Thông tin tài liệu
LINEAR/NONLINEAR ACOUSTIC WAVE PROPAGATION
THROUGH IDEAL FLUID WITH INCLUSION
BY
LIU GANG
(B.Eng, Civil Engineering, Harbin Engineering University, July, 2003)
(Ph.D, Solid Mechanics, Harbin Engineering University, July, 2007)
A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF ENGINEERING
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2011
Acknowledgements
I wish to express my deep gratitude and sincere appreciation to my supervisor,
Professor Khoo Boo Cheong, from the Department of Mechanical Engineering, NUS,
for his inspiration, support and guidance throughout my research and study. His broad
knowledge in many fields, priceless advices, and patience have played a significant
role in completing this work successfully.
I also wish to extend my sincere thanks to Dr. Pahala Gedara Jayathilake, from
the Department of Mechanical Engineering, NUS, for the discussion on life and
research issues. Special thanks are given to Mr. Karri Badarinath, Mr. Deepal Kanti
Das, Mr. Chen Yu, Ms. Wang Li Ping, Ms. Shao Jiang Yan, from Fluid Laboratory,
who shared their precious experience in life and offered generous support in my study
at NUS.
Finally, I would like to express my gratitude to the National University of
Singapore for offering me the opportunity to study, and providing all the necessary
resources and facilities for the research work.
National University of Singapore
September, 2011
Liu Gang
i
Table of Contents
Acknowledgements
i
Table of Contents
ii
Summary
iv
List of Figures
vii
Chapter 1 IntroductionEquation Chapter 1 Section 1Equation Section 1
1
1.1
Problem Definition, Motivation and Scope of Present Work
1
1.2
Outline of contents
7
Chapter 2 Mathematical FormulationEquation Chapter 2 Section 1Equation Chapter 2 Section 1Equation Section (Next)
2.1 Conformal transformation
2.2 On Perturbation Method
Chapter 3 Linear Acoustic Wave Scattering by Two Dimensional Scatterer with
Irregular Shape in an Ideal FluidEquation Section (Next)
3.1 Governing equations of linear acoustic wave
8
8
14
22
22
3.2 Conformal transformations of Helmholtz equation and corresponding physical
vector
24
3.3 Acoustic wave scattering by object with irregular across section
31
3.4 Results and Analysis
36
3.5 Conclusions
45
Chapter 4 An Analysis on the Second-order Nonlinear Effect of Focused Acoustic
Wave Around a Scatterer in an Ideal FluidEquation Section (Next)
46
4.1 Second order nonlinear solution for Westervelt equation
46
4.2 Perturbative method with small parameter for the nonlinear acoustic wave
50
4.2.1 Mathematical formulation of the nonlinear acoustic wave
50
4.2.2 Non-dimensional formulation of the governing equations
53
4.3 Analytical solution for the one-dimensionless equation
56
4.3.1 Analytical solution for plane wave
56
4.3.2 Analytical solution for cylindrical wave
56
4.3.3 Analytical solution for spherical wave
56
4.4 Results and Discussions
57
4.5 Conclusions
67
Chapter 5 Overall Conclusions and Recommendations
5.1 Conclusions
68
68
ii
5.2 Recommendations
70
Bibliography
71
Appendix A
80
Partial Coding for Linear Acoustic Wave Propagation Relate to Conformal Mapping
Method
80
Code 1:
80
Code 2:
80
Code 3:
81
Code 4
84
Appendix B
87
Analytical Solution for Plane wave
88
Analytical Solution for Cylindrical Wave
88
Analytical Solution for Spherical Wave
89
Appendix C
92
Analytical Solution for Plane Wave
92
Analytical Solution for Cylindrical Wave
93
Analytical Solution for Spherical Wave
94
Appendix D
Partial Coding for the Nonlinear Acoustic Wave Propagation
97
97
Code 1:
97
Code 2:
98
Code 3:
99
Appendix E
101
iii
Summary
This work proposed several analytical model for the linear/nonlinear acoustic
wave propagating through the ideal fluid with inclusion embedded. The conformal
mapping together with the complex variables method were applied to solve the linear
acoustic wave scattering by irregular shaped inclusion. Subsequently, we use the
perturbation method to analytically solve the nonlinear acoustic wave interact with the
regular shaped inclusion by expand the nonlinear governing equation into linear
homogeneous/non-homogeneous equations. In general, these two methods are
versatile to obtain the analytical solutions for two classes of problems: the linear
problems with complex boundary conditions and the nonlinear problem with more
complex governing equations.
For the linear model, we analytically obtained the two dimensional general
solution of Helmholtz equation, shown as Bessel function with mapping function as
the argument and fractional order Bessel function, to study the linear acoustic wave
scattering by rigid inclusion with irregular cross section in an ideal fluid. Based on the
conformal mapping method together with the complex variables method, we can map
the initial geometry into a circular shape as well as transform the original physical
vector into corresponding new expressions in the mapping plane. This study may
provide the basis for further analyses of other conditions of acoustic wave scattering
in fluids, e.g. irregular elastic inclusion within fluid with viscosity, etc. Our calculated
results have shown that the angle and frequency of the incident waves have significant
iv
influence on the bistatic scattering pattern as well as the far field form factor for the
pressure in the fluid. Moreover, we have shown that the sharper corners of the
irregular inclusion may amplify the bistatic scattering pattern compared with the more
rounded corners.
For the part of nonlinear acoustic wave propagation, we adopted two nonlinear
models to investigate the multiple incident acoustic waves focused on certain domain
where the nonlinear effect is not negligible in the vicinity of the scatterer. The general
solutions for the one dimensional Westervelt equation with different coordinates
(plane, cylindrical and spherical) are analytically obtained based on the perturbation
method with keeping only the second order nonlinear terms. Separately, introducing
the small parameter (Mach number), we applied the compressible potential flow
theory and proposed a dimensionless formulation and asymptotic perturbation
expansion for the velocity potential and enthalpy which is different from the existing
(and more traditional) fractional nonlinear acoustic models (eg. the Burgers equation,
KZK equation and Westervelt equation). Our analytical solutions and numerical
calculations have shown the general tendency of the velocity potential and pressure to
decrease w.r.t. the increase of the distance away from the focused point. At least,
within the region which is about 10 times the radius of the scatterer, the non-linear
effect exerts a significant influence on the distribution of the pressure and velocity
potential. It is also interesting that at high frequencies, lower Mach numbers appear to
bring out even stronger nonlinear effects for the spherical wave. Our approach with
small parameter for the cylindrical and spherical waves could serve as an effective
v
analytical model to simulate the focused nonlinear acoustic near the scatterer in an
ideal fluid and be applied to study bubble cavitation dynamic associated with HIFU in
our future work.
vi
List of Figures
Figure 2.1: Illustration of the conformal mapping that transforms the initial irregular
geometry into a circular one
8
Figure 3.1: The model for scattering of acoustic wave by rigid inclusion with irregular
across section
25
Figure 3.2: Illustration of the conformal mapping that transforms the initial irregular
geometry into a circular one
26
Figure 3.3: The geometry for the canonical ellipse based on the mapping
function w R p q , where r 1.0, R 0.75, p 1/3, q 1.0
37
Figure 3.4: The comparison of the present method with the T-matrix method at 2:1
aspect ratio ellipse for bistatic scattering pattern at (a) kr 2.0 and (b) kr 5.0
38
Figure 3.5: The comparison of the present method with the Fourier matching method
at 2:1 aspect ratio ellipse for the far-field form function
Figure
3.6:
The
geometries
for
the
scatterer
38
based
on
the
mapping
function w R p q , where r 1.3, R 0.7 and q 1 p 1.0 (ellipse),
2.0 (leaf clover) and 3.0 (rounded corner square)
39
Figure 3.7: The far field form function for the acoustic wave scattering by ellipse
cross section w R p q , where r 1.3, R 0.7, and q 1 p 1.0(a),
2.0(b) and 3.0(c)
40
vii
Figure 3.8: The bistatic scattering pattern for the model of slender ellipse, leaf clover
and approximate square at kr 2.0 and 5.0
42
Figure 3.9: The geometries for the scatterer based on the mapping function
w c1 n c2 2 n 1 c3 3n 2 with n 2
43
Figure 3.10: The bistatic scattering pattern for the scatterer with n-fold axes of
symmetry and sharp corners at kr 2.0
44
Figure 4.1: Schematic description of the model for multiple acoustic waves focused
around a scatterer inside an ideal fluid
47
Figure 4.2: The ratio of the pressure second harmonic to the fundamental term v.s. the
variation of the distance away from the focused point
58
Figure 4.3: The comparison between the analytical results of planar, cylindrical and
spherical wave including the fundamental and the second harmonic
59
Figure 4.4: The variation of pressure amplitude v.s. the distance away from the
focused point for the planar, cylindrical and spherical wave
60
Figure 4.5: The variation of the second order term of plane wave v.s. the variation of
wave number k , Mach number and the distance away from the focused point 62
Figure 4.6: The variation of the second order term for cylindrical wave v.s. the
different wave number, Mach number and the distance away from the focused point63
Figure 4.7: The variation of the second order term for spherical wave v.s. the different
wave number, Mach number and the distance away from the focused point
64
viii
Figure 4.8: The velocity potential distribution near the scatterer (the summation of the
first order term and the second order term) at k 2.0 and 0.3
65
ix
Chapter 1 Introduction
1.1 Problem Definition, Motivation and Scope of Present
Work
A better understanding of the physics of linear/nonlinear acoustic wave interact
with inclusion is important for a wide range of applications including underwater
detection, biomedical and chemical processes. On the aspect of linear acoustic wave,
considerable work has been done on the scattering by objects having regular cross
section. For instance, the separation of variables approach for the Helmholtz equation
has been shown for some particular shapes (Mclachlan 1954). For the two
dimensional problem, however, only the cross sections of the inclusion which are
circular (Liu et al. 2009; Liu et al. 2008), elliptic (Leon et al. 2004a) or parabolic in
shape can be applied using this method.
Apart from the separation of variables approach, there are other methods most of
which are limited to certain class of surfaces. An analytical approach that is formally
exact is the perturbation method (Skaropoulos & Chrissoulidis 2003; Yeh 1965). This
may be used for the penetrable or impenetrable boundary conditions, but is only valid
if the shape is close to one of the limited geometries employed in the above mentioned
separation of variable approach. Mathematically, an alternative method named
conformal mapping via the complex variables methods (Muskhelishvili 1975) has
been applied to study these kinds of problems. These complex variable methods are
1
proved to be rather versatile and have been used not only for linear elastostatic
problem involving cavities (Savin 1961), but also be used in, for example,
thermopiezoelectric
problems
involving
cavities
(Qin
et
al.
1999),
compressibility/shear compliance of pores having n-fold axes symmetry (Ekneligoda
& Zimmerman 2006; Ekneligoda & Zimmerman 2008) and others. On the
elastodynamic model, conformal mapping was applied to solve the in-plane elastic
wave propagation through the infinite domain with irregular-shaped cavity and
dynamic stress concentration (Liu et al. 1982), the anti-plane shear wave propagation
via mapping into the Cartesian coordinates (Han & Liu 1997; Liu & Han 1991) and
the anti-plane shear wave propagation via mapping of the inner/outer domain into
polar coordinates for ellipse (Cao et al. 2001; Liu & Chen 2004). Separately, Fourier
Matching Method (FFM) has been proposed which also involved mapping to study
the sound scattering by cylinders of noncircular cross sections (Diperna & Stanton
1994), the non-Laplacian growth phenomena (Bazant et al. 2003), as well as on
reinforcement layer bonded to an elliptic hole under a remote static uniform load
(Chao et al. 2009).
