Flaw Scattering Models
Learning Objectives
Far-field scattering amplitude
Kirchhoff approximation
Born approximation
Separation of Variables
Examples of scattering of simple shapes
(spherical pore, flat crack, side-drilled
hole)
Flaw Scattering
Fluid Model
Incident plane
wave
p0
p0 … pressure
amplitude
ei
es
xs
rs
n
y
flaw
At many wavelengths away from the flaw the
scattered waves are spherical waves
p scatt ( y , ω ) = p0 A ( ei ; e s )
exp ( ikrs )
rs
scattered pressure
A is called the plane wave far-field scattering amplitude
Flaw Scattering
Fluid Model
Incident plane
wave
p0
p0 … pressure
amplitude
ei
es
xs
rs
n
y
flaw
plane wave far field scattering amplitude of the flaw
−1 ⎡ ∂p%
⎤
%
e
n
A ( ei ; e s ) =
+
ik
⋅
p
( s ) ⎥ exp ( −ikx s ⋅ e s ) dS ( x s )
∫
⎢
4π S f ⎣ ∂n
⎦
p(x s ,ω )
p˜ =
p0
Flaw Scattering
−1 ⎡ ∂p%
⎤
%
A ( ei ; e s ) =
ik
p
e
n
+
⋅
( s ) ⎥ exp ( −ikx s ⋅ e s ) dS ( x s )
∫
⎢
4π S f ⎣ ∂n
⎦
p(x s ,ω )
p˜ =
p0
To obtain the far field scattering amplitude, we need to know the
pressure and velocity on the surface of the flaw due to the incident
and scattered waves
These quantities can be found by solving a flaw scattering
boundary value problem.
Flaw Scattering
Elastic Solid
u scatt ( y , ω ) = U 0
A ( eiβ ; e sP )
rs
scattered displacement
from a flaw
exp ( ik p rs ) + U 0
P-wave
A ( eiβ ; e Ss )
rs
S-wave
U0
eiβ
rs
flaw
xs
U0 … displacement
amplitude for wave
of type β (β = P,S) in
problem (1)
n
eαs
Scattered wave of
type α (α = P,S)
exp ( ik s rs )
Flaw Scattering
A ( eiβ ; eαs )
Vector scattering amplitude for a scattered wave of
type α due to a plane wave of unit displacement
amplitude and type β
A ( eiβ ; e sP ) = ( f P ; β ⋅ e sP ) e sP
elastic
constants
f
α ;β
= fl
α ;β
(β = P, S)
A ( eiβ ; e Ss ) = ⎡⎣f S ; β − ( f S ; β ⋅ e sS ) e sS ⎤⎦
⎡⎛ ∂u% p ⎞
⎤
α
α
%
exp
el = −
n
ik
e
n
u
ik
x
e
+
−
⋅
⎢
⎥
⎜
⎟
(
α sk p j
α s
s ) dS ( x s )
k
2 ∫ ⎜
⎟
4πρ cα S ⎢⎣⎝ ∂x j ⎠
⎥⎦
Clkpj el
displacements and displacement
gradients
(α = P,S ) (β = P, S)
Flaw Scattering
3-D scattering amplitude
u scatt ( y , ω ) = U 0
A ( eiβ ; e sP )
rs
exp ( ik p rs ) + U 0
A ( eiβ ; e Ss )
rs
exp ( iks rs )
The received voltage in some measurement models is related
to a specific component of the vector scattering amplitude
given by
A (ω ) = A ( eiβ ; eαs ) = A ( eiβ ; eαs ) ⋅ ( −dα )
polarization vector of an
incident wave from receiving
transducer (to be discussed
shortly)
Flaw Scattering
A(ω) can be obtained from:
Analytical/Numerical methods
Separation of variables
Boundary Elements (BEM)
Finite Elements (FEM)
Method of Optimal Truncation (MOOT)
T-matrix
Kirchhoff Approximation
Born Approximation
or A(ω) can be found experimentally through
VR (ω ) E * (ω )
A (ω ) =
deconvolution: VR (ω ) = E(ω ) A(ω )
2
2
E (ω ) + n
measure
model, measure
Flaw Scattering
Kirchhoff Approximation for a Volumetric Flaw
incident
wave
ei
tangent
plane P
er
n
reflected
wave
x
Slit
xs
D
O
on the "lit" surface: fields are those from plane waves
interacting with a plane surface
on the rest of the surface the fields are assumed to be zero
Flaw Scattering
It has been recently shown that the pulse-echo
