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Onipko, A.I., Berggren, K.-F., Klymenko, Yu O., Malysheva, L.I., Rosink, J.J.W.M.,
CHAPTER 8
RESONANCE ACOUSTIC SPECTROSCOPY
Farhang Honarvar and Esmaeil Enjilela
K.N. Toosi University of Technology, Faculty of Mechanical Engineering
P. O. Box 16765-3381, Tehran, Iran.
8.1 INTRODUCTION
Resonance acoustic spectroscopy (RAS) is the study of resonance effects
present in acoustic echoes of an elastic target. These resonance effects are
caused by the excitation of eigenvibrations of the target by an incident
acoustic wave. Such use of resonance effects for the purpose of
characterization of an elastic object (shape, size, composition, and integrity) is
similar to the way chemical elements are identified from their optical
properties. RAS is a powerful tool for monitoring various characteristics of
elastic components and in many engineering applications for quality control
purposes.
Resonances are intrinsic characteristics of the elastic object, characteristics
that depend only on its physical properties and are completely independent of
the source of excitation. The resonances of an elastic component are its
fingerprints. For a given geometry, resonances are indicators of the elastic
properties of the material such as density and elastic constants. Any defects,
inclusions, or inhomogeneities in the material will also affect its resonance
frequencies. Appropriate exploitation of the information contained within the
vibratory signals and identification of the resonance frequencies of the elastic
object can serve as a powerful tool for material characterization purposes.
In resonance acoustic spectroscopy, the target is insonified by a harmonic
acoustic wave. If the frequency of the wave coincides with one of the
resonance frequencies of the target, the corresponding eigenmode of the target
is excited and the target starts to vibrate, emitting energy into the surrounding
fluid. The vibrations of the target can then be measured by a sensing device.
By changing the carrier frequency of the incident wave, various eigen-
frequencies of the scatterer can be identified and measured. These
measurements can either be compared to predetermined reference values or
can be matched to a theoretical model. In the latter case, the physical
properties used in the mathematical model will correspond to those of the
elastic scatterer.
8. Resonance Acoustic Spectroscopy
Since RAS is based on the scattering of waves from elastic targets, in this
chapter we will first review the principles of wave scattering (Section 8.2).
This will be followed by a discussion of the mathematical models for acoustic
wave scattering from cylindrical components (Section 8.3). Experimental
methods developed for measurement of the scattered field, used for
“identification and “isolation” of resonances observed in the backscattered
theoretical capabilities of resonance scattering theory (RST) with these
experimental tools makes the application of RAS practically viable.
8.2 SCATTERING OF WAVES
The scattering and propagation of waves is a classical area that has engaged
the interest of many scientists in various disciplines. Extensive research in the
areas of acoustics, elastodynamics, and electromagnetism has resulted in
many mathematical, experimental, and computational techniques.
In acoustics, the medium in which the wave propagates is a fluid. Since
ideal fluids cannot support transverse stresses, the wavefield consists of
compression waves only. In the more general case of elastic waves, the
medium can be an elastic solid, which can support both normal and shear
stresses. Hence, the wavefield is composed of both compression and
transverse components. Moreover, there is also the possibility of “mode
conversion,” i.e., conversion between compression waves and shear waves
during the process of scattering.
The target can be either penetrable (transparent) or impenetrable (opaque).
For impenetrable targets, no internal fields can exist inside the body of the
scatterer. The impenetrable targets can be either soft (e.g., a bubble in water)
or rigid (e.g., an aluminum cylinder in air). On the other hand, in the case of
penetrable targets, the wave field can penetrate into the body of the target and
generate different types of waves on the surface and inside the body of the
target.
In this section, the scattering of acoustic waves from elastic targets is
considered. The geometry of interest is a cylinder; however, similar
theoretical and experimental procedures are available for spheres and
cylindrical shells as well.
8.2.1 Physics of Acoustic Resonance Scattering
The physics of the scattering of an acoustic wave from an elastic target is a
complicated phenomenon that involves the generation of different types of
surface waves outside and inside the elastic target. For example, in the case of
”
352
integrity, and health conditions of engineering components are presented.
echoes of elastic targets, are also discussed (Section 8.4). Combination of the
Sample applications of RAS for characterization of the shape, composition,
8.2 Scattering of Waves
an elastic cylindrical target, when a normally incident acoustic wave strikes
the boundary of the cylinder, in addition to the reflected specular echo,
surface waves are generated on the boundary of the cylinder. If the wave is
normally incident on the cylinder as shown in Figure 8.1, surface waves
circumnavigate the body in spiral trajectories, and if the wave is obliquely
incident, the travel paths are helical.
Figure 8.1 Insonification of an elastic cylinder by a normally incident plane acoustic wave.
These surface waves propagate near the boundary, either in the outer
medium, or in the body of the scatterer [1]. The waves that propagate on the
fluid side of the fluid-solid interface are Franz and Scholte-Stoneley waves
(also called creeping waves) [2, 3]. These waves are primarily related to the
geometry of the interface along which they travel and are present in the
extreme cases of rigid and soft cylinders. By contrast, at normal incidence,
two
distinct families of waves can propagate inside the elastic cylinder. The
first group comprises Rayleigh-type (or, more correctly, pseudo-Rayleigh-
type) waves and the other group comprises whispering gallery-type (WG)
waves [1]. Both of these two types of surface waves were first explained by
Rayleigh [4, 5], and were further studied by many other researchers [3, 6–10].
