ Z
−τ= =+τ
21 2
2ts
i
ZTZ Z
τ= =+τ
steel-water interface
( 6 246.5 10 kg/m ss scρ = × , 6 21.5 10 kg/m sw wcρ = × )
pi
pr
pt
pt
pr
pi
a)
b)
steel water
water steel
Power Coefficients: r t iP P P+ = (Instantaneous) Intensity:
r t iI I I+ =
Zτ = − v
2 2 2I Z v Z u= −τ = ωv =
2 22 1 1
1 2 12 1 1 2
2Z Z ZZ Z ZZ Z Z Z
⎛ ⎞ ⎛ ⎞−+ =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠
Special cases: solid/vacuum ( 2 0Z = )
1d sR R= = − , 2dT = , 1sR = − , 0sT =
solid/rigid ( 2Z → ∞ )
1d sR R= = , 0dT = , 2sT =
Shear wave at normal incidence: displacement:
2 12 1
r s sd
i s s
Z ZARZ ZA
−= =
+ 1
1 2
2t sd
i s s
ZATZ ZA
= =+
stress:
2 12 1
s srs
i s s
Z ZRZ Z
−τ= =+τ
21 2
2t ss
i s s
ZTZ Z
τ= =+τ
Impedance-Translation Theorem
d
incident wave
reflected wave
transmitted wave
Z1
Z2
Zo ,ko
Zload
Zinput
A+
A_ x
( ) exp( ) exp( )o ox A i k x A i k x+ −τ = + −
1( ) [ exp( ) exp( )]o oo o
xx A i k x A i k xi Z + −
∂ τ ∂= − = − − −
ωρ/
v
The input impedance of the layer:
input(0)(0) o
A AZ ZA A
+ −
+ −
τ += − =
−v
load( )( )
o o
o o
i k d i k do i k d i k d
d A e A eZ Zd A e A e
−+ −
−+ −
τ += − =
−v
load
load
o o
o o
i k d i k do
i k d i k do
Z e Z eAA Z e Z e
− −+
−
+=
−
Translation Formula:
loadinput
load
cos( ) sin( )cos( ) sin( )
o o oo
o o o
Z k d i Z k dZ ZZ k d i Z k d
−=
−
Reflection Coefficient:
input 1
input 1
Z ZR
Z Z−
=+
load 2Z Z=
Immersed/Embedded Layer:
2 1Z Z=
2 21
2 211
tan( )( )tan( )( ) 2
o o
o o o
i k d Z ZR
i k d Z Z Z Z−
=+ −
2(1 )T R= −
2 2
sin( )
sin ( ) 1o
o
k dR
k d
ξ=
ξ +
2 21
sin ( ) 1oT
k d=
ξ +
impedance contrast:
1 1½ o oZ Z Z Zξ = −/ /
Reflection/Transmission at a Layered Interface
ρ , c1 1
Incident Wave Reflection
ρ , c1 1
Transmission
ρ , co o
2 21
sin ( ) 1oT
k d=
ξ +
Thickness / Wavelength
Tran
smis
sion
Coe
ffic
ient
0
0.2
0.4
0.6
0.8
1
0 0.25 0.5 0.75 1 1.25
Plexiglas
Steel
in water
Reflectivity of Thin Cracks in Solids
2 2
sin( )
sin ( ) 1o
o
k dR
k d
ξ=
ξ +
0lim o
dR k d
→= ξ
log {Frequency x Thickness [MHz mm]}
Ref
lect
ion
Coe
ffic
ient
0
0.2
0.4
0.6
0.8
1
-10 -8 -6 -4 -2 0
air gap in steelwater-filled crack
in steel
Impedance Matching
loadinput
load
cos( ) sin( )cos( ) sin( )
o o oo
o o o
Z k d i Z k dZ ZZ k d i Z k d
−=
−
(2 1) 4od n= + λ /
(2 1)2ok d n π
= +
2
inputload
oZZZ
=
Perfect matching by quarter-wavelength layer:
1 2oZ Z Z= center frequency fo
4 4o o
o
cdf
λ= =
Bandwidth:
input 1
input 1
( )( )
( )Z f Z
R fZ f Z
−=
+
loadinput
load
2 2cos( ) sin( )( ) 2 2cos( ) sin( )
oo o
oo
o o
f fZ d i Z dc cZ f Z f fZ d i Z d
c c
π π−=
π π−
1 2oZ Z Z= and load 2Z Z=
2 1 2input 1 2
1 2 2
2 2cos( ) sin( )( ) 2 2cos( ) sin( )
o o
o o
f fZ d i Z Z dc cZ f Z Z f fZ Z d i Z d
c c
π π−=
π π−
( ) ( )o
o of f
RR R f f ff =
∂≈ + −
∂
input 1( )oZ f Z=
( ) 0oR f =