In order to solve the linear wave interaction problems with various surface that
is not close to a geometry amendable to separation of variable approach, one may still
need to resort to numerical technique. In general, the study of wave scattering by
non-circular shaped object by numerical methods can be broadly classified into three
groups: those concerned with elliptical cylinders based on expansions in Mathieu
functions (Barakat 1963; Sato 1975), those based on the null field method for which
2
any noncircular cylindrical geometries can be considered (Raddlinski & Simon 1993),
and those using Green’s function approach to obtain a governing Fredholm integral
equation (Veksler et al. 1999). There are some numerical methods which are formally
exact have been developed; those include the Mode Matching Method (Ikuno &
Yasuura 1978), the T-matrix Method (Lakhtakia et al. 1984), the Boundary Element
Method (Tobocman 1984; Yang 2002), as well as the Discontinuous Galerkin
Methods (Feng & Wu 2009).
On the aspect of engineering applications, the problems refer to linear sound
wave scattering from inclusions have been the subject of several studies usually
carried out either numerically or experimentally. For instance, the response of cylinder
with circular cross section in water (Billy 1986; Faran 1951; Mitri 2010a; Rembert et
al. 1992), and wave scattering from spherical bubble/shell (Chen et al. 2009; Doinikov
& Bouakaz 2010; Mitri 2005). On the other hand, the theoretical aspect of acoustic
study on inclusion with arbitrary cross sections in fluids are far fewer. Our proposed
method is an attempt to meet the need for various geometries and extend the classical
conformal mapping within the framework of complex variable methods for the
acoustic wave scattering problem in fluids. Incorporation of the mapping technique
into the scattering formulation allows one to analytically predict the far-field
(scattered) pressure results for penetrable or impenetrable scatterers.
In contrast to other methods, we only need to define the mapping functions for
our approach, by which we can transform the initial different geometries into a
circular one accordingly, together with the fractional order Bessel function (Liu et al.
3
2010) to satisfy the boundary conditions for other part of the geometry with regular
curve. Next, the general formation of the scattered wave can be obtained and only the
unknown coefficients need to be determined according to the different boundary
conditions. The distinct advantages of our proposed approach based on conformal
mapping with complex variables can be summarized as: (i) we can directly employ
the scheme according to the method of separation of variables via the argument of the
Bessel function for different curvilinear configurations in conjunction with the
selected mapping functions, see (Liu et al. 1982) for example. (ii) It is very expedient
for our method vis-à-vis some other numerical methods which need to discretize the
full domain, especially the regions at those nodes on the boundaries to accommodate
irregular curve. This leads to potential vast savings of computational resources and
memory. Our approach is possibly one of the first few to calculate the linear acoustic
wave scattering of noncircular cylinders with the use of conformal mapping within the
context of the complex variables method in the fluid. The results obtained are
validated against some special cases available in the literature, and then the effect of
different geometries of the solid inclusion with sharp corners is studied. (It may also
be remarked that our approach is based on the Schwarz-Christoffel mapping function
with the first two and three terms for irregular polygons (Ekneligoda & Zimmerman
2006; Ekneligoda & Zimmerman 2008)).
As is well known, linear acoustic theory is based on the assumption of small
amplitude waves and linear constitutive theory of the fluid medium (Whitham 1974).
Although these assumptions hold for many vibro-acoustic interactions, they become
4
inapplicable in sound fields with high amplitude pressure(Walsh & Torres 2007).
Unlike the linear acoustic wave equation, the nonlinear counterpart can handle waves
with large finite amplitudes, and allow accurate modeling of nonlinear constitutive
models in the fluid. Interesting phenomena unknown in linear acoustics can be
observed, for example, waveform distortion, formation of shock waves, increased
absorption, nonlinear interaction (as opposed to superposition) when two sound waves
are mixed, amplitude dependent directivity of acoustic beams, cavitation and
sonoluminescence (Crocker 1998).
As far as we are aware, there are various models to simulate the nonlinear
characteristic of the acoustic wave propagating through the fluid. For instance, the
one-dimensional Burgers equation has been found to be an excellent approximation of
the conservation equations for plane progressive waves of finite amplitude in a
thermoviscous fluid (Blackstock 1985). An effective model that combined effects of
diffraction, absorption, and nonlinearity in directional sound beams (i.e. radiated from
sources with dimensions that are large compared with the characteristic wavelength)
are taken into account by the Khokhlov-Zabolotskaya-Kuznestov (KZK) parabolic
wave equation (Bessonova & Khokhlova 2009; Liu et al. 2006). Several additional
models have been developed, usually in response to specific needs. For example, the
Westervelt equation is an almost incidental product of Westervelt’s discovery of the
parametric arrays (Kim & Yoon 2009; Norton & Purrington 2009; Sun et al. 2006).
In the past, much computational work have been done on nonlinear effects in
beams based on the KZK equations(Kamakura 2004; Kamakura et al. 2004; Lee &
5
Hamilton 1995; Tjotta et al. 1990). The one dimensional case of Westervelt equation
is studied extensively by finite element method (Pozuelo et al. 1999). However, when
acoustic fields of more complex conditions are considered, advanced numerical
calculation related to finite-differential becomes necessary (Vanhille & Pozuelo 2001;
Vanhille & Pozuelo 2004). In this work, our interest is to develop an analytical
solution of the multiple harmonic acoustic waves focused on the area near a scatterer
where the second order nonlinear effect dominates. Our study has important
implications for further work on bubble/nucleation cavitation by HIFU (high intensity
focused ultrasound) and others.
6
1.2 Outline of contents
This dissertation is divided into five chapters. Each of them consists of various
subsections. A brief summary for each chapter is given as follows.
In Chapter 1 (Introduction), the motivation and scopes of the present work are
presented. There is a brief presentation on the background of linear/nonlinear acoustic
wave propagation, in which attention is centered on using conformal mapping method
for linear acoustic wave scattering by the inclusion with irregular across section and
perturbation method for the nonlinear acoustic wave propagation.
In Chapter 2, we outline the mathematical background for the conformal
mapping method and perturbation method.
In Chapter 3, the basic theory of conformal mapping is reiterated with sufficient
details required for the development of the following section. Here, we present some
results of our method for comparison with other methods in term of accuracy and
efficiency.
In Chapter 4, the mathematical background for the perturbation method is
presented. Discussion on the second order nonlinearity shows clearly the nonlinear
effect for acoustic wave propagation. Numerical results for several examples are
presented to give the explicit comparison with linear condition.
Finally, conclusions are drawn and the directions for future work are discussed in
Chapter 5.
7
Chapter 2 Mathematical Formulation
2.1 Conformal transformation
In this part, we will recall the basic properties of conformal transformation. A
more detailed introduction of the relevant theoretical problems and the application to
the mathematical theory of elasticity can be found in I.I. Privalov’s (Privalov 1948) or
in the book of (Lavrentjev 1946) and Chapter 7 in (Muskhelishvili 1975).
Assume and be two complex variables such that
w ,
(2.1)
b
a
Medium I
Medium II
Medium II
Medium I
Figure 2.1: Illustration of the conformal mapping that transforms the initial
irregular geometry into a circular one
where w is a single valued analytic function in the region in the plane.
The equation (2.1) establishes the relation of every point in to some definite
point in the plane. Conversely, we can assume that each point of in
Eq.(2.1), corresponds to a definite point in . Consequently, we can said that Eq.(2.1)
determines an invertible single-valued conformal transformation or conformal
mapping of the region into the region (or conversely). In the sequel, when we
8
discuss about conformal transformation, it is always assumed to be reversible and
single valued.
The transformation is called conformal, because of the following property which
is characteristic for relations of the form (2.1), where w is a holomorphic
function. In other words, if in two linear elements be taken which extend from
some point and form between them an angle , the corresponding elements in
will form the same angle and the sense of the angle will be maintained which is
the basic definition of conformal mapping (as also shown in Chapter 11 of (Chiang
1997)).
Without special notification, the regions will be assumed to be rounded by one or
several simple contours. The region and may be finite or infinite (one of them
may be finite, while the other is infinite). If the region is finite and is infinite,
the function w must become infinite at some point of (as otherwise there
would not be some point of corresponding to the point at infinity in ). It is
easily proved that w must have a simple pole at that point, i.e. assuming for
simplicity that corresponds to 0 , then
w
c
a holomorphic function,
(2.2)
where c is a constant and no singularities will occur inside the domain of ;
otherwise the transformation can not be deemed as reversible and single-valued. If
and are both infinite and the points at infinity correspond to each other, the
function w must for the same reason have the form
9
w R a holomorphic function,
(2.3)
where R is a constant. It can be recalled that a function, holomorphic in an infinite
region, is understood to be one which is holomorphic in any finite part of this region
and which for sufficiently large
c0
c1
c2
2
... .
may be represented by a series as
Further, it may be shown that the derivative w cannot
become zero in ; otherwise the transformation would not be reversible and single
valued.
There also arises the following question: if two arbitrary regions and be
given, is it always possible to find a function w such that (2.1) gives a
transformation of into ? Here, only some general remarks will be made. First
of all, it is obviously impossible to obtain a (reversible and single-valued)
transformation of a simply connected region into a multiply connected one.
Consider the case when the two regions are simply connected and bounded by
simple contours. Then the relationship as shown in form (2.1), mapping one region
onto the other, can always be found and the function will be continuous up to the
contours. In addition, the function w may always be chosen so that an arbitrary
given point 0 of corresponds to an arbitrary given point 0 of and that the
directions of arbitrarily chosen linear elements, passing through 0 and 0 ,
correspond. These supplementary conditions will fully determine the function w .
For simplicity, suppose that is the unit circle with its centre at the origin.
Denote the circumference of the circle by , so that one has on
1 . Since the
10
transformation is to be continuous up to the contours, the function w will be
continuous on from the left (taking the anti-clockwise direction as positive); let its
boundary values be denoted by w , where ei is a point of .
Hereby, the behavior of the derivative w near and on will be interesting;
in particular, the question has to be considered whether w vanishes at any point
of the contour. If the coordinates of the points of the contour of have continuous
derivatives up to the second order along the arc (i.e. if the curvature of the contour
changes continuously), the function w is continuous up to and, denoting its
boundary values by w ,
w
dw
.
d
(2.4)
In addition,
w 0 everywhere on .
(2.5)
It is already known that w 0 inside . Further, if the coordinates of the points
of the contour of have also continuous derivatives up to the third order, the second
derivative w will be continuous on from the left and its boundary value
w is given by
w
dw
.
d
(2.6)
For this purpose it is sufficient to make the substitution 1 1 . In fact, when
covers the region 1 , 1 covers the infinite region with a circular hole 1 1 ,
11
and hence, considering
as a function of 1 , one obtains the required
transformation. So finite simply connected regions will almost always be mapped on
to the circle 1 , and infinite simply connected regions on to the region 1 . In
both cases one could limit oneself to transformations into the circle 1 , but the
stated convention is somewhat more convenient in practical applications.