scattering amplitude response, A(ω), for a stress-free
flaw (void or crack) in an elastic solid is identical to
the scalar (fluid) scattering amplitude:
A (ω ) = A ( ei ; −ei ) =
β
β
−ikβ
eiβ
e βs = −eiβ
2π
∫ (e
Slit
β
i
⋅ n ) exp ( 2ikβ x s ⋅ eiβ ) dS ( x s )
Flaw Scattering-Spherical Void
For the pulse-echo response of a spherical
void of radius a
⎡
sin ( ka ) ⎤
−a
A (ω ) ≡ A ( ei ; −ei , ω ) =
exp ( −ika ) ⎢exp ( −ika ) −
⎥
ka
2
⎣
⎦
0.7
0.6
2a
|A(ω)|
0.5
0.4
a = 1 mm
c = 5900m/s
0.3
0.2
0.1
0
0
5
10
15
20
frequency, MHz
25
30
Flaw Scattering-Spherical Void
Comparison with Kirchhoff solution with
exact separation of variables solution for the P-P scattering
amplitude of a void in an elastic solid (pulse-echo)
leading edge
response
1.4
exact solution (solid line)
1.2
2|A|/a
1
0.8
Kirchhoff solution
(dotted line)
0.6
"creeping" wave
0.4
0.2
0
0
5
10
15
20
frequency, MHz
25
30
Flaw Scattering-Spherical Void
Inverting the scattering amplitude (fluid model) into the
time domain
A ( t ) ≡ A ( ei ; −ei , t ) =
− a ⎡ ⎛ 2 a ⎞ c ⎛ −2 a
⎞⎤
+
−
,
0;
δ
t
U
t
⎜
⎟
⎜
⎟⎥
2 ⎢⎣ ⎝
c ⎠ 2a ⎝ c
⎠⎦
⎧1 t1 < t < t2
U ( t1 , t2 ; t ) = ⎨
⎩0 otherwise
"lit" surface
response
A(t)
c
4
t
leading edge
response
2a
c
Flaw Scattering-Spherical Void
Exact solution
for a void in an
elastic solid (time
domain)
vs the Kirchhoff
approximation solution
2.5
2
1.5
1
creeping wave
0.5
0
-0.5
-1
-1.5
-2
0
2
4
time, μsec
6
8
10
Flaw Scattering-Spherical Void
Although the Kirchhoff approximation is formally a high
frequency approximation ( ka >>1), numerical tests have
shown it capable of producing good agreement (<1 dB
difference) for the peak-to-peak time domain response of the
spherical void down to ka =1 provided the bandwidth is
sufficient
5
4.5
4
3.5
ka
white: differences <1 dB
gray: >1 dB but < 1.5 dB
black: > 1.5 dB
3
2.5
2
1.5
1
0.5
10
20
30
40
50
60
70
80
bandwidth, %
90
Flaw Scattering-Inclusion
Leading Edge Response - General Convex Inclusion
(scattered mode same as incident mode)
es tangent plane
nα
θi
stat
xs
α
S lit
es
ei
d
High frequency, stationary phase approximation
θi = α
A ( ei ; e s , ω ) = R12
Plane wave reflection coefficient
R1 R2
exp ⎡−
ik ei − e s d ⎤⎦
⎣
2
Gaussian curvature of flaw surface at
stationary phase point
Flaw Scattering-Inclusion
Leading Edge Response - General Convex Inclusion
(scattered mode same as incident mode)
es tangent plane
nα
θi
stat
xs
α
S lit
es
ei
d
In the time domain
A ( ei ; e s , t ) = R12
1
A (t ) =
2π
+∞
∫ A (ω ) exp ( −iωt ) dω
−∞
R1 R2
δ ( t + ei − e s d / c )
2
Flaw Scattering-Crack
Kirchhoff Approximation – Flat Elliptical Cracks
x3
es
plane normal to e q
ei
n
a1
u1
a2
x2
u2
re
−ia1a2 (e i ⋅ n)
A(e i ;e s ) =
J1 (k e i − e s re )
ei − e s re
x1
re =
eq
ei − es
eq =
ei − es
a12
(eq ⋅ u1 )
2
+ a22
(e q ⋅u 2 )
2
Flaw Scattering-Crack
General pitch-catch crack response-circular crack
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
frequency, MHz
The magnitude of the P-wave pulse-echo far-field scattering amplitude versus frequency for a 1 mm radius circular crack in steel
with an angle of incidence of from the crack normal.