2 1r Z Z= /
input 1
2 2cos( ) sin( )( ) 2 2cos( ) sin( )
o o
o o
f fr d i r dc cZ f Z f fd i r d
c c
π π−=
π π−
sin( ) 1, and cos( )2
oo o
o
f fk d k df−π
≈ ≈ Δ =
input 1( ) r i rZ f Zi r
Δ −≈
Δ −
1 1
1 1
( 1)( )( 1) 2
r i rZ Zri rR f
r i r r i rZ Zi r
Δ −−
Δ −Δ −≈ ≈Δ − Δ + −
+Δ −
1 1( )
421
o
o
f fr rR f ifr rr i
−− − π≈ ≈
+ −Δ
22 2
2energy
( 1) ( 1)1 14 4 2
oo
f fr rTr r f
⎛ ⎞−− − π≈ − Δ = − ⎜ ⎟
⎝ ⎠
2 1r
4 21 1.8( 1) 1o
rf f rBQ f r r
−= = ≈ ≈
π − −
f1 and f2 are the half-power (-3 dB) points
Quarter-Wavelength Matching Layer
quarter-wavelength matching layer between quartz and water
Thickness / Wavelength
Ene
rgy
Tran
smis
sion
0
0.2
0.4
0.6
0.8
1
0 0.25 0.5
exact
approximate
unmatched
quarter-wavelength matching layer between steel and water
Thickness / Wavelength
Ener
gy T
rans
mis
sion
0
0.2
0.4
0.6
0.8
1
0 0.25 0.5
exact
approximate
unmatched
Continuous Transition
solid rock
mud
ultrasonic transducer
incident wave
echo from the bottomclear water
ρ , c1 1
ρ , coj oj
j = 1
j = N
ρ , c2 2 For the jth layer:
oj oj ojZ c= ρ , 2oj
oj
fkcπ
= , (j = 1, 2, ... N)
dN
=
Recursive relationship:
load1 2 2 2Z Z c= = ρ
loadinp
load
cos( ) sin( )cos( ) sin( )
j oj oj ojj oj
oj oj j oj
Z k d i Z k dZ Z
Z k d i Z k d−
=−
load 1 inp j jZ Z+ =
Reflection coefficient:
inp 1
inp 1
N
N
Z ZR
Z Z−
=+
, where 1 1 1Z c= ρ
Imperfect Interface, Finite Interfacial Stiffness
1( )ei x tki iu A − ω= 1( )1 1ei x tki i iu Z i A Z −ωτ = − = ω
1( )ei k x tr ru A − − ω= − 1( )
1 1ei k x tr r ru Z i A Z − − ωτ = = ω
2( )ei x tkt tu A − ω= 2( )2 2 ei x tkt t tu Z i A Z −ωτ = − = ω
ρ , c2 2
ρ , c1 1
Incident Wave Reflection
Transmission
K
x
Boundary Conditions:
i r tu u u u+ + Δ =
i r tτ + τ = τ
i r tuK K
τ + τ τΔ = =
K denotes the normal Kn or transverse Kt interfacial stiffness
Slip boundary conditions: n tK K → ∞/
Low-density interphase layer: 3 6n tK K ≈ −/
Kissing bond: 2 3n tK K ≈ −/
Partial bond: 0.5 1n tK K ≈ −/
Reflection and Transmission Coefficients
Continuity of displacement:
ti r tA A A
Kτ
− = −
21i r ti ZA A A
Kω⎛ ⎞− = −⎜ ⎟
⎝ ⎠
Continuity of stress:
1 1 2i r tA Z A Z A Z+ =
Stress reflection and transmission coefficients:
Imperfect interface:
2 1 1 22 1 1 2
//
tri i
A Z Z i Z Z KRA Z Z i Z Z K
− + ωτ= = =+ − ωτ
2 21 1 2 1 2
2/
t ti i
A Z ZTA Z Z Z i Z Z K
τ= = =+ − ωτ
Ideal interface (K→∞):
2 10
2 1ri
Z ZRZ Z
−τ= =+τ
20
1 2
2ti
ZTZ Z
τ= =+τ
00
lim R Rω→
= and 00
lim T Tω→
=
lim 1Rω→∞
= − and lim 0Tω→∞
=
Frequency Dependence
Moduli of the reflection and transmission coefficients of an imperfect steel-aluminum bond of 14 310 N mK = / for longitudinal wave at normal incidence
Frequency [MHz]
Ref
lect
ion
and
Tran
smis
sion
Coe
ffic
ient
s
00.10.20.30.40.50.60.70.80.9
1
0 2 4 6 8 10
Reflection