Following on, a few remarks will be made regarding the condition of multiconnected regions. For example, a doubly connected region (i.e. a region, bounded
by two contours, regions of more general shape will not be considered here) may
always be mapped on to a circular ring. It is different from the simply connected
regions; this ring may not be chosen quite arbitrarily. The ratio between the radii of
the inner and outer circles will depend on the shape of .
Two simple theorems will be stated here (Muskhelishvili 1975)*:
(i)
Let be a finite or infinite (connected) region in the plane, bounded by a
simple contour , and let w be a function, holomorphic in and
continuous up to the contour. Further, let the points, defined by w ,
describe in the plane (moving always in one and the same direction) some
simple contour L , when describes ( where it is assumed that different
points of correspond to different point of L ). Then w gives the
conformal transformation of the region , contained inside L , on the region
.
*
The more detailed characteristics of conformal mapping functions are introduced in the Chapter 7 of this reference
12
(ii)
Let be a finite or infinite (connected) region, bounded by several contour
1 , 2 ,…, k (having no points in common). Let w be a function,
holomorphic in and continuous up to the boundary, and let the point ,
defined by w , describe in the plane the simple contours L1 ,
L2 , …, Lk (not having common points), bounding some (connected) regions
, when describes the contour 1 , 2 ,…, k . When describes the
boundary of in the positive direction (i.e. let all the time on the left),
the corresponding point describes the boundary of likewise in the
positive direction. Under these conditions
w
represents the
conformal transformation of on to .
It is clear to us that, if and are conformally transformed into one another
by a relation of the form (2.1), the point will move in the positive direction along
the boundary of , when describes the boundary of in the positive direction.
13
2.2 On Perturbation Method
For most of the real problems, the techniques of getting exact solutions are very
restrictive. Definitely, those problems that the exact solutions can be obtained must be
sufficiently idealized for the technique to be appreciable (Chiang 1997). For more
practical models whereby either the boundary geometry or the governing equations
are more complex, the approximate solutions become imperative. If the problem is
close to one that is solvable, perturbation methods are effective methodologies to get
the analytical answers. However, if the problem is very complicated to accord an
exact solution, the numerical methods via discretization must be employed. Generally
speaking, the analytical perturbation methods are much more versatile to gain
qualitative insight, while numerical methods are much better to produce quantitative
information. Sometimes the two categories of methods can be employed together to
get the semi-analytical solutions for some problems with small departures from the
real phenomenon.
Subsequently, we will give brief introductions to the analytical perturbation methods.
On the methodology, let us first outline the typical ideas and procedure for
perturbation analysis.
1) Identify a small parameter. This is the important first step by which we can
recognize the physical scales relevant to the problem. After that, we can normalize
all variables with respect to these characteristic scales. In the normalized form, the
governing equations will contain dimensionless parameters, each of which stands
14
for certain physical mechanisms. If one of the parameters, say , is much smaller
than unity (if the parameter happens to be large, we can choose the reciprocal as
the small parameter), then can be chosen as the perturbation parameter.
2) Expand the solution as an ascending series with respect to the small parameter .
For instance, a power series u u0 u1 2u2 ..., where un is named as the
nth-order term. The series may vary according to the manner that appears in
the equations. If 1 2 , ,… are present, we can employ a series in integral powers
of 1 2 . If only 2 , 4 ,… appear, try a series referring to integral powers of
2 ,etc.
3) Combine terms of the same order in the governing equations and auxiliary
conditions, and get perturbation equations at each order.
4) Calculating from the lowest order, solve the equations at each order successively,
up to certain order, at say O m .
5) By substituting the results for un , n 0,1, 2,... back into perturbative expansions
u u0 u1 2u2 ..., we can get the final solution, which is accurate to the
desired order, say O m .
This straightforward procedure mentioned above is generally known as regular
perturbation. There are many problems where the regular perturbation series may fail
in the independent variable. So, we will introduce the singular perturbation analysis,
which involves the following additional steps:
(i)
Diagnose the reason why the regular expansion is unreasonable. Check
15
which of the original assumptions are incorrect so that failure occurs.
Which are the terms that we initially supposed to be small or to be large?
(ii)
Choose new normalization parameters and accord the magnitudes to the
terms that should be important and commence a new perturbation analysis.
Sometimes the new solution may reveal the need to expand the solution
with the ordering terms such as ln , 2 ln ,…, etc.
The above procedures can be suitable for most problems. Generally speaking, the
governing equations can be algebraic equations, ordinary or partial differential
equations or integral equations. We need to emphasize that the importance of
identifying the correct small parameter by finding the relevant scales of the physics.
Without the physical foresight, it is very difficult for us to make effective use of the
mathematics to simplify the real problem. In general, the execution of perturbation
analysis can be tedious, however, for the mathematical elegancy, we should have a
spirit to persist on this approach if this method is deemed feasible to handle the
research problem.
Subsequently, we would like to introduce the categories of perturbation method
that can be widely applied. The first one is the regular perturbations of algebraic
equations. Let us examine the quadratic equation:
u 2 2 u 1 0,
(2.7)
where is much smaller than unity. The exact solution is well known. We shall use
this simple example to illustrate the procedure that can be extended to equations that
16
cannot be solved exactly.
Let us propose to find the solution as a perturbation series
u u0 u1 2u2 3u3 ...,
(2.8)
and substitute this series into (2.7)
u
u1 2u2 3u3 ... 2 u0 u1 2u2 3u3 ... 1 0,
2
0
(2.9)
Expanding the square and collecting terms of equal powers, we get
u
2
0
1 2u0 2u0u1 2 2u1 u12 2u0u2 ... 0,
(2.10)
With the coefficient of each power of set to zero, a sequence of perturbation
equations is obtained at various orders
0 : u02 1 0,
(2.11)
: 2u0 2u0u1 0,
(2.12)
2 : 2u0u2 u12 2u1 0,
(2.13)
From (2.11) the lowest order solution is
u0 1,
(2.14)
With this result higher-order problems are solved successively
u1 1,
And u2
(2.15)
u12 2u1
1
. Summarizing, the approximation up to 2 is
2u0
2
17
u 1
2
2
... ,
(2.16)
Clearly, this result confirms that the perturbation series will guarantee the accuracy.
Note that the perturbation equation at the leading order for u 0 is still quadratic and
has two solutions. Higher order solutions simply improve the accuracy of the two.
This feature is typical of regular perturbations.
Beyond the regular perturbations, there are another kind of perturbation method
named as singular perturbation method. The following is the cubic equation:
u 1 2 u 3 ,
(2.17)
It can also be solved exactly. For small let us try the straightforward expansion
u u0 u1 2u2 3u3 ... ,
(2.18)
Substituting this series into (2.17) and expand the cubic term, we get
u0 u1 2u2 3u3 O 4
1 2 u03 3u02u1 2 3u02u2 3u0u12 O 3 ,
(2.19)
Equating like powers of yields the perturbation equations
O 0 : u0 1,
(2.20)
O : u1 2u03 ,
(2.21)
O 2 : u2 6u02u1 ,
(2.22)
The solutions are obviously u0 1 , u1 2 , u2 12 , … , hence the final solution is
18
u 1 2 12 2 O 3 .
(2.23)
Why did the two other solutions of the original cubic equation disappear? The reason
is that in (2.17) the term u 3 of highest power is multiplied by the small parameter.
The straightforward perturbation series causes the highest power at the leading order
to vanish, hence only one solution is left; higher order analysis merely improves the
accuracy of this solution. In similar situations the problem is called singular, and the
straightforward expansion is sometimes called the naive expansion.
After checking the source of error, we seek a ‘cure’ by rescaling the unknown so
as to shift the small parameter to a lower order term in the new equation. Let u x m ,
where m is yet unknown. Equation (2.17) then becomes
x m 1 2 x3 3m 1.
(2.24)
We now face several choices: (i) All three terms are equally important. This choice
would require m 0 and 3m 1 0 , which cannot be satisfied at the same time. (ii)
Only one of the three terms dominates. Clearly, the results are full of inconsistencies.
(iii) Two out of three terms dominate over the remaining one. One must now identify
the pair by trial and error.
Let us assume that the second and third terms in (2.24) are more important than
the first. Equating the powers of , we get 3m 1 0 , implying that m 1 3 . But
(2.24) becomes x 1 3 1 2 x3 , where the first term appears to be the greatest,
thereby contradicting the original assumption. Hence the choice is not acceptable. The
second choice that the first and second terms dominate must also be ruled out, since
19
this corresponds to the naïve expansion. The remaining choice is to balance the first
and third. Equating their powers of , we get m 3m 1 or m 1 2 . Equation
(2.24) becomes
x 1 2 1 2 x3 1 2 .
(2.25)
Indeed, the second term is much smaller. Substituting the new expansion
x x0 1 2 x1 x2 3 2 x3 ...
(2.26)
into (2.25) and collecting like powers of , we get the perturbation equations
O 1 2 : 2 x03 x0 0,
(2.27)
O 0 : 6 x02 x1 x1 1 0,
(2.28)
O 1 2 : 6 x02 x2 6 x0 x12 x2 0,
(2.29)
The solutions at successive orders are x0 0, 2 2, 2 2 , x1
x2
1
6x 1
2
0
and
6 x0 x12
. To find x1 , x2 , … explicitly the three solutions for x0 must taken
6 x02 1
one at a time. For x0 0 , we have x1 1 and x2 0 , hence x 1 2 O 3 2 and
u 1 2 x 1 O
(2.30)
This result is just the solution found by naïve expansion. For the second root
x0 2 2 , x1 1 2 and x2 3 2 8 , hence x 2 2
1 3 2 12
u 1 2 x 2 2 1 2
... .
2
8
1 2
2
3 2
... and
8
(2.31)
20
For
the
third
x 2 2
1 2
2
root
x0 2 2 ,
x1 1 2
and
x2 3 2 8 ,
hence
3 2
... and
8
1 3 2 12
u 2 2 1 2
...
2
8
(2.32)
Improvement for all the roots can be calculated in the above procedures. There are
some interesting applications of perturbation methods to heat transfer as shown in the
book by Aziz and Na (Aziz & Na 1984).
21
Chapter 3 Linear Acoustic Wave Scattering
by Two Dimensional Scatterer with
Irregular Shape in an Ideal Fluid
3.1 Governing equations of linear acoustic wave
The propagation of linear sound waves in a fluid can be modeled by the equation
of motion (conservation of momentum) and the continuity equation (conservation of
mass). With some simplifications by taking the fluid as homogeneous, inviscid, and
irrotational, acoustic waves can serve to compress the fluid medium in an adiabatic
and reversible manner. The linear acoustic wave equation is written as:
2
1 2
.
c02 t 2
(3.1)
The pressure p and small fluctuating on quantity of the static mass density are
expressed as:
p 0
or 20
c0 t
t
where c0
(3.2)
p . Here, c0 is the wave speed in the fluid, 0 is the static mass
density of the medium and is the small fluctuating quantity of the static mass
density.
We set the Laplace variable as s i , and as such the direct and inverse
Laplace transforms are given by (Pablo & Felipe 2009):
22
i t e t e it dt ,
0
t
et
2
i eit d .