Flaw Scattering-Crack
Comparison of the Kirchoff approximation and MOOT for
a 0.381 mm radius circular crack at an incident angle of 45
degrees (pulse-echo)
0.25
KIR
MOOT
inc = 45
amplitude
0.2
0.15
0.1
0.05
0
0
5
10
15
frequency, MHz
20
25
Flaw Scattering-Crack
When eq is parallel to the crack normal, n :
−ik (e i ⋅ n)a1a2
A(e i ;e s ) =
2
eq
n
ei
Special case - pulse echo:
ika1a2
A(e i ;−ei ) =
2
ei
16
14
12
10
8
6
4
2
0
0
5
10
15
20
25
30
frequency, MHz
The magnitude of the P-wave pulse-echo far-field scattering amplitude versus frequency for a 1 mm
radius circular crack in steel at normal incidence.
es
n
es
Flaw Scattering-Crack
Comparison of the Kirchoff approximation and MOOT for
a 0.381 mm radius crack at normal incidence (pulse-echo)
2
KIR
MOOT
1.8
1.6
inc=0
amplitude
1.4
numerical
errors in
MOOT
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
frequency, MHz
20
25
Flaw Scattering-Crack
Inverting these results into the time domain
e i ⋅n ≠ 0
⎧ a1a2 c ( ei ⋅ n )
t
⎪⎪−
2 2
2
−
π
r
e
e
A ( t ) ≡ A ( ei ; e s , t ) = ⎨
i
s
e
( ei − es re / c ) − t 2
⎪
0
otherwise
⎪⎩
re
t < ei − e s
c
Example: pulse-echo
flash points
t
Flaw Scattering-Crack
At normal incidence, for pulse-echo
− a1a2 d δ ( t )
A ( ei ; −ei , t ) =
dt
2c
A(e i ; es )
t
Flaw Scattering-Crack
A planar crack is a very “specular” reflector
θ
eiβ
3
2.5
2
1.5
1
0.5
0
0
10
20
30
40
50
angle, degrees
60
70
80
90
Flaw Scattering-Crack
Although the Kirchhoff approximation is formally a high
frequency approximation ( ka >>1), numerical tests have
shown it capable of producing good agreement (<1 dB
difference) for the peak-to-peak time domain response of a
circular crack at normal incidence down to ka =1.5 provided
the bandwidth is sufficient
5
4.5
4
3.5
ka
white: differences <1 dB
gray: >1 dB but < 1.5 dB
black: > 1.5 dB
3
2.5
2
1.5
1
0.5
10
20
30
40
50
60
70
bandwidth, %
80
90
Flaw Scattering-Crack
Comparison of synthesized waveforms scattered from a 0.381
mm radius crack using the Kirchhoff approximation and
MOOT using frequencies from 0-25 MHz approximately
(pulse echo)
1
0.9
0.8
amplitude
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
frequency, MHz
20
25
Flaw Scattering-Crack
0-15 degrees from crack normal
20
KIR
MOOT
0
5
15
10
5
0
-5
-10
-15
-20
10
15
Flaw Scattering-Crack
20-35 degrees from crack normal
KIR
MOOT
20
4
25
3
30
2
1
0
-1
-2
-3
-4
-5
35
Flaw Scattering-Crack
40-55 degrees from crack normal
KIR
MOOT
40
1.5
45
50
1
0.5
0
-0.5
-1
-1.5
55
Flaw Scattering-Crack
60-70 degrees from crack normal
1
0.8
KIR
MOOT
60
65
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
70
Flaw Scattering-Crack
75-85 degrees from crack normal
0.8
KIR
MOOT
75
0.6
80
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
85
Flaw Scattering-Crack
Comparison of peak-to-peak values of MOOT and
Kirchhoff versus angle of incidence for the 0.381mm
radius crack (pulse-echo)
The arrow shows where agreement is within 1 dB
25
MOOT
KIR
20
15
10
5
0
0
10
20
30
40
50
60
70
80
90
Flaw Scattering-Crack
Now consider the scattering amplitude with narrow band
Gaussian window
The central frequency is 10 MHz, and the bandwidth is 1 MHz
Radius of the crack a = 0.381mm
1
0.9
0.8
amplitude
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
frequency, MHz
20
25
Flaw Scattering-Crack
the range of angles where the agreement is good is
significantly reduced
3.5
KIR
MOOT
3
2.5
2
1.5
1
0.5
0
0
10
20
30
40
50
60
70
80
90
Flaw Scattering-Crack
Now consider the scattering amplitude with wider band Gaussian
window
The central frequency is 10 MHz, and the bandwidth is 6 MHz
Radius of the crack a = 0.381mm
1
0.9
0.8
amplitude
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
frequency, MHz
20
25
Flaw Scattering-Crack
The range of excellent agreement now is back to
angles as great as 45 degrees or more
18
KIR
MOOT
16
14
Amplitude
12
10
8
6
4
2
0
0
10
20
30
40
50
Angle, degree
60
70
80
90
Flaw Scattering-Crack
Scattering
max. incident angle, degrees
Maximum incident angle where the peak-to-peak time
domain agreement between the Kirchhoff approximation
and the exact solution for a circular crack is less than 1 dB
for ka =5.0 (other ka values shown same trend).
