aeem-7028 lecture, part 3 reflection and transmissionpnagy/classnotes/aeem7028 ultrasonic nde... ·...
TRANSCRIPT
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Part 3
REFLECTION AND TRANSMISSION
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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+ =
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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
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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
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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
− −+
−
+=
−
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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 = −/ /
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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
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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
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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−
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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 −
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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
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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
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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
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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 ≈ −/
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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→∞
=
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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
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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
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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
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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]
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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
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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
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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
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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
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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° = ° = −
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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
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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
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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
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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
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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
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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
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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
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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
⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
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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
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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/
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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
=+
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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