Transmission
For similar materials ( 1 2Z Z Z= = ):
21 2 1
i Z K iRi Z K iω ω Ω
= =− ω − ω Ω
/ // /
1 11 2 1
Ti Z K i
= =− ω − ω Ω/ /
2K ZΩ = / is the characteristic transition frequency
Oblique Incidence, Snell’s Law
c2θ2
c1 θ1
λ1
λ2Λ
c2θ2
c1 θ1
λ1
λ2Λ
1 21 2sin sin
λ λΛ = =
θ θ
1 2
1 2sin sinc c
f f=
θ θ
1 2
1 2sin sinc c
=θ θ
Reflection and Transmission
θdi
solid 1
Rd
Rs
Id
Td
solid 2
Ts
z
yθs1
θd1
θs2
θd2
solid 1
Rd
Rs
solid 2
TdTs
θsi
z
yIs θs1
θd1
θs2
θd2
Snell's Law:
1 1 2 2
1 1 1 1 2 2
sin sin sin sin sin sindi si d s d s
d s d s d sc c c c c cθ θ θ θ θ θ
= = = = =
Constitutive relationships:
( 2 ) yzyy
uuz y
∂∂τ = λ + λ + μ
∂ ∂
( )y zzy
u uz y
∂ ∂τ = μ +
∂ ∂
2 2
1 1 1 1 11 1, 2 ,s dc cμ = ρ λ + μ = ρ 2 22 2 2 2 22 2, and 2s dc cμ = ρ λ + μ = ρ
Boundary Conditions
both normal and transverse velocity and stress components must be continuous at the interface
(2) (1)
(2) (1)
(2) (1)
(2) (1)
0000
y y
z z
yy yy
zy zy
u u
u u
⎡ ⎤−⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥τ − τ⎢ ⎥ ⎢ ⎥
⎣ ⎦⎢ ⎥τ − τ⎢ ⎥⎣ ⎦
( 1) ( 2) ( 1) ( 2) ( )
( 1) ( 2) ( 1) ( 2) ( )
( 1) ( 2) ( 1) ( 2) ( )
( 1) ( 2) ( 1) ( 2) ( )
d d s s iy y y y yd d s s i
z z z z zd d s s i
yy yy yy yy yyd d s s i
zy zy zy zy zy
u u u u u
u u u u u
⎡ ⎤ ⎡ ⎤− + − +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− + − +⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥−τ + τ −τ + τ τ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−τ + τ −τ + τ τ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
longitudinal incidence:
Id = 1, Is = 0 shear incidence
Is = 1, Id = 0
11 12 13 14 1 1
21 22 23 24 2 2
31 32 33 34 3 3
41 42 43 44 4 4
or
d
d
s
s
a a a a R b ca a a a T b ca a a a R b ca a a a T b c
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
longitudinal [b] or shear wave incidence [c]
The matrix elements aij, bi, and ci can be easily calculated from simple geometrical considerations:
1 2 1 2
1 2 1 2
1 1 2 2 1 1 2 2
1 21 1 2 2 1 1 2 2
1 2
cos cos sin sinsin sin cos cos
cos2 cos2 sin 2 sin 2
sin 2 sin 2 cos2 cos2
d d s s
d d s s
d s d s s s s s
s ss d s d s s s s
d d
Z Z Z Zc cZ Z Z Zc c
− θ − θ − θ θ⎡ ⎤⎢ ⎥− θ θ θ θ⎢ ⎥
= ⎢ ⎥− θ θ − θ − θ⎢ ⎥⎢ ⎥− θ − θ θ − θ⎢ ⎥⎣ ⎦
a
(the common - iω factor was omitted in the last two rows)
11
11 1
1
cossin
sincos
andcos2sin 2
sin 2 cos2
disi
disi
d sis si
ss di s si
d
ZZ
cZ Zc
− θ⎡ ⎤θ⎡ ⎤⎢ ⎥θ ⎢ ⎥⎢ ⎥ θ⎢ ⎥= =⎢ ⎥θ
⎢ ⎥− θ⎢ ⎥⎢ ⎥⎢ ⎥− θ − θ⎣ ⎦⎢ ⎥⎣ ⎦
b c
Cramer's rule:
(1) (2) (3) (4)det[ ] det[ ] det[ ] det[ ], , ,det[ ] det[ ] det[ ] det[ ]d d s sR T R T= = = =
a a a aa a a a
Special Cases
a) fluid-vacuum b) fluid-fluid (cd2 > cd1)
fluid
vacuum
θi θr
Id Rdd
fluid 1
θi θr
Id
Tdd
Rdd
fluid 2
θd2
c) solid-vacuum d) solid-vacuum (longitudinal incidence) (shear incidence)