60
55
50
45
40
35
30
25
20
15
10
10
20
30
40
50
60
bandwidth, %
70
80
90
Flaw Scattering-SDH
Kirchhoff approximation for pulse-echo scattering of a
side-drilled hole (incident direction in plane
perpendicular to the hole axis)
eiβ
L
2b
A ( ei ; −ei
β
β
)
k b) L
(
⎡J
=
β
2
⎣
1
( 2k b ) − iS ( 2k b )⎤⎦ +
β
Bessel function
1
β
i ( kβ b ) L
π
Struve function
Flaw Scattering-SDH
Comparison with exact 2-D separation of variables
solution for P-waves (pulse-echo)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
The three-dimensional normalized pulse-echo P-wave scattering amplitude versus normalized wave number for
a side drilled hole in the Kirchhoff approximation (solid line) and from the exact two-dimensional separation
of variables solution (dashed line).
Flaw Scattering-SDH
Comparison with exact 2-D separation of variables
solution for S-waves (pulse-echo_)
1.1
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
The three-dimensional normalized pulse-echo SV-wave scattering amplitude versus normalized wave number for
a side drilled hole in the Kirchhoff approximation (solid line) and from the exact two-dimensional separation of
variables solution (dashed line).
Flaw Scattering
Kirchhoff approximation - Summary
For volumetric flaws -the Kirchhoff approximation
properly models the leading edge signal as long as
ka >1 approximately but does not model other
waves (creeping waves, etc.)
For cracks – the Kirchhoff approximation models
the flash point signals properly in pulse-echo as
long as ka > 1 approximately and the incident
angle is less than about 50 degrees for wide band
responses. For narrow band responses this angle is
considerably reduced to as little as 15-20 degrees.
Flaw Scattering -Inclusion
The Born Approximation
The Born approximation assumes that the material of
an inclusion differs little from the host material so that
to first order the incident wave passes through the
inclusion unchanged (weak scattering, low frequency
approximation)
ein
c
incident
wave
Flaw Scattering -Inclusion
The Born approximation generally is developed from a
volume integral expression for the scattering amplitude
density difference
difference in elastic constants
⎡
∂u%m ⎤
α
α
2
%
A ( ei ; e s ) =
u
ik
e
C
ik
Δ
+
Δ
−
⋅
ρ
ω
exp
x
e
(
⎢
⎥
α sk
α
q
k qm j
s ) dV ( x )
2 ∫
∂x j ⎥⎦
4πρ cα V f ⎢⎣
β
α
−d qα
fields in the flaw replaced by incident fields
u%q = u%
incident
q
∂u%m ∂u%mincident
=
∂x j
∂x j
Flaw Scattering -Inclusion
For a spherical inclusion in pulse-echo
spherical Bessel function
A ( e i ; −e i ) = − 4 k β b F
β
where
β
2 3
j1 ( 2kβ b )
2k β b
1 ⎛ Δρ Δc ⎞
F= ⎜
+
⎟
⎜
2 ⎝ ρ cβ ⎟⎠
relative density relative wave speed
difference
difference
Flaw Scattering -Inclusion
Corresponding time domain impulse response for a
spherical inclusion (pulse-echo)
front surface
response
−2b
cα
back surface
response
2b
cα
t
Flaw Scattering -Inclusion
Comparison with "exact" separation of variables solution
for the pulse-echo response of a spherical inclusion by
synthesizing a time-domain response from frequencies
ranging from 0-20 MHz approximately for a 1 mm
radius inclusion (10 % differences in properties)
1.5
amplitude
1
0.5
0
-0.5
-1
-0.5
0
0.5
1
time, μsec
The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where the density
and compressional wave speed are both ten percent higher than the host steel. Solid line:
Born approximation, dashed line: separation of variables solution.
