Vij D.R. Handbook of Applied Solid State Spectroscopy
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8. Resonance Acoustic Spectroscopy
2
șș
2
ı
ı 1 ı 1
(ı ) ȡ
ș
r
rr rz r
rr c
u
rr zr t
T
w
ww w
www w
)(8.13
2
șșș ș ș
ș
2
ıı ı
12
ıȡ
ș
rz
rc
u
rr r z t
ww w w
ww w w
(8.14)
2
ș
zz
2
ı
ı 11ı
ȡ
ș
z
rz z
rz c
u
rr r z t
w
www
ww w w
(8.15)
where
ȡ is the cylinder density and
ș
,,
r
uu and
z
u are the displacements in
the
, ș,r and z directions, respectively. Substituting equation (8.12) in
equations (8.13) – (8.15) gives,
22
44
2
zr
uu
c
z
rz
§·
ww
¨¸
ww w
©¹
2
22
șș
11
22222
22
șș
12 13
2
13 1 1
2 ș 2 ș 2 ș
11
ȡ , (8.16)
2 șș
rr rr
rzr
c
uu
uu uu
c
rrrr r r rr
uu
uuu
cc
rrrrzrt
§·
ww
ww w
¨¸
ww w w ww
©¹
ªw º
ww ww
§·
¨¸
«»
©¹
ww w ww w
¬¼
ș
44
1
ș
z
u
u
c
zzr
ªw º
ww
§·
¨¸
«»
©¹
ww w
¬¼
22
2
șșș ș
11
22 2 2 2
111311
ș 222ș 2 ș 2
rr
uuu u
uu
c
rrrrrrrr
§·
www
ww
¨¸
www www
©¹
2
2
13 ș
2
ȡ ,
ș
z
c
cu
u
rz t
w
w
ww w
(8.17)
2
22 2
ș
44
22 2
1111
șș
zz z rr
u
uu u uu
c
r r rz rr rz zr
§·
w
ww w ww
¨¸
w w ww w w ww
©¹
22
ș
13 33
22
1
ȡ .
ș
rrzz
c
u
uuuu
cc
zr r r z t
ªwº
wwww
§·
¨¸
«»
©¹
wwwww
¬¼
(8.18)
The displacement vector can be written in terms of three scalar potentials,
I
,
Ȥ
, and
ȥ
[49],
ˆˆ
(Ȥ )(ȥ ),
zz
ea e Iu uuU
(8.19)
where a is the radius of the cylinder. Substituting equation (8.19) in equations
ı
ı
(8.16) – (8.18) results in the following set of equations:
358
c
8.3 Mathematical Models
222
22
11 13 44 11
222
(2 ) ȡ
c
cccc
zzt
§·
wwIwI
I
®
¨¸
www
©¹
¯
22
2
11 13 44 13 44 11
22
ȥȥ
()ȥ (2 ) 0
c
accc ccc
zzt
½
ªº
www
¾
«»
www
¬¼
¿
(8.20)
22
2
13 44 33 13 44
22
(2) ( 2)
c
cc ccc
zzt
ªº
wwIwI
I
«»
www
¬¼
222
22
44 33 13 44
222
ȥȥ
ȥ (2)ȡ 0
c
acccc
zzt
§·ª º
www
«»
¨¸
www
©¹
¬¼
(8.21)
222
22
11 12 11 12
44
222
() ()ȤȤ
Ȥȡ0
22
c
cc cc
c
zzt
§·ª º
w ww
§·
¨¸
«»
¨¸
©¹
www
©¹
¬¼
(8.22)
The first two equations, (8.20) and (8.21) indicate that the compression
wave P (represented by
I) and the vertically polarized shear wave SV
(represented by
ȥ ) are coupled. Equation (8.22) shows that the horizontally
polarized shear wave SH (represented by
Ȥ
) is decoupled. Equations (8.20) –
(8.22) should be solved for
I, ȥ, and
Ȥ
. Applying the separation of
variables technique and taking advantage of irrotational and solenoidal
properties of the longitudinal and transverse fields, the solution to
equation
(8.20) can be obtained by setting the terms in curly brackets equal
to zero
[50],
22
2
11 13 44 11
22
(2 ) ȡ
c
dilational
cccc
zt
ªº
«»
wI wI
«»
I
ww
«»
«»
¬¼
22
2
11 13 44 13 44 11
22
ȥȥ
()ȥ (2 ) ȡ 0,
c
accc ccc
zzt
ªº
ªº
«»
«»
www
«»
«»
www
«»
«»
«»
«»
¬¼
¬¼
(8.23)
22
2
11 13 44 11
22
(2 ) ȡ 0,
c
cccc
zt
wI wI
I
ww
(8.24)
ȡ
ȡ
rotational
359
that is,
8. Resonance Acoustic Spectroscopy
22
2
11 13 44 13 44 11
22
ȥȥ
()ȥ (2 ) ȡ 0.