(3.3)
(3.4)
Here is the angular frequency and is a stability constant. It can be noted that
when 0 , Eqs. (3.3) and (3.4) correspond to the Fourier transforms. In other words,
the Laplace transform can be simply obtained by applying the Fourier integral
to t exp t , i.e., a damped version of t . As such, is also known as the
damping constant. The integral transforms (Eqs.(3.3) and (3.4)) also provide a
possible means to solve for the incident wave modeled as pulse, shock wave or any
other prescribed waveform.
By substituting (3.4) into Eq.(3.1), we can obtain the corresponding Helmholtz
equation in the frequency domain as below:
i .
2
2
2
c0
(3.5)
Here, we will define i c0 ik , and the reduced Helmholtz equation becomes:
2 k 2 0,
(3.6)
where k can be reformulated as k i c0 . The solution of in Eq.(3.1) can
also be assumed as:
e i t .
(3.7)
Here can stand for pressure, velocity or velocity potential, and we can choose the
variables according to the facility for expressing the boundary conditions. Moreover,
23
if we set 0 , the wave number k will be the same as the harmonic wave
k c0 . In this paper, the outgoing scattered wave will be combined with Hankel
function of the first kind and the time term e i t .
3.2 Conformal transformations of Helmholtz equation and
corresponding physical vector
For the model of acoustic wave scattering by three dimensional inclusion with
arbitrary geometry embedded inside the water, analytical approach is deemed rather
restrictive and confined to simple geometry. However, it is conceivable to solve for
the ‘degraded’ two dimensional model with irregular cross section by applying the
method of conformal mapping in the complex plane. Firstly, we will introduce the
degraded two dimensional Helmholtz equation in the Cartesian coordinates shown as:
2 2
2 k 2 0.
2
x
y
(3.8)
Here, we shall introduce the complex variables x yi and x yi into the
Helmholtz equation (3.6) and re-write as:
2
1
k 2 0.
4
(3.9)
For wave scattering problems involving non-circular objects in the complex
,
plane (as shown in Figure 3.1), it is possible to map the internal/external
region of the irregular shaped object (in the , plane) onto the inside/outside
region of the circle (in the , plane). In addition, it is taken that the conformal
24
mapping function w should be an analytic function to ensure the configuration in
the plane is locally similar to its image in the mapped plane. In other words,
the first order derivative of the mapping function w at any point is neither 0
and . We also note that since d w d , when we transform the initial
infinite-small element d
into d , there can be an expansion of length of
magnitude w and a rotation of Arg w .
y
x
Solid
Incident Acoustic wave
Infinite fluid
Figure 3.1: The model for scattering of acoustic wave by rigid inclusion with
irregular across section
Consequently, the corresponding governing equation (3.9) in , plane takes
on the following form:
w w 2
2
k 0.
4
(3.10)
Equation (3.10) is a general expression for the spatial linear acoustic wave in the
,
plane. It needs to be pointed out that
x yi , d d ei ;
r ei , d d ei ; w , d w d , d w' d . That is, we can
obtain the velocity components U x and U y as expressed by the mapping function
25
as well as mapping coordinates , :
Ux
1
,
w w'
1
1
1
Uy i
w w'
(3.11)
.
(3.12)
The corresponding vector U r and U inside the mapping plane as expressed
by the coordinates , are:
Ur
1
,
r w'
(3.13)
U
.
r w
(3.14)
i
'
b
a
Medium II
Medium II
Medium I
Medium I
Figure 3.2: Illustration of the conformal mapping that transforms the initial
irregular geometry into a circular one
It should be noted that the mapping function can transform the initial geometry
into a circular shape (as shown in Figure 3.2), and the corresponding physical vector
is changed (shown as Eq.(3.13) and Eq.(3.14)) inside the mapping plane , and
is different from the original expressions for the physical vector as presented in Eq.
26
(3.11) and Eq. (3.12).
As we know, Eq.(3.10) is the fundamental equation for solving acoustic wave
scattering around a two dimensional object with any cross section. They can be solved
by separation of variables , 1 2 (Liu et al. 1982), and this leads to
Eq.(3.10) taking on the following:
1 2
w w 2
k 1 2 0.
4
(3.15)
The linear combination of 1 and 2 corresponding to various values of
(the separation constant) would then be the general solution of Eq.(3.10):
, A exp ik w w 2 d .
W
(3.16)
The path W is any path of integration in the plane, which makes the expression
convergent.
Furthermore,
we
can
set
exp it ,
w w exp i ,
where t is a complex variable, w is the norm and is the phase angle of w .
Consequently, we can obtain the following expression for Eq.(3.16):
,
a
m
m
eim e
im ik w sin
W
d
(3.17)
where t 2 , and am are arbitrary constants. Here, m is an integer.
Denoting the integral by m , we have
m k w( ) e
W
im ik w sin
d .
(3.18)
We can further express Eq.(3.17) as follows:
27
w
, am m k w( )
.
w
m
m
(3.19)
Expression (3.19) is the general solution of Eq.(3.10). It is possible to introduce the
fractional order Bessel/Hankel function of Helmholtz equation via the method of separating
variables without the mapping function. In our approach, we propose the formulation for the
general solution of Helmholtz equation: the Bessel function with fractional order mp
previously or the mapping function k w( ) as the argument, respectively, by which we
may extend the analysis for the in-plane elastic wave propagating through the solid or
acoustic wave transmitting through the fluid with complex boundary conditions. We would
like to point out that the general solution depends on the choice of path for integration. Along
certain path, the function can be Bessel or Hankel function. etc. In the case of circular region
where polar coordinates are adopted, the expression (3.18) turns out to be the Bessel function;
for the model of elliptical domain with elliptic coordinate system, it gives rise to the Mathieu
functions. Moreover, one should take note that the singularity of the Hankel function at zero
point allows the construction of the standing wave inside the circular/fan-shaped domain as
expressed by Bessel function and ring-shaped domain by the Hankel function, respectively.
For the acoustic wave propagating through the fluid medium without the
transmission into the inclusion, the general mapping function for the numerical
calculation can be cast as (Muskhelishvili 1975):
w R p q ， R 0 ，0 p 1 q .
(3.20)
The above mapping function thus map the outside of the curve in the ,
plane
on to the region 1 . For the condition of q 1 , the contour will be an ellipse;
28
when
q 1 p 2 or q 1 p 3 , the corresponding contours have three or four
cusps, respectively, and they resemble the shape of a triangle or square. Circles with
radii r 1 in the transformed plane correspond in the plane to hypotrochoids,
which likewise for r near 1 resemble triangles or squares with rounded corners. If in
Eq.(3.20) is replaced by 1 , one obtain the transformation of the region such
that 1 with w R 1 p q .
For the holomorphic series w 1 ck k , we can map the region
k 1
outside of in the , plane into the interior of the unit circle in the ,
plane. However, it is deemed not quite logical to cast the problem for the wave
propagation through the fluid inside the scatterer in the transformed plane. Separately,
it may be noted that in the method of Schwarz-Christoffel Formula (successive
approximate solution(Mikhlin 1947)), by expanding the holomorphic mapping
function w into the polynomial series w ck k , it is possible to map
k 1
the region outside of in the , plane into the exterior of the unit circle in the
,
plane. If there are only two non-zero terms in the mapping, i.e.
w cn n , the hole is a hypotrochoid that is a quasi-polygon having n 1
equal ‘sides’(England 1971; Zimmerman 1986). In order for the mapping to be
single-valued, and for not to contain any self-intersections, cn must satisfy the
restriction 0 cn 1 n . The choice of cn 0 gives rise to a circle, whereas the
limiting value of cn 1 n gives a scatterer with n 1 pointed cusps. For the
particular choice of cn 2 n n 1 , the mapping coincides with the first two terms of
29
the Schwarz-Christoffel mapping for an
n 1 -sided
equilateral polygon and
resembles a polygon with slightly rounded corners (Levinson & Redheffer 1970;
Savin 1961; Zimmerman 1991).
In the literature, there are suggested mapping functions for the inclusion with
sharp corners but these possess an
n 1 -fold axis of symmetry with the application
of the Schwarz-Christoffel mapping function (see Ekneligoda etc. (Ekneligoda &
Zimmerman 2006; Ekneligoda & Zimmerman 2008)). For our problem, we would like
to propose the mapping function as below:
w c1 n c2 2 n 1 c3 3n 2 .
(3.21)
For the mapping to be conformal, and for the contour not to have any
self-intersections, it is clear that w 0 be avoided along the contour. This poses
some restrictions on the allowable range of values for the ci coefficients. If
w 0 for some values of on the unit circle, as the ci values increase, this
will first occur at
n 1
equally spaced points that include the point corresponding
to 1 . Hence, the restrictions for the ci can be found by setting w 1 0 . For
the two term mapping, this leads to the restriction that c1 1 n . For the three-term
mapping, the condition is obtained as nc1 2n 1 c2 1 , and for the four-term
mapping, the condition is provided as nc1 2n 1 c2 3n 2 c3 1 . (see also
(Ekneligoda & Zimmerman 2006; Ekneligoda & Zimmerman 2008)).
30
3.3 Acoustic wave scattering by object with irregular across
section
Consider a rigid object with irregular cross section immersed in an ideal
compressible fluid, such that both the viscous and thermal effects can be neglected.
Here, we shall assume that there is no standing wave inside the rigid body. In other
words, we are only concerned with the scattering wave generated by the impenetrable
rigid body, from which we can calculate the radiation acting on the scatterer as well as
the far field scattered pressure generated by the scattering. (For other conditions of
acoustic wave scattering by elastic object immersed in Newtonian fluid, there will be
the standing wave inside the elastic object and the linear shear force is generated by
the movement of fluid. Under such assumptions, the unknown coefficients of the
scattering wave and standing wave inside the elastic inclusion can be obtained
through the continuity of stress and velocity along the interface. This is to be pursued
in future work in which our proposed method is fully capable of solving.)
For the plane acoustic wave equation in Eq.(3.6), the expression for acoustic
waves in the complex plane as well as in the mapping plane is shown below
ik
i A0 eiknr A0 eik x cos y sin A0 e 2
as:
e
i
ei
ik
A0 e 2
w e
i
w ei
.
(3.22)
where is the angle between the positive x -axis and the direction in which the
wave travels.
We
can
also
use
the
Laurent
expansion
in
the
form
of
31
exp 1 2 m J m m , and assume that k
and ie i
.
In this manner, we can obtain another format for the incident potential i
expressed by the Bessel function as below:
A0 i Jm k
m
i
m
m
im
m
e A0 i Jm k w
m
w im
e .
w
m
(3.23)
According to the Sommerfield radiation condition, the outgoing scattered wave
that is generated by the rigid body:
s
w
Bm H m k w( )
,
w
m
m
1
(3.24)
where H m1 is Hankel function of the first kind of order m th. The total velocity
potential field in the fluid domain t should be the summation of the incident wave
i and the scattered wave s .