solid
vacuum
θi θr
θs
Rdd
Rds
Idsolid
vacuum
θi θr
θd
Is
Rsd
Rss
e) fluid-solid
fluid
solid
θi θr
θs
θd
Id
Tdd
Rdd
Tds
f) solid-fluid g) solid-fluid (longitudinal incidence) (shear incidence)
solid
fluid
θi
Id
Tdd
Rdd
Rdsθs1
θd2
θd1θ =r
solid
fluid
θi
Tsd
Rsd
RssIs
θd1
θd2
θs1θ =r
h) solid-solid i) solid-solid (longitudinal incidence) (shear incidence)
θi
solid 1
Id Rdd
Rds
Tdd
solid 2
Tds
θs1
θs2
θd2
θd1θ =r
solid 1
θiIs
Rsd
Rss
solid 2
TsdTss
θs2
θd2
θd1
θs1θ =r
Fluid-vacuum: 1 ,dd r iR ≡ =θ θ Fluid-fluid:
22 1
sin sin, d ir i
d dc cθ θ= =θ θ
22
1sin sind
d id
cc
=θ θ
2 1 2thend d d ic c< <θ θ
2 1 2thend d d ic c> >θ θ There exists one critical angle ( 2 2sin 1, 90d d→ → °θ θ )
11
2sin d
crd
cc
=θ
Solid-Vacuum Interface, Mode Conversion P-wave incident (no critical angle):
sinsin( ) , s ir d i
s dc cθθ= = =θ θ θ
S-wave incident:
sin sin( ) , d ir s i
d sc cθ θ
= = =θ θ θ
There exists one critical angle (sin 1dθ → or 90dθ → ° )
1sin scr
d
cc
=θ
The boundary conditions require that both normal
and transverse stress disappear at the surface.
cos2 sin 2 cos2
sin 2 cos2 sin 2
d s s s d sdd
s ss d s s s dds
d d
Z Z ZR
c cZ Z ZRc c
− θ − θ θ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ ⎥− θ θ − θ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
22
2
22
2
cos 2 sin 2 sin 2
cos 2 sin 2 sin 2
ss s d
ddd
ss s d
d
cc
Rcc
θ − θ θ
= −θ + θ θ
depends on the Poisson ratio of the solid
(0 ) (90 ) 1dd ddR R° = ° = −
Longitudinal and Shear Wave Reflection Coefficients
ν = 0.3 (solid) and ν = 0.35 (dashed)
Angle of Incidence [deg]
Ref
lect
ion
Coe
ffic
ient
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40 50 60 70 80 90
longitudinal-to-longitudinal
longitudinal-to-shear
Angle of Incidence [deg]
Ref
lect
ion
Coe
ffic
ient
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30 35
shear-to-longitudinal
shear-to-shear
Polar diagrams
longitudinal incidence
0o15o
45o30o
60o
90o
75o
90o
75o
60o
45o
30o15o longitudinal
shear
shear incidence
0o15o
45o30o
60o
90o
75o
90o
75o
60o
45o
30o15o longitudinal
shear
Fluid-Solid Interface
( 1) ( 2) ( 2) ( )
( 1) ( 2) ( 2) ( )
( 2) ( 2) 00
d d s iy y y yd d s i
yy yy yy yyd s
zy zy
u u u u⎡ ⎤ ⎡ ⎤− + +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−τ + τ + τ = τ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥+τ +τ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
11 12 14 1
31 32 34 3
42 440 0
dd
dd
ds
a a a R ba a a T b
a a T
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
2 2
1 2 2 2 2 1
22 2 2 2
2
cos cos sin coscos2 sin 2
00 sin 2 cos2
i d s dd i
d d s s s dd d
s dss d s s
d
RZ Z Z T Z
c TZ Zc
⎡ ⎤⎢ ⎥− θ − θ θ − θ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− θ − θ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎢ ⎥− θ − θ⎢ ⎥⎣ ⎦