c
ccc c cc
zt
ww
ww
(8.25)
By solving the resulting separated partial differential equations, the scalar
potential functions for an infinite cylinder are found to be of the form [50],
0
,ș,, ( )cos(ș)
z
kzi
nn
n
rzt BJsr ne
f
I
¦
, (8.26)
0
ȥ ,ș,, ( )cos(ș),
z
kzi
nn
n
rzt CJsr ne
f
¦
(8.27)
0
Ȥ ,ș,, ( )sin(ș).
z
kzi
nn
n
rzt DJsr ne
f
¦
(8.28)
where s is an unknown parameter that must be determined. Substituting
equations (8.26) – (8.28) in equations (8.20) – (8.22) gives the A
1u3
B
1u3
=
1u3
matrix equation:
2222
11 13 44 44
22 2 2
44 33 13 44
(( ) ( Ȧ ())
((ȡȦ ()))
0
zcz
cz
aik s c c c s c k
as c s c c c k
U
2222
11 12 44
00
00.
0
(( ) 2(ȡȦ ))
n
n
n
cz
B
C
D
sc cs ck
º
ªº
ª
º
»
«»
«
»
»
«»
«
»
»
«»
«
»
¬
¼
¬¼
¼
(8.29)
For a nontrivial solution, the coefficient determinant in equation (8.29) must
be zero. This yields the following characteristic equation:
42 2 2 2
11 12
44 11 44
( ȟȗ)
((
)
ȡȦ
)
0,
2
cz
cc
ccs s s ck
(8.30)
where
22 2 2 2 2
13 44 11 33 44 44
ȟ () (ȡȦ )(ȡȦ ),
zc z c z
cckc ck c ck
(8.31)
2222
44 33
ȗ (ȡȦ )(ȡȦ ).
czcz
ck ck
(8.32)
Three roots
,,
21
ss and
3
s of equation (8.30) correspond to the wave
numbers of P, SV, and SH waves, respectively, and can be written as,
2
11 44
2
1
11 44
ȟȟ4 ȗ
,
2
cc
s
cc
(8.33)
22 2 2
11 13 44
222
13 44 33
((ȡȦ (2)))
(( 2 ) (ȡȦ ))
0
cz
zcz
scs c c k
ik c c s c k
ª
«
«
«
¬
and,
O
360
8.3 Mathematical Models
2
11 44
2
2
11 44
ȟȟ4 ȗ
,
2
cc
s
cc
(8.34)
22
2
44
3
11 12
2(ȡȦ )
,
cz
ck
s
cc
(8.35)
which implies that the potential functions should be of the form [45, 50]:
>@
12 2
0
,ș,, ( ) ( )cos(ș)
z
kzi
nn nn
n
rzt BJsrqCJsr ne
f
I
¦
, (8.36)
>@
11 2
0
ȥ ,ș,, ( ) ( )cos(ș)
z
kzi
nn nn
n
rzt qBJsrCJsr ne
f
¦
, (8.37)
3
0
Ȥ(,ș,,) ( )sin(ș).
z
kzi
nn
n
rzt DJsr ne
f
¦
(8.38)
and the coupling coefficients are
22 2
11 1 c 13 44
1
222
11 13 44 1 44
((ȡȦ (2)))
,
(( ) (ȡ ())
z
zcz
cs c c k
q
aik c c c s c k
(8.39)
22 2
11 13 44 2 44
2
22 2
11 2 c 13 44
(( ) (ȡȦ ())
.