By substituting expressions (3.22) and (3.24) into (3.13) and (3.14), we can obtain
the corresponding velocity components as follows:
'
'
ikA0 ik2 w e i w ei w i w i
e
e
e ,
'
r w'
2
r
w
(3.25)
'
'
kA0 ik2 w e i w ei w i w i
e
e
e ,
'
r w'
2
r
w
(3.26)
U r
i
i
U
k
Ur Bm Hm1 1 k w
2 m
s
ik
U Bm Hm11 k w
2 m
s
m1
w w'
Hm1 1 k w
w r w'
w w'
Hm1 1 k w
'
w r w
m1
m1
w w'
, (3.27)
w r w'
m1
w w'
. (3.28)
w r w'
32
Here, the unknown coefficient Bm of the radiation velocity potential in Eq. (3.24)
can be determined according to the radial component of the velocity U r set to be
zero along the interface of the fluid and the rigid inclusion (i.e. a Neumann condition
on the velocity potential U r t t ; see also (Mitri 2010b)):
U r i
r a
U r s
0.
r a
(3.29)
This assumption has been recognized to properly model the radiation force on liquid
drops in air and under reduced gravity environments (Mitri & Fellah 2007). If the
density of cylinder is taken as zero (or very small in comparison to the fluid density),
the solution applies to the scattering by a soft cylinder (satisfying the Dirichlet
condition on the pressure quantity P t 0 t t 0 at the cylinder radius r a ;
see also (Flax et al. 1981)). Furthermore, this boundary condition with infinite series
can be written as:
B
m
m
m 0,
(3.30)
where
m Hm1 k w
1
iA0 e
w
w
ik
w e i w ei
2
m1
w
w'
1
Hm 1 k w
'
w
r w
m1
w'
, (3.31)
r w'
w' i w' i
e
e .
'
r w'
r
w
The periodic function can be expanded via the Fourier series (
(3.32)
n
where n
1
2
2
0
n
ein ,
e in d ). Similarly, the unknown coefficients Bm in Eq.(3.30)
can be determined via Fourier series expansion of Eq.(3.30) in the matrix form where
33
n is equal to m ( ein on the both side are omitted):
B
n
n m
n .
(3.33)
m e in d ,
(3.34)
nm
n
where
nm
n
1
2
1
2
2
0
2
0
e in d .
(3.35)
Equation (3.33) could then be reduced to a series of algebraic equations by truncating
to finite terms accordingly. The number of truncated terms employed depends on the
frequency and accuracy requirement. The results obtained are calculated by a code
written in MATLAB 7.0 interpreter language and employment of 19 terms in the
Fourier series which means n and m range from -9 to 9.
Based on the above coefficients Bm , we can substitute these into the expressions
for the velocity as well as the pressure in the fluid domain as follows (the time
term e i t is omitted here for simplicity):
m
ik
w ei wei
w
t
1
P P P 0
0 i A0 e 2
Bm Hm k w()
.
t
m
w
t
i
s
(3.36)
When the variable tends to infinity, the asymptotic expression of Hankel function is
such that (Mow & Pao 1971):
Hm1 k w()
2
2
m
i k w( ) m 2 4
i k w( ) 4
e
i
e
.
k w()
k w()
(3.37)
34
Consequently, we can obtain the expression for the far-field scattered pressure from
the infinitely long non-circular cylinder due to an incident plane wave of unit
amplitude:
fs
P
w
.
w
m
2
i k w( ) 4
0 i Bm i
e
k w()
m
m
(3.38)
The wave number of incident wave can be expressed as the ratio of the inclusion
2r to the wave length of the incident wave, namely,
Wave number kr i r c0 2 r
(3.39)
which represents a dimensionless frequency 2 r . In this work, the amplitude of the
incident acoustic wave A0 1 , and the speed of the acoustic wave propagating through
the fluid is c0 1480 m s .
It is noted that the relations widely applied for the derivation of the complex
variables together with mapping function are shown as below:
1
H m k w
1
H m k w
1
H m k w
m
w k 1
k w
H
w 2 m 1
w
w
m
w
w
k H m1 1 k w
2
m
w
k 1
H m 1 k w
2
w
m 1
w' ,
w
w
w
w
(3.40)
m 1
w' ,
(3.41)
m 1
w' ,
(3.42)
35
m
w k 1
1
H m k w
k w
H
w 2 m 1
w
w
m 1
w' .
(3.43)
Furthermore, if we set w z , the equations are reduced to the
corresponding derivations found in the literature (Liu et al. 2010; Liu et al. 2009; Liu
et al. 2008) which were obtained by the complex variable method without the local
conformal mapping function. (In other words, those models can also stand for the
mapping of the circular cavity into circular one).
3.4 Results and Analysis
All results in this section were generated using the boundary condition
summarized in Eq.(3.30) to Eq.(3.32), with Eq.(3.23) for the incident field and
Eq.(3.24) for the scattered field. The pressure field given by Eq.(3.36) is applicable to
the fluid domain as well as the far field expression for the scattered pressure shown in
Eq.(3.38). The damping constant (as shown in Eq.(3.3)) is assumed to be zero for
the present model of ideal fluid.
In order to verify the accuracy of the results obtained with our proposed
employment of the conformal mapping in conjunction with the complex variables
method, the computed results are to be compared to published work (Pillai et al. 1982)
for kr ranging from 0.1 to 5. To this end, the following far-field form function f
is considered (Leon et al. 2004b):
f 2daeff
P s
a
i
P
2
(3.44)
36
with an effective radius aeff defined by aeff
a
2
b 2 2 . Here, d is the
distance of observation, a stands for the major semi-axis of the ellipse, and b is the
minor semi-axis. The major semi-axis of the ellipse is set to be a 1 and the minor
semi-axis b 0.5 (as applied to the mapping function w R p q ,where
the employment of R a b 2 , p a b a b and q 1 transforms the
ellipse into a circle with the radius of one (as shown in Figure 3.3).We calculate the
bistatic scattering pattern along the loop of the inclusion, as well as the equivalent
far-field form factor f .
2
Ellipse x/y=2.0
Y Axis
1
0
-1
-2
-2
-1
0
1
2
X Axis
Figure 3.3: The geometry for the canonical ellipse based on the mapping
function w R p q , where r 1.0,
R 0.75, p 1/3, q 1.0.
Figure 3.4 represents the polar plots of the bistatic scattering pattern at fixed
frequencies of kr 2.0 and 5.0, respectively, with different incident angles at 00,
450, 900. It is evident that our results for the bistatic scattering pattern concur very
well with the results obtained by T-matrix method† (Pillai et al. 1982)).
†
The detailed results from Pillai et al. 1982 can be found from the Appendix E
37
90
1.2
0.8
120
(a)
=0
60
=90
0
2.0
120
(b)
=0
60
1.5
1.0
0.4
0
=45
0
=90
0
30
150
0.5
0
180
0.0 180
0
0.5
0.4
0.8
90
0
30
150
0
=45
330
210
1.0
330
210
1.5
1.2
240
300
240
2.0
300
270
270
Figure 3.4: The comparison of the present method with the T-matrix method (Pillai et al. 1982)
at 2:1 aspect ratio ellipse for bistatic scattering pattern at (a) kr 2.0 and (b) kr 5.0
If∞I
1
0.1
0.01
0.1
0
α =0
0
α =30
0
α =60
0
α =90
Diperna & Stanton
1
10
kr
Figure 3.5: The comparison of the present method with the Fourier matching method
(Diperna & Stanton 1994) at 2:1 aspect ratio ellipse for the far-field form function
In Figure 3.5, we have plotted the far-field form function for wave incidences at
00 (along the major axis of the cross section), 300, 600 and 900 to the acoustically hard
cylinder. It is evident that our results concur well with the Fourier matching method
38
(FMM) at the condition of 00 (Diperna & Stanton 1994). We also show the
variation of incident angle with respect to the horizontal axis; it is interesting to note
that the scattering far-field form factor for the hard cylinder with horizontal incidence
00 is slightly higher than that of oblique incidences (for instance, 300, 600,
900). The far-field form factor decreases with respect to the increase of the incident
angle . However, as the frequency increases the form factor becomes slightly
higher for the condition of vertical incidence at 900. According to the mapping
function w R p q , we set R 0.7, q 1 p 1.0, 2.0 and 3.0 to take
on the shape of ellipse, leaf clover and approximate square, respectively (see Figure
3.6). Here, we have assumed the mapped circle to be r ei with r 1.3,
corresponding to the ellipse, triangle or square with rounded corners.
2
Ellipse
Leaf Clover
Approximate Square
Y A x is
1
0
-1
-2
-2
-1
0
1
2
X Axis
Figure 3.6: The geometries for the scatterer based on the mapping function
w R p q , where r 1.3, R 0.7 and
q 1 p 1.0 (ellipse), 2.0 (leaf clover) and 3.0 (rounded corner square)
39
If∞I
1
(a)
0.1
α
α
α
α
0.01
0.1
1
0
=0
0
=30
0
=60
0
=90
10
kr
If∞I
1
(b)
0.1
0.01
0.1
If∞I
1
1
10
1
10
kr
(c)
0.1
0.01
0.1
kr
Figure 3.7: The far field form function for the acoustic wave scattering by various cross section
w R p q , where r 1.3, R 0.7 and q 1 p
(a) 1.0 (ellipse), (b) 2.0 (leafclover), (c) 3.0 (rounded corner square)
Figure 3.7-a plots the far-field form factor for the (relatively more slender)
ellipse, while Figure 3.7-b and Figure 3.7-c show the far-field form factor of the
40
incident acoustic wave scattering by the leaf clover and rounded corner square for
various incident angles and frequencies, respectively. This scattered form function is
plotted versus kr , where r is the radius of the circular rigid inside the mapping
plane. It is clear that the distribution of the form factor takes on essentially the same
variation for the scattering by the ellipse. However, there appears to exhibit a sharp
decrease at around the wave number of kr 4 for the leaf clover (Fig.3.7-b) and
square (Fig.3.7-c). Compared with previous published literature, our method is able to
match better with the experimental results obtained from the physical optics
approximation (Diperna & Stanton 1994). It is also observed that the general far-field
form factor for the square and leaf clover shapes are larger than the slender ellipse
which may be attributed to the smaller cross section of the latter.
The bistatic scattering pattern at kr 2.0 and 5.0 for the model of slender ellipse,
leaf clover and approximate square along the loop of the inclusion are calculated and
shown in Figure 3.8. It is clear that the angle and frequency of the incident waves
have significant influence on the bistatic scattering pattern. It is also obvious that the
bistatic scattering pattern with solid line is symmetrical to the horizontal for the
condition of incident angle at 00. In general, the increase of the incident
frequency will result in higher bistatic scattering pattern for all the shapes examined.
In particular, among the geometries, the slender ellipse seemingly can amplify
relatively more the bistatic scattering pattern for these selected frequencies, which
may be attributed to the relatively sharper corners on the left and right edge of the
slender ellipse as compared to the leaf clover or square.