Flaw Scattering -Inclusion
Comparison with "exact" separation of variables solution
for the pulse-echo response of a spherical inclusion by
synthesizing a time-domain response from frequencies
ranging from 0-20 MHz approximately for a 1 mm
radius inclusion (50 % differences in properties)
8
6
amplitude
amplitude
errors
time of arrival errors
4
2
0
-2
-4
-1
-0.5
0
0.5
1
time, μsec
The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where the density
and compressional wave speed are both fifty percent higher than the host steel.Solid line: Born approximation,
dashed line: separation of variables solution.
Flaw Scattering -Inclusion
Doubly Distorted Born Approximation
ABorn ( eiβ ; −eiβ ) = −4kβ2 b3 F
j1 ( 2kβ b )
1 ⎛ Δρ Δc ⎞
F= ⎜
+
⎟⎟
⎜
2 ⎝ ρ cβ ⎠
2k β b
replace wave speeds of host material by that of the
flaw in the Born approximation
for the pulse-echo response of a spherical
inclusion
A ( e i ; −e i
β
β
)
DDBA
= −4k b F%
2
fβ
3
j1 ( 2k f β b )
2k f β b
where
1 ⎛ Δρ Δc
%
F= ⎜
+
⎜
2 ⎝ ρ f cf β
⎞
⎟⎟
⎠
Flaw Scattering -Inclusion
Front surface amplitude response is improved, and
relative time of arrival of back surface response is
correct, but there is an error in absolute time of
arrivals (50% differences in properties)
8
amplitude
6
4
2
0
-2
-4
-1
-0.5
0
0.5
1
time, μsec
The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where the density and
compressional wave speed are both fifty percent higher than the host steel. Solid line: Doubly Distorted Born
approximation, dashed line: separation of variables solution.
Flaw Scattering -Inclusion
Why is the front surface response amplitude improved
by the Doubly Distorted Born Approximation ?
2
1.8
1.6
F
1.4
1.2
1
R12
0.8
0.6
F%
0.4
0.2
0
1
1.2
1.4
1.6
1.8
Δρ
ρ
Answer: because
=
2
Δc
c
2.2
2.4
2.6
2.8
3
F% ≅ R12β ;β (plane wave reflection coefficient)
Flaw Scattering -Inclusion
This suggests we apply a phase correction to the doubly
distorted Born approximation and replace F% by R12β ;β
resulting in the modified Born approximation (MBA)
for the pulse-echo response of a spherical
inclusion
A ( e i ; −e i
β
β
)
MD 2
β ;β
= −4k b R12
2
fβ
3
exp ⎡⎣ 2ik f β b (1 − c f β / cβ ) ⎤⎦
j1 ( 2k f β b )
2k f β b
phase correction
Flaw Scattering -Inclusion
The MBA
(50 % differences in properties)
6
5
amplitude
4
3
2
1
0
-1
-2
-1
-0.5
0
0.5
1
time, μsec
The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where the density and compressional wave
speed are both fifty percent higher than the host steel.Solid line: MBA approximation, dashed line: separation of variables solution.
Flaw Scattering -Inclusion
The MBA
(100 % differences in properties)
20
amplitude
15
10
5
0
-5
-10
-1.5
-1
-0.5
0
0.5
1
1.5
2
time, μsec
The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where
the density and compressional wave speed are both one hundred percent higher than the host steel.
Solid line: MBA approximation, dashed line: separation of variables solution.
Flaw Scattering
The Method of Separation of Variables
The sphere and the cylinder are the only two geometries
where we can obtain exact separation of variables solutions for
elastic wave scattering problems. These are commonly used as
"exact" solutions to test more approximate theories and
numerical methods.