((ȡȦ (2)))
zcz
z
aik c c c s c k
q
cs c c k
(8.40)
For the submerged cylinder, the boundary conditions at
ar are [51]
is is ș
2
w
1
();ı ();ı 0; ı 0.
ȡȦ
rij r rz
pp u pp
r
w
w
(8.41)
Expanded expressions for displacements and stresses in terms of potential
functions can be found in [45]. By inserting the potential functions from
equations (8.36) – (8.38) into equation (8.41), a system of four linear algebraic
equations is obtained for each value of n:
11 12 13 14
1
21 22 23 24
2
31 32 33 34
41 42 43 44
0
0
n
n
n
n
aaaa A
b
aaaa B
b
aaaaC
aaaa D
ªºªº
ª
º
«»«»
«
»
«»«»
«
»
«»«»
«
»
«»«»
«
»
¬
¼
¬¼¬¼
. (8.42)
Elements
ij
a and
i
b of the matrices in equation (8.42) are given in [45].
Equation (8.42) can be solved for
n
A for any specified frequency using
Cramer’s rule.
Ȧ
361
8. Resonance Acoustic Spectroscopy
1121314
2222324
32 33 34
42 43 44
11 12 13 14
21 22 23 24
32 33 34
42 43 44
0
0
()
0
0
n
ba a a
ba a a
aaa
aaa
Aka
aaaa
aaaa
aaa
aaa
. (8.43)
The scattered pressure field is usually evaluated in the far-field
)( ar !! at a
fixed angle
ș for a range of frequencies. The resulting far-field amplitude
spectrum, which is called a form function, is obtained from the equation [19],
1/2
s
o
p
2
(, ) .
p
kri
r
fka e
a
A
f
§·
§·
T
¨¸
¨¸
©¹
©¹
(8.44)
The total form function can be written as the sum of individual normal
modes,
0
(ș,) (,)
n
n
fka fka
f
f
T
¦
, (8.45)
where the individual normal modes are defined as,
2
(ș,) İ cos( ).
ʌ
nnn
fka A n
ika
A
T(8.46)
For solving the corresponding problem for isotropic cylinders, it is a
common practice to use the Helmholtz decomposition method in which the
displacement field is written in terms of two distinct components [49], i.e.,
IuU ȥ , (8.47)
where
I is the rotation-free component corresponding to the longitudinal
wave and
ȥ is the solenoidal term, which represents the S-wave. This
approach is easier but cannot be used in the case of anisotropic materials since
in these materials the wave modes are not of a purely compression or shear
nature any more.
It can be shown that the model developed in the previous section can also
be applied to isotropic cylinders [45]. For an isotropic cylinder, the values of
the elastic constants
ij
c can be written in terms of the Lame constants Ȝ and
11 33 12 13 44
Ȝ 2ȝ; Ȝ; ȝ.cc cc c (8.48)
362
ȝ such that,
8.3 Mathematical Models
By substituting the above values into equations (8.36), (8.37) and (8.38)
these equations are reduced to
L
0
,ș,, ( )cos(ș)
z
kzi
nn
n
rzt BJkr ne
f
I
¦
, (8.49)
T
0
ȥ ,ș,, ( )cos(ș)
z
kzi
nn
n
rzt CJkr ne
f
¦
, (8.50)
T
0
Ȥ
(,ș,,) ( )sin(ș).
z
kzi
nn
n
rzt DJkr ne
f
¦
(8.51)
where,
12 12
22
22
LT
22
LT
ȦȦ
;
zz
kkkk
cc
§· §·
¨¸ ¨¸
©¹ ©¹
. (8.52)
Equations (8.49) to (8.51) are the corresponding potential functions for an
isotropic cylinder.
8.3.1 Resonance Scattering Theory (RST)
The formalism of resonance scattering theory (RST) for acoustic resonance
scattering is based on the corresponding theory in scattering [52]. RST
provides a physical explanation of the wave phenomena that are observed in
the acoustic-vibrational interaction. RST states that in acoustic scattering, the
spectrum of the returned echo from a submerged elastic cylinder consists of
two distinct parts: a smooth background plus a resonant part. The background
varies smoothly with frequency and is present even for impenetrable
scatterers and the resonant part consists of a number of resonance peaks that
coincide with the eigenfrequencies of the vibrations of the cylinder. As
mentioned by its originators, RST carries out two major functions:
(1) It separates the resonance components from the background signal.