41
90
3
2
(a) Ellipse 120
kr=2.0
=0
60
90
0
=45
0
=90
0
2
30
150
1
0
330
210
240
2
120
3 (b) Leaf clover
kr=2.0
=0
60
330
210
240
300
90
0
=45
0
=90
0
30
150
120
3 (e) Leaf clover
kr=5.0
2
0
=0
60
0
=45
0
=90
30
150
1
0 180
0
0 180
1
0
1
330
210
3
240
2
(c) Square 120
3
kr=2.0
330
210
3
300
270
90
240
300
270
90
0
=0
60
=45
0
=90
0
30
150
(f) Square 120
3
kr=5.0
=0
60
0
0
=45
0
=90
2
1
30
150
1
0 180
0
1
3
0
270
1
2
0
30
150
3
300
270
90
2
0
=90
1
3
2
0
=45
0 180
1
2
=0
60
1
0 180
2
(d) Ellipse 120
3
kr=5.0
0 180
0
1
330
210
240
300
2
3
330
210
240
270
300
270
Figure 3.8: The bistatic scattering pattern for the model of slender ellipse, leaf clover and
approximate square at kr 2.0 and 5.0
Next, we calculate the far-field form function f for the irregular geometries
with sharp corners based on the novel mapping function taken from Eq.(3.21)
w c1 n c2 2 n 1 c3 3n 2 with the first two or three terms of the
42
Schwarz- Christoffel mapping function. Figure 3.9 show the irregular scatterer have
threefold symmetry and sharp cornners (i.e. n 2 ). The calculated results depicted in
Figure 3.10 correspond to the geometries illustrated in Fig.3.9. According to Fig.3.8-b
and Fig.3.10-a1, we found that the bistatic scattering pattern are similar, though the
mapping function applied are vastly different from each other. This attests to the
viability and robustness of the present approach with the proper employment of
conformal mapping with complex variables. From Figs.3.10-a3,b1,b2,b3, it is clear
that the n-fold symmetric objects with sharp corners can greatly amplify the bistatic
scattering pattern compare with the smooth corners as shown in Fig.3.10-a1,a2. Our
calculations have clearly shown that the present method can easily take on
geometrical shape possessing sharp corners.
2
2
(a)
1)
2)
3)
1
0
-1
-2
-2
1)
2)
3)
(b)
Y A x is
Y A x is
1
c1=1/5,c2=0,c3=0
c1=1/3,c2=0,c3=0
c1=1/2,c2=0,c3=0
c1=1/3,c2=1/15,c3=0
c1=1/6,c2=2/15,c3=0
c1=1/9,c2=7/45,c3=0
0
-1
-1
0
X Axis
1
2
-2
-2
-1
0
1
2
X Axis
Figure 3.9: The geometries for the scatterer based on the mapping function
w c1 n c2 2 n 1 c3 3n 2 with n 2
43
90
3
2
(a1)
120
=0
60
90
0
=45
0
=90
0
2
30
150
1
0
330
210
240
(a2)
2
120
=0
60
0
0
=90
0
330
210
240
300
(b2)
120
=0
60
0
0
=45
0
=90
30
150
1
0 180
0
0
1
2
330
210
240
240
300
120
300
270
270
90
(a3)
330
210
3
3
=0
60
=45
=90
0
3
2
30
150
90
0
0
(b3)
120
=0
60
0
=45
0
=90
0
30
150
1
1
0 180
0
0 180
0
1
1
3
3
2
30
150
1
2
=90
0
0
=45
0 180
2
0
270
90
1
3
=45
30
150
3
300
270
90
2
0
1
3
2
=0
0 180
1
3
(b1)
60
1
0 180
2
3
120
330
210
2
3
240
330
210
300
270
240
300
270
Figure 3.10: The bistatic scattering pattern for the scatterer with n-fold axes of symmetry and
sharp corners at kr 2.0
44
3.5 Conclusions
In this study, we have analytically obtained the two dimensional general solution
of Helmholtz equation, shown as Bessel function with mapping function as the
argument and fractional order Bessel function, to study the linear acoustic wave
scattering by rigid inclusion with irregular cross section in an ideal fluid. Based on the
conformal mapping method together with the complex variables method, we can map
the initial geometry into a circular shape as well as transform the original physical
vector into corresponding expressions in the mapping plane. This study may provide
the basis for further analyses of other conditions of acoustic wave scattering in fluids,
e.g. irregular elastic inclusion within fluid with viscosity, etc. Our calculated results
have shown that the angle and frequency of the incident waves have significant
influence on the bistatic scattering pattern as well as the far field form factor for the
pressure in the fluid. Moreover, we have shown that the sharper corners of the
irregular inclusion may amplify the bistatic scattering pattern compared with the more
rounded corners.
45
Chapter 4 An Analysis on the Second-order
Nonlinear Effect of Focused Acoustic Wave
Around a Scatterer in an Ideal Fluid
4.1 Second order nonlinear solution for Westervelt equation
The widely used linear wave equation used to describe the small signal (infinitesimal
amplitude) propagation is the Helmholtz equation(Liu et al. 2009):
2
1 2
0.
c2 t 2
(4.1)
Here, can be pressure, velocity or velocity potential in the field of fluids, or
displacement and displacement potential for solids (Liu et al. 2010); c is the small
signal sound speed since there are no shear waves inside the ideal fluid. For the linear
wave propagation in elastic solids, the in-plane longitudinal and transverse waves (Liu
et al. 1982) as well as anti-plane transverse wave (Liu et al. 2008) can co-exist and
governed by the same Helmholtz equation. If we neglect the time term in Eq.(4.1), the
Laplace equation is also widely employed for the incompressible ideal fluid
(Doinikov & Zavtrak 1995; Wang et al. 1996) and static linear solid mechanics
(Ballarini 1990).
In the past, much computational work have been done on nonlinear effects in acoustic
wave beams based on the KZK equations (Kamakura 2004; Kamakura et al. 2004;
Lee & Hamilton 1995; Tjotta et al. 1990). The one dimensional case of Westervelt
46
equation is studied extensively by finite element method (Pozuelo et al. 1999).
However, when acoustic fields of more complex conditions are considered, advanced
numerical calculation related to finite-differential becomes necessary (Vanhille &
Pozuelo 2001; Vanhille & Pozuelo 2004). In this work, our interest is to develop the
analytical solution of the multiple harmonic acoustic waves focused on the area near a
scatterer where the second order nonlinear effects dominate (see Figure 4.1). Our
study has important implications for further work on bubble/nucleation cavitation by
HIFU (high intensity focused ultrasound) and others.
O
Rs
Re
Multiple incident acoustic waves
Figure 4.1: Schematic description of the model for multiple acoustic waves focused
around a scatterer inside an ideal fluid
On the problem of multiple acoustic waves focusing on intensive area, one model
governing the dynamics is the Westervelt equation,
2 p
1 2 p
2 p 2 3 p
4 3 0.
c2 t 2 c4 t 2
c t
(4.2)
Here, the third term is the nonlinear term and the forth term stands for the
47
thermoviscous effect. is the static medium density, c is the wave speed of the
acoustic wave in the undisturbed liquid, 1 2 is the coefficient of
nonlinearity, and
is defined as the ratio of specific heats of the gas.
1 Pr
is
the
diffusivity
associated
with
sound
absorption
where is the kinematic viscosity, 4 3 B is the viscosity number,
B is named as the bulk viscosity coefficient and Pr is the Prandtl number. If we
consider the case of ideal fluid, the forth term can be set equal to zero, and the
reduced Westervelt equation is,
2 p
1 2 p
2 p2
0.
c2 t 2 c4 t 2
(4.3)
To solve the second-order nonlinear equation, the successive approximations
method can be applied. This method assumes a solution for the acoustic pressure
consisting of the addition of two terms in the form of p p1 p2 where p1
represents the first order approximation and p2 the second order correction, p2
being much smaller than p1 p2 p1 . Neglecting the terms of third and higher
orders, we can obtain the approximate equation as:
2
1 p1 p2
2 p12
p1 p2 2
0.
c
c4 t 2
t 2
2
(4.4)
According to Eq.(4.4), the second order solution for the acoustic wave can be written
as
p P1 exp it P2 exp 2it .
The correct analytical solutions of the one dimensional Eq.(4.4) for plane, cylindrical
48
and spherical waves are obtained, respectively, as shown below. It is pointed out,
however, that the coefficients on the right hand side of the initial governing equation
for the second-order nonlinear term shown as Eq.10 in (Pozuelo et al. 1999) are
deemed incorrect. In particular, Eq.10 in reference (Pozuelo et al. 1999) is shown
with the term 2k 2
4k 2
c .
2
c
2
, which we reckoned should be reflected as
Therefore, the analytical results that follow for the second-order
nonlinearity given in Eq.16, Eq.27 and Eq.19 for plane wave p p , cylindrical wave pc
and spherical wave ps , respectively, in (Pozuelo et al. 1999) may contain some
errors/ inaccuracies. Finally, we substituted the derived solutions reflected as Eqs. 16,
27 and 19, correspondingly for p p , pc and ps into Eq.10 of reference (Pozuelo
et al. 1999) and found some degree of inconsistency.
Shown in Appendix B are the details for our derivations in which the results are given
below as:
p p P0 e ikx eit
pc P0
P02
1 2ikx e2ikx e 2it ,
2
2 c
2
i kr
e
kr
2
P0 H 0
kr e
i t
4
eit
4i P02 i 2 2ikr 2it
e e
e
0c2
2
2i krP02
2
H
kr
e 2it
0
2
c
P0 ikr it i kP02 ln r 1 n 2 ! 2ikr 2it
e e
e .
e
c2 r n 2 4ik n 1 r n
r
(4.5)
(4.6)
n
ps
(4.7)
It is clear that the first term in Eqs. (4.5), (4.6) and (4.7) are the general solutions for
the one dimensional linear acoustic wave with time harmonic condition, and the
49
second terms are the approximate solutions for the second-order nonlinearity which
satisfy the Sommerfield radiation condition at the far field. Here, P0 stands for
harmonic excitation of the wave amplitude, and k c 2 , where is the
circular frequency of the incident wave, and stands for the wave length.
4.2 Perturbative method with small parameter for the
nonlinear acoustic wave
4.2.1 Mathematical formulation of the nonlinear acoustic
wave
The analytical solutions in Section 4.1 for the one dimensional Westervelt
equation are obtained by the perturbation method without any small parameter.
Furthermore, we can only get the pressure inside the fluid domain without the explicit
results for the velocity and velocity potential. Below, we will propose a novel
mathematical formulation and a different model for the nonlinear acoustic wave that
goes beyond the traditional models based on Burgers equation, KZK equation and
Westervelt equation. The compressible potential flow theory is invoked together with
the dimensionless formulation and asymptotic perturbation expansion referring to the
small parameter Mach number. It is to be found that our approach allows the
decoupling of the potential and enthalpy terms to the second order, and which accord
much flexibility for the calculation of the other physical quantities inside the fluid
domain.
For our model, we shall take the liquid as inviscid and compressible. As such,
50
the liquid flow is governed by the equation of mass conservation:
u 0,
t
(4.8)
and the momentum conservation equation:
u
1
u u p.
t
(4.9)
In many fluid dynamic problems, significant liquid compressibility due to high
speed motion occurs when thermal effects in the liquid are unimportant. We therefore
assume that thermal effects in the liquid are insignificant. With this assumption, the
liquid state is defined by a single thermodynamic variable. The sound speed c and
the enthalpy h of the liquid can be defined as follows(Wang & Blake 2010):
c2
p dp
dp
, h
,
p
d
where the reference pressure
(4.10)
p is the pressure in the undisturbed liquid
(hydrostatic pressure).
By assuming that the flow is irrotational, we may introduce a velocity potential
such that u . By applying c 2 dp d , c 2 p , c 2 p , u ,
where the dot “ ” stands for differential w.r.t time, we can re-write Eq.(4.8) as:
2
1 h
h 0.