Flaw Scattering-Void
Example 1: pulse-echo P-wave scattering of a spherical void
−1 ∞
n
A ( e ; −e ) =
An
−
1
(
)
∑
ik p n =0
p
i
p
i
An =
E3 E42 − E4 E32
E31 E42 − E41 E32
{
}
E3 = ( 2n + 1) ⎡⎣ n 2 − n − ( k s2b 2 / 2 ) ⎤⎦ jn ( k pb ) + 2k p bjn +1 ( k p b )
{
}
E4 = ( 2n + 1) ( n − 1) jn ( k pb ) − k p bjn +1 ( k p b )
E31 = ⎡⎣ n 2 − n − ( k s2b 2 / 2 ) ⎤⎦ hn(1) ( k p b ) + 2k p bhn(1+)1 ( k p b )
E41 = ( n − 1) hn(1) ( k p b ) − k p bhn(1+)1 ( k p b )
E32 = − n ( n + 1) ⎡⎣( n − 1) hn(1) ( ks b ) − ks bhn(1+)1 ( k s b ) ⎤⎦
1
1
E42 = − ⎡⎣ n 2 − 1 − ( k s2b 2 / 2 ) ⎤⎦ hn( ) ( k s b ) − k s bhn( +)1 ( ks b )
Flaw Scattering-Void
The normalized P-wave pulse-echo scattering amplitude 2A/b
for a spherical void of radius b.
1.4
1.2
2|A| / b
1
0.8
0.6
0.4
0.2
0
0
5
10
15
frequency, MHz
20
25
30
Flaw Scattering-Void
Using this separation of variables solutions at many frequencies
to synthesize a P-wave impulse time domain solution (pulseecho)
amplitude
6
0
-4
-8
-12
-1
-0.5
0
0.5
1
1.5
time, μsec
The time-domain pulse-echo P-wave response of a 0.5 mm radius spherical void in steel ( cp = 5900 m/s, cs = 3200 m/sec)
obtained by applying a low-pass cosine-squared windowing filter between 10 and 20 MHz to the separation
of variables solution and then inverting the result into the time domain with the inverse Fourier transform.
Flaw Scattering-Void
Example 2: pulse-echo SV-wave scattering of a spherical void
−1 ∞ ( −1) ( 2n + 1) Bn
s
s
A ( ei ; −ei ) =
∑
ik s n =1
2
H J − H 43 J12
J
Bn = 13 42
− 41
H13 H 42 − H 43 H12 H 41
n
J12 = n ( n + 1) ⎡⎣( n − 1) jn ( ks b ) − ks b jn +1 ( k s b ) ⎤⎦
H12 = n ( n + 1) ⎡⎣( n − 1) hn(1) ( ks b ) − k s b hn(1+)1 ( ks b ) ⎤⎦
H13 = ⎡⎣ n 2 − n − ( ks2b 2 / 2 ) ⎤⎦ hn(1) ( k p b ) + 2k p b hn(1+)1 ( k p b )
J 41 = ( n − 1) jn ( k s b ) − ks b jn +1 ( ks b )
H 41 = ( n − 1) hn(1) ( ks b ) − ks b hn(1+)1 ( ks b )
J 42 = ⎡⎣ n 2 − 1 − ( ks2b 2 / 2 ) ⎤⎦ jn ( ks b ) + ks b jn +1 ( ks b )
1
1
H 42 = ⎡⎣ n 2 − 1 − ( ks2b 2 / 2 ) ⎤⎦ hn( ) ( k s b ) + ks b hn( +)1 ( k s b )
H 43 = ( n − 1) hn(1) ( k p b ) − k p b hn(1+)1 ( k p b )
Flaw Scattering-Void
The SV-wave pulse-echo scattering amplitude
0.7
0.6
amplitude
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
20
frequency, MHz
The magnitude of the pulse-echo SV-wave response, , versus frequency for a 0.5 mm radius spherical void in steel
( cp = 5900 m/s, cs = 3200 m/sec) as calculated by the method of separation of variables
Flaw Scattering-Void
amplitude
Using this separation of variables solutions at many frequencies
to synthesize an SV-wave impulse time domain solution (pulseecho)
6
0
-4
-8
-12
-1
-0.5
0
0.5
1
1.5
time, μsec
The time domain pulse-echo SV-wave response for the same void considered in the P-wave case by applying a low-pass cosinesquared windowing filter between 10 and 20 MHz to the separation of variables solution and then inverting the result into the time
domain with the inverse Fourier transform.