In the case of scattering from an elastic cylinder in vacuum, a real Rayleigh-
type wave exists, which becomes a true Rayleigh wave in the limit when the
cylinder radius tends to infinity. Rayleigh and WG waves have been
experimentally photographed inside submerged glass cylinders [11, 12].
When the wave is obliquely incident on the cylinder, additional types of
waves are generated in the cylinder, and correspondingly additional resonance
modes of the cylinder are excited. These additional (surface) waves which
travel on helical paths are called (axially) guided waves (GW) [13–15] and are
mostly of a transverse nature. The angle of each helix depends on the phase
velocity of the corresponding surface wave [16].
A close relationship exists between the resonances of an elastic scatterer
and the circumferential waves that travel along its periphery [17]. These
353
8. Resonance Acoustic Spectroscopy
surface waves and their relationship with the natural resonances of the
scatterer have been extensively studied for the case of elastic cylinders. If the
frequency of the incident wave coincides with one of the vibrational
eigenfrequencies of the cylinder, surface waves match phases upon
circumnavigation so as to build up to the resonance by constructive
interference. It was the resonance scattering theory (RST) formulated by
Uberall and his coworkers [18, 19] that physically described this phenomenon
for the first time. RST will be discussed in more detail in further sections.
8.2.2 Acoustic Wave-Scattering from Elastic Targets
The dynamic scattering response of impenetrable scatterers, such as soft or
rigid cylinders, to an incident wavefield has been studied since the pioneering
works of Rayleigh [4]. In scalar form, the wave equation can be separated and
solved in terms of infinite Rayleigh series in eleven coordinate systems [20].
Due to slow convergence of Rayleigh series, this is usually is a time-
consuming process. The Watson-Summerfeld method was suggested as a
better solution technique in 1968 [21]. In this method, the standard contour of
integration for the slowly convergent Rayleigh series is transformed such that
the new contour results in a quickly convergent series. The new path encloses
new poles, which are associated with a quickly convergent residue series. The
Watson-Summerfeld method has been most successful in explaining the
physical behavior of the waves generated around impenetrable targets (called
Franz waves).
In the case of elastic targets, the scattering equations are more complicated,
and even in the case of isotropic materials they can be solved in a closed form
for only a few geometries. If either the scatterer or the surrounding medium
(or both) is elastic, then the wave equations can only be solved in three
coordinate systems; viz. in spherical, in two-dimensional cylindrical, and in
Cartesian (for an infinite plane fluid layer) coordinate systems [20]. The only
true 3D target with closed form solution is the elastic sphere. Numerical
schemes have been used for treating elastic scatterers of nonspherical
geometry [22, 23].
The first pioneering study of acoustic wave scattering from submerged
solid elastic circular cylinders and spheres based on normal-mode expansion
is due to Faran [24]. In the case of an elastic cylinder insonified by a normally
incident, plane acoustic wave, with the assumption that both the incident
plane wave and the cylinder be infinite, the problem has no dependence on the
axial coordinate of the cylinder (2D problem). Hickling [25] formulated the
problem of acoustic wave scattering from elastic spheres and carried out
experiments on different spherical targets.
Although White [26] formulated the problem of the scattering of an
obliquely incident wave from an infinite elastic cylinder embedded in another
354
8.2 Scattering of Waves
elastic medium in 1958, it was not until 1980 that the problem of scattering of
an obliquely incident acoustic wave from a solid elastic cylinder was
specifically addressed. Flax et al. [27] formulated this problem and numeri-
cally evaluated the far-field backscattered amplitude spectrum for an isotropic
aluminum cylinder at various angles of incidence. They noted the appearance
of the resonance effects in the numerically calculated spectra. The same
problem was also studied by Li and Ueda [28] who carried out experimental
measurements on aluminum and steel rods at small angles of incidence. Bao
et al. [29] studied the relationship between resonances and surface waves in
the scattering of an obliquely incident wave from an infinite elastic cylinder.
They also evaluated the dispersion curves of Rayleigh-type helical surface
waves.
Development of the resonance scattering theory by Uberall and his
observed in scattering of waves from elastic targets.
Surface waves generated by a plane acoustic wave obliquely incident on
a
cylinder were studied by Nagl et al. [30], basing their work on RST for elastic
cylinders and the Watson-Summerfeld transformation for impenetrable
cylinders. Uberall [22] studied the helical surface waves on cylinders and
cylindrical cavities of finite length and found that a refraction effect takes
place between incident and helical wave directions. His calculations for finite
length cylinders were based on a transition matrix (or T-matrix) approx-
imation [16]. Veksler [31] computed the modal resonances for an aluminum
cylinder insonified by an obliquely incident wave and showed their
correspondence to the extrema of the backscattered spectrum. A geometrical
interpretation of the phase matching of the helical surface waves was given by
Conoir et al. [32, 33], who also showed that the resonances shift towards
higher frequencies when the angle of incidence is increased. Further
explanations on this phenomena were presented by Ying et al. [34].