2
c t
(4.11)
Similarily for Eq.(4.9), the integration leads to the unsteady Bernoulli equation,
1
2
h h .
t 2
(4.12)
51
Here, h may be set equal to zero since the enthalpy is referenced to the undisturbed
fluid at infinity. To find the expressions for the sound of speed c and enthalpy h ,
we use the Tait model equation of state to relate the pressure and density as follows
(Lezzi & Prosperetti 1987; Prosperetti & Lezzi 1986):
p
p B
n
n B.
(4.13)
The values B 3049.13 bars, n 7.15 give an excellent fit to experimental
pressure-density relation for water up to 105 bars (Fujikawa & Akamatsu 1980). Here,
p and are the pressure and density in the undisturbed liquid (hydrostatic model).
From Eq.(4.13) together with c 2 dp d , we get
c2
n p B
n
p B p B
1/ n
n 1 / n
(4.14)
and
n 1 / n
c 2 c2
c2 p B
1
,
h
n 1
n 1 p B
where the definition of c2
n p B
(4.15)
is the wave speed of the acoustic wave in
the undisturbed liquid.
Then we expand the enthalpy h and the sound speed c around the equilibrium
pressure p using a Taylor series expansion as follows (Prosperetti & Lezzi 1986):
h
p p
2
1 p p
2
...,
2c
(4.16)
52
1
1
p p
2 n 1 4 ....
2
c
c
c
4.2.2
Non-dimensional
(4.17)
formulation
of
the
governing
equations
We would like to introduce two typical scales Rs and U , where Rs denotes
the radius of the scatterer and U p
1/ 2
stands for the liquid velocity near the
scatterer. The dimensionless quantities indicated by asterisk, are given as below:
r
Rs
r* , t
Rs
t* , RsU * , h U 2 h*
U
c c c* , p p U 2 p*
(4.18)
(4.19)
where the Mach number U c (is also the small parameter to be used for the
perturbation analysis). It is evident that 0 , or otherwise the dimensionless
formulation in Eq.(4.18) will have a singularity.
From the above, we have:
2
*
*
2
2 U *
,
,
U2
U
2
2
Rs r*
r
r
r*
t
t*
(4.20)
h
U 2 h* h U 3 h*
,
.
r
Rs r* t Rs t*
(4.21)
Then substituting Eqs.(4.20), (4.21) and (4.19) into (4.11) and (4.12), respectively,
we obtain the governing equations in terms of dimensionless variables:
2 * 1 h*
* h*
2
2
0
2
r*
r* r*
c* t*
(4.22)
53
2
* 1 2 *
h* 0.
t* 2
r*
(4.23)
We stress that the dimensionless variable r* also stands for the space vector
dependent on the dimensional coordinates system be planar, cylindrical or spherical.
Next, from Eq.(4.15), the dimensionless speed of sound of the liquid is,
c*2 1 2 n 1 h* ,
(4.24)
and the dimensionless enthalpy of the liquid in Eq.(4.15) is given by
n 1 / n
1 p B
h* 2
1 .
n 1 p B
1
(4.25)
Substituting p p U 2 p* and h U 2 h* into Eq.(4.16) and Eq. (4.17), and
referring to the Taylor series expansion, we can get
1
h* p* 2 p*2 o 2 ,
2
(4.26)
1
1 n 1 2 p* o 2 .
2
c*
(4.27)
By substituting Eq.(4.26) and Eq.(4.27) into Eq.(4.22), we obtain:
h
2 * h*
h
2 * * n 1 h* * o 2 0.
2
r*
t*
t*
r* r*
(4.28)
Next, the perturbation expansions for the potential * and enthalpy h* are defined
as follows:
* r* , t* *0 r* , t* 2 *1 r* , t* o 4 ,
(4.29)
h* r* , t* h*0 r* , t* 2 h*1 r* , t* o 4
(4.30)
54
where they are introduced into Eq.(4.23) and Eq.(4.28), respectively. We can get the
separate governing equations for the first and second order w.r.t. 2 :
*0
h*0 0
t*
(4.31)
2
*1 1 *0
h*1 o 4 0
2 r*
t*
(4.32)
2 *0 h*0
0
r*2
t*
(4.33)
h
2 *1 h*1 *0 h*0
n 1 h*0 *0 o 4 0.
2
r*
t*
r* r*
t*
(4.34)
By combining Eq.(4.31) and Eq.(4.33),
Eq.(4.32) and Eq.(4.34), we obtain the
fundamental equation for the second order nonlinear acoustic wave (dimensionless
formulation) as
2 *0
*0
0
t*2
2
*
*2 *1
(4.35)
*0 2 *0
2 *1 1
2
n
o 4 0. (4.36)
1
* *0
* *0
* *0
2
2
t*
t*
t* t*
2 t*
where the Hamilton operator * is defined in terms of r* . As elucidated before, the
parameter n is assumed to reasonably match the experimental pressure-density
relation for water up to 105 bars at n 7.15 (Fujikawa & Akamatsu 1980).
55
4.3 Analytical solution for the one-dimensionless equation
4.3.1 Analytical solution for plane wave
The analytical solutions for the dimensional second-order nonlinear plane wave
are shown below (the detailed derivation to obtain the analytical solutions are
provided in the Appendix C):
0 A cos kr eit
1 A cos 2kr e2it
(4.37)
i A2
n 1
n 1 kr sin 2kr
cos 2kr n 3 e2it ,
2
8U
4
(4.38)
where A , k , are the amplitude, wave number, frequency, respectively.
4.3.2 Analytical solution for cylindrical wave
The analytical solutions for the dimensional second-order nonlinear cylindrical
wave are shown as below:
0 A H 0 2 kr eit
1 A H0 2 2kr e2it
(4.39)
J 2kr n 1 H 2 kr 2 2 H 2 kr 2 Y 2kr rdr
0
0
1
2it
i A23 2 0
e ,
2
2
2U 4
2
2
Y0 2kr n 1 H0 kr 2 H1 kr J0 2kr rdr
(4.40)
4.3.3 Analytical solution for spherical wave
The analytical solutions for the dimensional second-order nonlinear spherical
wave are shown as below:
56
0
ARs
cos kr eit ,
r
ARs
cos 2kr e 2it
r
r
2it
i A2 Rs2 kr n 1 Ci 4kr 2Ci 2kr ln sin 2kr
Rs
2 2 2
e .
8U r
2
kr n 1 Si 4kr 2Si 2kr cos 2kr 8cos kr
(4.41)
1
(4.42)
By substituting Eqs.(4.37) and (4.38), Eqs.(4.39) and (4.40), and Eqs.(4.41)
and (4.42) into 0 2 1 O 4 , respectively, we obtain the asymptotic
expression for the velocity potential up to and including O 2 .
4.4 Results and Discussions
The obtained analytical solutions are analyzed with regard to the nonlinear
effects around the area where the multiple incident waves are focused. It is important
to emphasize that the initial interactions among the multiple sources will not be taken
into account since each source is not sufficiently large intensity to induce nonlinear
deformation. If one assume the scatterer to be the equilibrium bubble, the
Rayleigh-Plesset equation (Franc & Michel 2004) or modified Rayleigh model (Gong
et al. 2010) as well as the relationship between the pressure in the fluid and the gas
pressure inside the bubble (Matula et al. 2002) (where the nonlinear effects have not
taken into account previously), can be employed to describe the bubble
cavitation/deformation. Alternatively, we may use the general dimensionless
equation dr dt together with Eqs.(4.41) and (4.42) to construct and express the
nonlinear variation of the bubble radius as well. In this work, we shall assume the
57
radius of the scatterer Rs 1 . Since we know that the nonlinear acoustic wave effect
will only dominate especially the fluid region and its vicinity outside the scatterer, the
distance defined as x or r should larger than the radius of the scatterer ( Rs 1 ); in
Second harmonic to fundamental ratio
the subsequent analysis the effective distance Re will range from 1 to 100.
0.12
0.09
Planar coordinate
Cylindrical coordinate
Spherical coordinate
0.06
0.03
0.00
1
2
3
4
Distance from the focused point
5
Figure 4.2: The ratio of the pressure second harmonic to the fundamental
term v.s. the variation of the distance away from the focused point
By using the analytical solutions for the one dimensional Westervelt equation,
the pressure distribution around the scatterer for the fundamental and the second order
harmonic of planar, cylindrical and spherical coordinates in water are calculated. Here,
in our calculation, the wave speed c is assumed to be 1500m/s, water density
is 1000 kg/m3, nonlinear coefficient is 3.5, initial pressure P0 is 4 105 Pa, and
frequency is f 15000 hz (Pozuelo et al. 1999). (For example, one can interpret the
presence of four incident waves at pressure intensity of Pi 105 Pa each. As such,
there is the multiple incident waves focusing with linear superposition giving rise to
P0 4 105 Pa at the vicinity of the scatterer.) The calculated results for the ratio
58
between the second order harmonic to fundamental (first order term) are displayed in
Figure 4.2. It is evident that the general variation depicted in Fig.4.2 are only similar
in trend but not in magnitude to that found in reference (Pozuelo et al. 1999). From
Eq.(4.5) and Figure 4.2, it is observed that the ratio between second harmonic to first
order harmonic for plane wave increase linearly with increasing distance away from
the focused point. For the second harmonics of spherical and cylindrical waves, the
increase with distance from the focused point, is less important for the waves for the
same vibration amplitude of the excitation source.
280
Pressure amplitude(dB)
260
Fundamental-planar
Fundamental-cylindrical
Fundamental-spherical
Second order-planar
Second order-cylindrical
Second order-spherical
240
220
200
180
160
140
20
40
60
80
Distance from the focused point
100
Figure 4.3: The comparison between the analytical results of planar, cylindrical
and spherical wave including the fundamental and the second harmonic
Figure 4.3 presents a more explicit comparison between the fundamental and the
second order harmonic pressure amplitudes in planar, cylindrical and spherical
coordinates. According to the definition of the pressure level (dB) set at
20 log10 p / preference , we can plot the pressure level distribution w.r.t. the distance
59
away from the focused point (the reference pressure is assumed to be 10-6 Pa). The
calculated pressure amplitude variation shown in Fig. 4.3 can be compared to the
results obtained by (Pozuelo et al. 1999). From Fig.4.3, it is observed that the second
order harmonic distribution for the planar wave will increase, the second order
harmonic distribution of cylindrical wave seems to be constant while the second order
for the harmonic spherical wave will initially increase slightly and then decrease with
the increment of the distance away from the focused point. The results from (Pozuelo
et al. 1999) indicate fairly similar trend but the respective magnitudes are quite
different.
Pressure amplitude(dB)
250
240
230
Summation for planar wave
Summation for cylindrical wave
Summation for spherical wave
220
210
200
190
20
40
60
80
Distance from the focused point
100
Figure 4.4: The variation of pressure amplitude v.s. the distance away from the
focused point for the planar, cylindrical and spherical wave
The summation of the fundamental and the second order term are shown in
Figure 4.4. It is observed that the amplitude for the cylindrical and spherical wave
decrease monotonically. On the contrary, the pressure amplitude will increase with
60
distance away the focused point for the model of planar wave. Hence, accordingly, it
is deemed not reasonable to describe the focused HIFU around the scatterer by the
planar wave. It is clear that within the distance of less than 10, the decrease of the
aggregate pressure amplitudes of cylindrical and spherical wave appear to be very fast
and the tendency to moderate beyond this point for the region between 10 to 100.