Flaw Scattering- SDH
Example 3: pulse-echo P-wave scattering of a cylindrical void
A3 D ( e ; −e
p
i
p
i
L
)=
i
2π
1/ 2
∞
∑ ( 2 − δ )( −1)
n
0n
n =0
⎛ 2iπ ⎞
A2 D (ω ) = ⎜
⎟
k
⎝ α2 ⎠
Fn
⎧1 n = 0
⎩0 otherwise
δ 0n = ⎨
Fn = 1 +
Cn( 2) ( k p b ) Cn(1) ( ks b ) − Dn( 2) ( k p b ) Dn(1) ( ks b )
Cn(1) ( k p b ) Cn(1) ( ks b ) − Dn(1) ( k p b ) Dn(1) ( ks b )
(
)
(
Cn( ) ( x ) = n 2 + n − ( k s b ) 2 H n( ) ( x ) − 2n H n( ) ( x ) − x H n( +)1 ( x )
i
2
i
(
i
i
Dn( ) ( x ) = n ( n + 1) H n( ) ( x ) − n 2n H n( ) ( x ) − x H n( +)1 ( x )
i
i
i
i
)
)
A3 D (ω )
L
Flaw Scattering-SDH
Recall, the scattering amplitude for the pulse-echo P-wave
case was:
0.9
0.8
0.7
0.6
A3 D / L
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
k pb
8
10
Flaw Scattering- SDH
Using this separation of variables solutions at many frequencies
to synthesize a P-wave impulse time domain solution (pulseecho)
30
amplitude
20
creeping wave
10
0
-10
-20
-30
-1
-0.5
0
0.5
1
1.5
time, μsec
The time-domain pulse-echo P-wave response of a 0.5 mm radius cylindrical void in steel (cp = 5900 m/s, cs = 3200 m/sec) obtained by
applying a low-pass cosine-squared windowing filter between 10 and 20 MHz to the separation of variables solution and then inverting the
result into the time domain with the inverse Fourier transform.
Flaw Scattering - SDH
Example 4: pulse-echo SV-wave scattering of a cylindrical void
A3 D ( eisv ; −eisv )
L
i
=
2π
∞
∑ ( 2 − δ 0n )( −1)
n=0
n
Gn
⎧1 n = 0
⎩0 otherwise
δ 0n = ⎨
( ks b ) Cn(1) ( k pb ) − Dn( 2) ( ks b ) Dn(1) ( k pb )
Gn = 1 + (1)
1
1
1
Cn ( k p b ) Cn( ) ( ks b ) − Dn( ) ( k p b ) Dn( ) ( ks b )
Cn(
2)
Flaw Scattering -SDH
Again, the scattering amplitude for the pulse-echo SVwave case was:
1.1
1
0.8
0.6
A3 D / L
0.4
0.2
0
0
2
4
ks b
6
8
10
Flaw Scattering - SDH
Using this separation of variables solutions at many frequencies
to synthesize an SV-wave impulse time domain solution
(pulse-echo)
30
creeping wave
amplitude
20
10
0
-10
-20
-30
-40
-1
-0.5
0
0.5
1
1.5
time, μsec
The time-domain pulse-echo SV-wave response of a 0.5 mm radius cylindrical void in steel (cp =5900 m/s, cs = 3200 m/sec)
obtained by applying a low-pass cosine-squared windowing filter between 10 and 20 MHz to the separation of variables solution and
then inverting the result into the time domain with the inverse Fourier transform.
Flaw Scattering - SDH
Experimentally determined scattering amplitude by
deconvolution (side-drilled hole)
⎡ A (ω ) ⎤
VR (ω ) = G (ω ) ⎢
⎥
L
⎣
⎦
G (ω ) = s (ω ) E (ω )
⎡ ˆ (1)
⎤ ⎡ 4πρ 2 cα 2 ⎤
( 2)
ˆ
E (ω ) = ⎢ ∫ V0 ( z , ω ) V0 ( z , ω ) dz ⎥ ⎢
T ;a ⎥
−
ik
Z
⎣L
⎦ ⎣ α2 r ⎦
Flaw Scattering -SDH
0.9
0.8
0.7
|A/L|
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
frequency, MHz
12
14
16
18
20