The theoretical explanations of RST were later complemented by
experimental procedures of the method of isolation and identification of
resonances (MIIR) proposed by French researchers [35–44]. RST and MIIR
provide the necessary theoretical and experimental tools for resonance
acoustic spectroscopy applications.
The mathematical models developed for scattering of waves from isotropic
cylinders for normally [24] and obliquely [27] incident waves were improved
by Honarvar and Sinclair [45] to include both isotropic and transversely
isotropic cylinders. The mathematical model developed by Honarvar and
Sinclair was the first attempt in considering anisotropic scatterers. Ahmad and
Rahman suggested an alternative solution for the problem of scattering of
waves from transversely isotropic cylinders [46]. Since the mathematical
model for wave scattering from anisotropic cylinders is more general and
includes the simpler case of isotropic cylinders, this model will be discussed
in the next section.
coworkers [18, 19] provided a physical explanation for the phenomena
355
8. Resonance Acoustic Spectroscopy
The mathematical model for the scattering of a plane harmonic acoustic wave
obliquely incident on a transversely isotropic cylinder will be discussed in this
section. Transversely isotropic materials have hexagonal symmetry, and in the
from elastic properties in the transverse plane.
Figure 8.2 shows an infinite monochromatic plane acoustic wave of
frequency
/2ZS incident at an angle D on a submerged circular cylinder of
infinite length and outer radius a, and density
c
U . The cylinder is submerged in
a fluid of density
w
U .
Figure 8.2 Geometry used for formulating the problem.
A cylindrical coordinate system (,,)rzT is chosen with the z-direction
coincident with the axis of the cylinder. The pressure
p
i
of the incident plane
compression wave, external to the cylinder at an arbitrary point M
(,,)rzT is
defined by,
i0
,ppe (8.1)
where p
0
is the incident wave amplitude, k is the wave vector, and r is the
position vector of the point M. Moreover,
k,c Z where Z is the angular
frequency and c is the compression wave velocity in the fluid medium
surrounding the cylinder. In a Cartesian coordinate system, one can write:
ˆˆˆ
cosș sin ș ,rrz
xyz
re ee (8.2)
and
ˆˆ ˆ
(cosĮ 0sinĮ ).k
xy z
keee (8.3)
Therefore,
cosĮ cosș sin Į.kr kz kr (8.4)
After introducing the notation
sin Į;cosĮ,
z
kk k k
A
(8.5)
356
8.3 MATHEMATICAL MODELS
it()k.rZ
case of a cylinder, it is assumed that the axial elastic properties are different
8.3 Mathematical Models
cosș ( Ȧ )
i
.
z
ik r i k z t
ppe e
A
(8.6)
The first exponential term in equation (8.6) can be written in terms of
cylindrical functions [47, (Formula 10.1.47)],
cosș
0
İ ()cos(ș),
ik r
n
nn
n
eiJkrn
A
f
A
¦
(8.7)
where
İ
n
is the Neumann factor (
0
İ 1 and İ 2
n
for 1tn ) and
n
J is the
( Ȧ )
i
0
İ ()cos(ș).
z
ikz t
n
nn
n
pp iJkr ne
f
A
¦
(8.8)
We consider the steady state situation and drop the time dependence term
from now on. The outgoing scattered wave pressure p
s
at point M satisfies the
wave equation,
2
2
s
22
1
0,p
ct
§·
w
¨¸
w
©¹
(8.9)
where
2
is the Laplace operator. In a cylindrical coordinate system,
equation (8.9) can be expanded to
2222
s
222222
11 1
0.
ș
p
rrrr zct
§·
wwww w
¨¸
wwww w
©¹
(8.10)
By using the method of separation of variables, the scattered pressure field
will be found as follows,
(1)
s0
0
İ ( ) cos( ș).
z
kzi
n
nnn
n
pp iAHkr ne
f
A
¦
(8.11)
where
n
A
for a transversely isotropic
material is,
11 12 13
șș șș
12 11 13
13 13 33
ș ș
44
44
ș ș
11 12
İ
00
İ
00 0
İ
00 0
,
2İ
000 0 0
2İ
0000 0
2İ
00000( )/2
rr rr
zz zz
z z
rz rz
r r
ccc
ccc
ccc
c
c
cc
½ ½
ªº
«»
°° ° °
«»
°° ° °
«»
°° ° °
°° ° °
®¾ ® ¾
«»
«»
°° ° °
«»
°° ° °
«»
°° ° °
«»
°° ° °
¬¼
¯¿ ¯ ¿
V
V
V
V
V
V
(8.12)
where
ı
ij
are stress components, İ
ij
are strain components, and
ij
c are the
elastic constants. In a continuum, the equations of motion in the absence of
body forces are [48],
0
0
357
one can write,
are the unknown scattering coefficients and
law
H
(1)
n
0
is the Hankel Function
of the First kind of order n. The general Hooke's
Bessel function of the first kind of order n. Thus, equation (8.6) can be written as,