2 2
1 2 2 2 2
22 2 2 2(1) 2
2 2
1 2 2 2 2
22 2 2 2
2
cos cos sincos2 sin 2
0 sin 2 cos2det[ ]
det[ ]cos cos sin
cos2 sin 2
0 sin 2 cos2
i d s
d d s s s
ss d s s
ddd
i d s
d d s s s
ss d s s
d
Z Z ZcZ Zc
R
Z Z ZcZ Zc
− θ − θ θθ − θ
− θ − θ= =
− θ − θ θ− θ − θ
− θ − θ
aa
cos cos sin
cos2 sin 2
0 sin 2 cos2cos cos sin
cos2 sin 2
0 sin 2 cos2
i d s
f d s s s
s d d sdd
i d s
f d s s s
s d d s
c c c
c cR
c c c
c c
θ θ − θρ θ − θ
θ θ=
θ θ − θ−ρ θ − θ
θ θ
1 2ρ = ρ ρ/
1 2,f d d dc c c c= = , 2s sc c= , 1i di dθ = θ = θ , 2d dθ = θ , and 2s sθ = θ
2 2 2
2 2 2cos ( cos 2 sin 2 sin 2 ) ( cos2 cos sin 2 sin )
cos ( cos 2 sin 2 sin 2 ) ( cos2 cos sin 2 sin )i s s d s f d s d s d sd
ddi s s d s f d s d s d sd
c c c c cR
c c c c c
θ θ + θ θ − ρ θ θ + θ θ=
θ θ + θ θ + ρ θ θ + θ θ
Displacement, Stress, Intensity, and Power Coefficients
( ) ( )
1
jstress displacement ZZ
βαβ αβ
αΓ = Γ
( )
1
jstress ZZ
βαβαβ
αΓ = Γ
Γ stands for either R (j = 1) or T (j = 2)
α and β are either d or s
( ) ( ) ( ) 21
jintensity displacement stress ZZ
βαβ αβ αβ αβ
αΓ = Γ Γ = Γ
( ) ( ) 21 1 1
j j jpower intensity cos Z coscos Z cos
β β βαβ αβ αβ
α α α
θ θΓ = Γ = Γ
θ θ
( ) ( ) ( ) ( ) 1power power power powers sd dR R T Tα αα α+ + + ≡
Law of reciprocity:
( ) ( )power powerαβ βαΓ ≡ Γ
Energy Reflection and Transmission Coefficients
aluminum in water
Angle of Incidence [deg]
Ener
gy R
efle
ctio
n an
d Tr
ansm
issi
onC
oeff
icie
nts
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
reflection
longitudinaltransmission shear
transmission
steel in water
Angle of Incidence [deg]
Ener
gy R
efle
ctio
n an
d Tr
ansm
issi
onC
oeff
icie
nts
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
reflection
longitudinaltransmission
sheartransmission
Energy Reflection and Transmission Coefficients
Plexiglas/aluminum interface
Angle of Incidence [deg]
Ener
gy R
efle
ctio
n C
oeff
icie
nts
00.10.20.30.40.50.60.70.80.9
1
0 10 20 30 40 50 60 70 80 90
longitudinalreflection
shear reflection
Angle of Incidence [deg]
Ener
gy T
rans
mis
sion
Coe
ffic
ient
s
0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60 70 80 90
longitudinalshear
transmissiontransmission
Slip Boundary Conditions
( 1) ( 2) ( 1) ( 2)
( 1) ( 2) ( 1) ( 2)
( 1) ( 2) ( 1) ( 2)
( 1) ( 2) ( 1)
normal displacementtangential displacement
normal tractiontangential traction
d d s sy y y yd d s s
z z z zd d s s
yy yy yy yyd d s
zy zy zy
u u u u
u u u u
− + − +⎧ ⎫⎪ ⎪ − + − +⎪ ⎪⎨ ⎬
−τ + τ −τ +τ⎪ ⎪⎪ ⎪⎩ ⎭ −τ + τ −τ +τ
( )
( )
( )
( 2) ( )
iyi
ziyy
s izy zy
u
u
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥τ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥τ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
11 12 13 14 1 1