(2) It mathematically expresses the resonance amplitude in a form where
its resonance nature can be observed explicitly.
RST has been discussed in detail in many references [18–19, 39, 53–54]. In
this section, principles of RST are briefly discussed. Let’s start with consi-
dering an infinite plane acoustic wave
( Ȧ )
io
p
p,
ikx t
e
(8.53)
with a propagation constant
Ȧ /kc incident along the x-axis (Į 0
D
) on a
solid elastic cylinder (see Figure 8.1).
We follow closely the derivation presented by Flax et al. [19]. At a point
s
363
M (,r ș) the incident wave produces the following scattered field
p ,
8. Resonance Acoustic Spectroscopy
(1)
so
0
İ () ( )cos(ș),
n
nn n
n
p
piAxHkrn
f
¦
(8.54)
where
.
)()(
)()(
)(
)1()1(
xxHFxH
xJxFxJ
xA
nnn
nnn
n
c
(8.55)
The quantities F
n
, related to the modal mechanical impedance of the
cylinder [55], are the quotient of two
22 u determinants,
(1)
2
w
T
(2)
c
ȡ ()
() ,
ȡ ()
n
n
n
Dx
Fx x
Dx
(8.56)
where
22 23
(1)
32 33
n
aa
D
aa
and
12 13
(2)
32 33
.
n
aa
D
aa
(8.57)
Expressions for a
ij
are given in [19]. It should be noted that the matrices
given in equation (8.57) are similar to those given in equation (8.43),
however, isotropy of the material and normalcy of the incident wave has
resulted in much smaller matrices. The argument x of the Bessel and Hankel
functions in equation (8.56) is
w
Ȧ /xka ac , where
w
c is the speed of
sound in the surrounding fluid. The prime denotes differentiation with respect
to the argument. In the far-field where
)(,
)1(
xHar
n
!! may be written in its
asymptotic form [47],
(1 )
1/ 2
() (2/ʌ ).
nix
n
Hx xie
| (8.58)
Then the far-field scattered pressure becomes
1/2
so
0
(ș)(2/ʌ ) İ ()cos(ș),
ikr
nn
n
ppeikr Axn
f
¦
(8.59)
and the form function
f
f of the scattered pressure is
s
o
2
(ș).
ikr
p
r
fe
ap
f
(8.60)
0
2
(ș) İ ()cos(ș).
ʌ
nn
n
fAxn
ix
f
f
¦
(8.61)
where individual normal modes or partial waves which make up the form
functions are defined as
2
(ș) İ ()cos(ș),
ʌ
nnn
fAxn
ix
(8.62)
so that,
0
(ș)(ș).
n
n
ff
f
f
¦
(8.63)
The form function can also be represented by using equation (1.59),
364
8.3 Mathematical Models
There are two limiting cases of these results. If
c
ȡ ,of the solution
represents the scattering by a rigid cylinder:
),(/)()(
)1(
xHxJxA
nn
r
n
c
c
(8.64)
and if
c
ȡ
s(1)
() ()/ ().
nnn
A
xJxHx (8.65)
The scattered pressure of equation (8.54) may be written as a normal mode
series (setting
0
1p ),
(1)
s
0
1
(,ș) İ (1)()cos(ș)
2
n
nn n
n
p
riSHkrn
f
¦
(8.66)
where the scattering function of the nth mode with a constant unit magnitude
containing the scattering phase shifts
į
n
is introduced as follows:
2 į
.
n
i
n
Se (8.67)
In the present case,
12().
nn
SAx (8.68)
For rigid and soft cylinders, the scattering functions are, respectively,
r
2 į
r(2) (1)
()/ ()
n
i
nn n
SHxHxe
cc
(8.69)
and
s
2 į
s(2) (1)
()/ ()
n
i
nn n
SHxHxe (8.70)
Superscripts r and s have been used for terms corresponding to rigid and
soft cylinders, respectively. The corresponding phase shifts can be shown to
be the real quantities,
r
tan į ()/ (),
nn n
JxYx
cc
and
s
tan į ()/ ().