21 22 23 24 2 2
31 32 33 34 3 3
41 42 43 44 4 4
or
d
d
s
s
a a a a R b ca a a a T b ca a a a R b ca a a a T b c
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
Slip boundary conditions:
continuity of the normal displacement and traction
vanishing tangential traction on both sides
11 12 13 14 1 1
31 32 33 34 2 2
41 43 4 4
42 44
or0 0
0 0 0 0
d
d
s
s
Ra a a a b cTa a a a b cRa a b cTa a
⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
Angle-Beam Transducers
transducer
specimen
couplant
θs
θiwedge
sinsin
s s
i i
cc
θ=
θ
Plexiglas/Aluminum, longitudinal-to-shear transmission
Angle of Refraction [deg]
Ener
gy T
rans
mis
sion
00.10.20.30.40.50.60.7
30 40 50 60 70 80 90
"slip" boundary
"rigid" boundary
SH Wave Reflection and Transmission at a Solid-Solid Interface
solid 1
R
solid 2
T
θiI
z
yθi=θs1
θs2
( ) ( ) ( )i r tx x xu u u+ = and ( ) ( ) ( )i r t
xy xy xyτ + τ = τ
( ) ( ) ( )
( ) ( ) ( )
r t ix x xr t i
xy xy xy
u u u⎡ ⎤ ⎡ ⎤− +⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥−τ + τ τ⎣ ⎦ ⎣ ⎦
or 11 12 1
13 14 2
a a cRa a cT
⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦
All displacements are in the x direction only (without the common i te− ω term):
1 1( ) ( cos sin )i s i si i k y k zxu e − θ + θ=
1 1( ) ( cos sin )i s i si i k y k zxu e − θ + θ=
2 2( ) ( cos sin )t s t st i k y k zxu T e − θ + θ=
2t sθ = θ , 2 1sin sint s s ic cθ = θ/
Stress components:
22xy xy s xc u yτ = με = ρ ∂ ∂/
1 1( ) ( cos sin )1cos i s i si i k y k z
xy s ii Z e − θ + θτ = − ω θ
1 1( ) (cos sin )1cos i s i sr i k y k z
xy s ii Z Re θ + θτ = ω θ
2 2( ) ( cos sin )2 cos t s t st i k y k z
xy s ti Z T e − θ + θτ = − ω θ
s sZ c= ρ is the specific acoustic impedance of the medium
1 2 1
1 1 1cos cos coss i s t s i
RZ Z ZT
−⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥θ θ θ⎣ ⎦⎣ ⎦ ⎣ ⎦
(the second row was divided by - iω )
(Displacement) reflection and transmission coefficients:
1 2 1 2
1 2
1 2
1 1cos cos cos cos1 1 cos cos
cos cos
s i s t s i s t
s i s t
s i s t
Z Z Z ZRZ Z
Z Z
θ θ θ − θ= =
− θ + θθ θ
1 1 1
1 2
1 2
1 1cos cos 2 cos1 1 cos cos
cos cos
s i s i s i
s i s t
s i s t
Z Z ZTZ Z
Z Z
−θ θ θ
= =− θ + θ
θ θ
“Normal component” of the acoustic impedance ' coss sZ Z= θ
' '1 2
' '1 2
s s
s s
Z ZRZ Z
−=
+ and
'1
' '1 2
2 s
s s
ZTZ Z
=+
Rayleigh Wave Solid-vacuum interface (free surface):
cos2 sin 20
sin 2 cos2 0
d s s sd
ss d s s s
d
Z ZR
cZ Z Rc
− θ − θ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥− θ θ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦
Nontrivial solution exists if:
22
2cos 2 sin 2 sin 2 0ss s d
d
cc
θ + θ θ =
sin sin 1s d
s d Rc c cθ θ
= =
Relative velocities:
1 2( )2(1 )
s
d
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 1.121+ ν
η ≈+ν