nn n
JxYx (8.71)
A rigid or soft scattering function may be factored out from the elastic
scattering function as follows:
r1(2)1 1(1)1
()/()
nnnn nn
SSFz Fz
(8.72)
or
s(2) (1)
()/(),
nnnn nn
SSFz Fz (8.73)
where the quantities
() () ()
()/ (), 1,2
ii i
nn n
zxHxHx i
c
(8.74)
are related to the modal specific acoustic impedances [55].
equation (8.73) can be approximated as,
365
o 0, the solution represents the scattering by a soft cylinder:
After some mathematical manipulations [18–19], it can be shown that
8. Resonance Acoustic Spectroscopy
rr
r
2 įį
r
rr
1
1
2
12 sinį ,
1
2
nn
nl
ii
nn
l
nl nl
Sie e
xx i
f
ªº
*
«»
«
»
«»
*
¬¼
¦
(8.75)
or
ss
s
2 įį
s
ss
1
1
2
12 sinį ,
1
2
nn
nl
ii
nn
l
nl nl
Sie e
xx i
f
ªº
*
«»
«»
«»
*
¬¼
¦
(8.76)
where
r
nl
x and
s
nl
x are the lth resonance frequencies of the nth mode and
r
nl
*
and
r
nl
* are the resonance widths related to radiation damping. Based on
expressions (8.75) and (8.76), resonance scattering theory argues that the
scattered field is a summation of two components, i.e., a resonance
component, which is the first term of equations (8.75) and (8.76), and a
smooth background component, which is the second term of these equations.
Following the approach proposed by Rhee and Park [56], the scattering
function
n
S can be written as,
r*
,
nnn
SSS (8.77)
where
*1(2)11(1)1
()/().
nnn nn
SFz Fz
(8.78)
Equation (8.77) states that
n
S is the product of a rigid background
r
n
S and
the remaining term
*
n
S , which includes resonances. However,
*
n
S is not a pure
resonance term. It has a constant term, which hides resonances unless it is
removed. To see this,
*
n
S can be written as
)/()(
1)1(11)2(1*
nnnnn
zFzFS
(1) 1 (2) 1 1 (1) 1 res
()/()11,
nn nn n
zz Fz S
(8.79)
where
res
n
S is defined as the resonance scattering function that consists purely
of resonance information. From equation (8.77), the resonance scattering
function
res
n
S can be expressed as
(1) 1 (2) 1
res
r1(1)1
12,
12
r
nnn nn
n
r
nnn n
Szz AA
S
SFz A
(8.80)
where
(1/ 2)( 1)
nn
AS and
rr
(1 / 2)( 1).
nn
AS
366
8.3 Mathematical Models
In equation (8.66), 1
n
S contributes to both the background and
resonance parts of the scattered field.
res
n
S is the component that only contains
resonance effects of
n
S . Therefore, from equation (8.66), the scattered wave
due to resonance can be obtained by replacing
1
n
S with
res
n
S :
res res (1)
s
0
1
(,ș) İ () ()cos(ș)
2
n
nn n
n
p
riSHkrn
f
¦
r
(1)
r
0
İ ()cos(ș)
12
n
nn
nn
n
n
AA
iHkrn
A
f
¦
(8.81)
and the individual normal mode for resonances in the far-field can be written
as
r
res
r
2
(ș) İ cos( ș).
12
ʌ
nn
nn
n
AA
fn
A
ix
(8.82)
The expressions containing the corresponding soft background parameters
are analogous,
*(2)(1)
()/()
nnn nn
SFz Fz
(1) (2) (1) res
()/()11
nn nn n
zz Fz S (8.83)
(1) ( 2) s
res
s(1)s
12
12
nnn nn
n
nnn n
SzzAA
S
SFz A
(8.84)
s
res
s
2
(ș) İ cos( ș).
12
ʌ
nn
nn
n
AA
fn
A
ix
(8.85)
The expressions in equations (8.82) and (8.85) are also applicable to
acoustic wave scattering from shell structures or fluid cylinders with
appropriate changes in F
n
. Using equations (8.82) and (8.85), the resonance
part of the backscattered field can be completely separated from the
background. A typical extraction of the resonance spectrum from the form
function based on RST is shown in Figure 8.3.
367