Being so, one can suggest that the domain encompassing less than ten times of the
radius of the scatterer, the non-linear effect exert a significant influence on the
focused high intensity acoustic wave.
Next, we shall discuss the nonlinear effect for the focused acoustic wave with the
small parameter Mach number by compressible potential flow theory. We will assume
the amplitude of velocity potential A 1 , and radius of the scatter Rs 1 (the details
for the planar, cylindrical and spherical wave can be found in the Appendix C). As
interpreted earlier, *r* kr is defined as the dimensionless wave number and the
velocity potential is only depend on the variable U in the frequency domain and the
small parameter U c ( c is a constant at 1500m/s ). For the numerical
calculations, we shall take the dimensionless wave number as kRs 1.0, 3.0 and 5.0
and the small parameter Mach number as 0.1, 0.3 and 0.5, so that U c
becomes 150, 450 and 750, respectively. Figure 4.5 shows the variation of the second
order term of plane wave w.r.t. the increase of the distances from the focused point for
different parameters. It is evident that with the increase of the wave number
k (shown as the horizontal columns), we can find that the general tendency of the
amplitude for the second order term to increase and the variation takes on shorter
61
periodic time. For the fixed low wave number of k 1.0 (first column in Fig.4.5), the
increase of the Mach number ( 0.1, 0.3 and 0.5) can bring about comparatively
large amplitude for the second order term. However, the maximum amplitude for each
second term appears to be constant without any variation w.r.t. the increase of
distance from the focused point. This phenomena can be interpreted as that the low
frequency stands for quasi-static model, and thus the variation of the distances have
lesser influence on the second-order nonlinear term. At the high frequency of k 5.0
(third column in Fig.4.5), the maximum amplitudes tend to increase w.r.t. the position
away from the focused point.
0.35
0.20
0.15
0.10
0.30
0.25
0.20
0.15
0.10
0.05
0.05
0.00
0.00
20
0.40
k=1.0, =0.3
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.40
20
0.20
0.15
0.10
0.05
5
10
15
Distance from the focused point
20
0.00
0.40
k=3.0, =0.3
0.30
0.25
0.20
0.15
0.10
0.00
0.40
k=1.0, =0.5
0.35
0.30
S e con d o rde r term
S e co n d o rd e r te rm
0.35
0.25
0.35
0.05
5
10
15
Distance from the focused point
0.25
0.20
0.15
0.10
0.15
0.10
0.00
20
20
20
k=5.0, =0.3
0.30
0.25
0.20
0.15
0.10
0.00
0.40
k=3.0, =0.5
0.20
0.05
5
10
15
Distance from the focused point
5
10
15
Distance from the focused point
0.25
0.00
5
10
15
Distance from the focused point
0.05
0.30
0.05
k=5.0, =0.1
0.30
0.35
S econ d o rd er term
0.35
5
10
15
Distance from the focused point
0.35
S ec on d ord er te rm
0.25
k=3.0, =0.1
S e c o n d o rd e r te rm
S ec o nd o rd e r te rm
0.30
0.40
S eco nd o rd er term
k=1.0, =0.1
S econ d o rd er term
S econ d o rd e r te rm
0.35
0.40
0.40
0.40
5
10
15
Distance from the focused point
20
k=5.0, =0.5
0.30
0.25
0.20
0.15
0.10
0.05
5
10
15
Distance from the focused point
20
0.00
5
10
15
Distance from the focused point
20
Figure 4.5: The variation of the second order term of plane wave v.s. the
variation of wave number k , Mach number and the distance away
from the focused point.
62
Second order term
0.15
(a)
0.10
0.05
0.00
Second order term
0.15
5
10
15
Distance from the focused point
(b)
20
k=3.0, =0.1
k=3.0, =0.3
k=3.0, =0.5
0.10
0.05
0.00
5
10
15
Distance from the focused point
0.15
(c)
Second order term
k=1.0, =0.1
k=1.0, =0.3
k=1.0, =0.5
20
k=5.0, =0.1
k=5.0, =0.3
k=5.0, =0.5
0.10
0.05
0.00
5
10
15
Distance from the focused point
20
Figure 4.6: The variation of the second order term for cylindrical wave
v.s. the different wave number, Mach number and the distance away
from the focused point.
63
1.0
(a)
Second order term
0.8
k=1.0, =0.1
k=1.0, =0.3
k=1.0, =0.5
0.6
0.4
0.2
0.0
5
10
15
Distance from the focused point
7
(b)
Second order term
6
5
20
k=3.0, =0.1
k=3.0, =0.3
k=3.0, =0.5
4
3
2
1
0
7
Second order term
6
5
5
10
15
Distance from the focused point
(c)
20
k=5.0, =0.1
k=5.0, =0.3
k=5.0, =0.5
4
3
2
1
0
5
10
15
Distance from the focused point
20
Figure 4.7: The variation of the second order term for spherical wave v.s. the different
wave number, Mach number and the distance away from the focused point.
Figure 4.6 illustrate the variation of the nonlinear effect for cylindrical wave at
different wave number, Mach number and distance away from the focused point. The
64
general tendency is that with the increase of the distance away from the focused point,
this result in ever smaller amplitude for the second order term. For the fixed wave
number k , the increase of the Mach number will bring about a larger amplitude for
the second order term. With the fixed higher Mach number ( 0.5), the increment of
the wave number can also weaken the amplitude for the second order term.
Velocity potential distribution
2.5
2.0
Summation of planar wave
Summation of cylindrical
Summation of spherical
1.5
1.0
0.5
0.0
5
10
15
Distance from the focused point
20
Figure 4.8: The velocity potential distribution near the scatterer (the summation of
the first order term and the second order term) at k 2.0 and 0.3
Figure 4.7 shows the variation of the nonlinear effect for the spherical wave
for the different wave number, Mach number and distance away from the focused
point. Similar to that observed for the cylindrical wave, the general tendency is that
the increase of the distance away from the focused point will result in smaller
amplitude for the second order term. At the fixed low frequency of k 1.0, the
increase of the Mach number ( 0.1, 0.3 and 0.5) will bring about a larger amplitude
for the second order term. Conversely, at the higher frequencies of k 3.0 and
65
k 5.0, a lower Mach number results in stronger nonlinear effects and this trend is
very different from the cylindrical wave.
Finally, Figure 4.8 give the total velocity potential (the summation of the first
order term and the second order term) at k 2.0 and 0.3. We can observe the
variation of the velocity potential for both the cylindrical wave and spherical wave
appear to be consistent with the conclusion from the Westervelt model for the
pressure distribution. The general tendency of the velocity potential will decrease w.r.t.
the increase of the distance away from the focused point while the amplitude of the
velocity potential for the planar wave seem to be periodic constant. For the spherical
wave, since our analytical solution is expressed by cosine/sine function (or
cosine/sine integral function) rather than the exponential function, a characteristic
feature of this functional is a periodic rather than monotonic curve as from the
Westervelt equation.
66
4.5 Conclusions
In this study, we adopted two nonlinear models to investigate the multiple
incident acoustic waves focused on certain domain where the nonlinear effect is not
negligible in the vicinity of the scatterer. The general solutions for the one
dimensional Westervelt equation with different coordinates (plane, cylindrical and
spherical) were analytically obtained based on the perturbation method with keeping
only the second order nonlinear terms. By introducing the small parameter (Mach
number), we applied the compressible potential flow theory and proposed a novel
dimensionless formulation and asymptotic perturbation expansion for the velocity
potential and enthalpy which is different from the existing fractional nonlinear
acoustic models (eg. the Burgers equation, KZK equation and Westervelt equation).
Our analytical solutions and numerical calculations have shown the general tendency
of the velocity potential and pressure to decrease w.r.t. the increase of the distance
away from the focused point. At least, within the region which is about 10 times the
radius of the scatterer, the non-linear effect exerts a significant influence on the
distribution of the pressure and velocity potential. It is also interesting that at high
frequencies, lower Mach numbers appear to bring out even stronger nonlinear effects
for the spherical wave. Our novel approach with a small parameter for the cylindrical
and spherical waves could serve as an effective analytical model to simulate the
focused nonlinear acoustic near the scatterer in an ideal fluid and be applied to study
bubble cavitation dynamic associated with HIFU in our future work.
67
Chapter 5 Overall Conclusions and
Recommendations
5.1 Conclusions
In this thesis, we analytically applied two novel methods shown as conformal
mapping and perturbation to solve linear/nonlinear acoustic wave propagating through
ideal fluid with inclusion inside. The robustness and versatility of our methods were
verified by comparing with some calculated results from other published literatures.
The major contributions of this thesis can be summarized as below:
First of all, we analytically obtained the two dimensional general solution of
Helmholtz equation, shown as fractional order Bessel function with mapping function
as the argument, to study the linear acoustic wave scattering by rigid inclusion with
irregular cross section in an ideal fluid. Based on the conformal mapping method
together with the complex variables method, we can map the initial geometry into a
circular shape as well as transform the original physical vector into corresponding
new expressions in the mapping plane. This study may provide the basis for further
analyses of other conditions of acoustic wave scattering in fluids, e.g. irregular elastic
inclusion within fluid with viscosity, etc. Our calculated results have shown that the
angle and frequency of the incident waves have significant influence on the bistatic
scattering pattern as well as the far field form factor for the pressure in the fluid.
Moreover, we have shown that the sharper corners of the irregular inclusion may
amplify the bistatic scattering pattern compared with the more rounded corners.
68
Separately, we further use the perturbation method to solve the one dimensional
nonlinear acoustic wave propagating through the infinite domain with a submersible
inclusion. In fact, we adopted two nonlinear models to investigate the multiple
incident acoustic waves focused on certain domain where the nonlinear effect is not
negligible in the vicinity of the scatterer. The general solutions for the one
dimensional Westervelt equation with different coordinates (plane, cylindrical and
spherical) are analytically obtained based on the perturbation method with keeping
only the second order nonlinear terms. By introducing the small parameter (Mach
number), we applied the compressible potential flow theory and proposed a novel
dimensionless formulation and asymptotic perturbation expansion for the velocity
potential and enthalpy which is different from the existing more fractional nonlinear
acoustic models (eg. the Burgers equation, KZK equation and Westervelt equation).
Our analytical solutions and numerical calculations have shown that the general
tendency of the velocity potential and pressure to decrease w.r.t. the increase of the
distance away from the focused point. At least, within the region which is about 10
times the radius of the scatterer, the non-linear effect exerts a significant influence on
the distribution of the pressure and velocity potential. It is also interesting that at high
frequencies, lower Mach numbers appear to bring out even stronger nonlinear effects
for the spherical wave. Our approach with small parameter for the cylindrical and
spherical waves could serve as an effective analytical model to simulate the focused
nonlinear acoustic near the scatterer in an ideal fluid and be applied to study bubble
cavitation dynamic associated with HIFU in future work.
69
5.2 Recommendations
To the best of our knowledge, we would like to point out that the conformal
mapping together with complex variables method can only handle the two
dimensional problem, however, it may be an effective approach to solve the
geometries with sharp corners. On the perturbation method, it is an effective means to
obtain the asymptotic solution of some complex problem, in particularly, to provide
the physical analysis to give qualitative insight. At this stage, we would like to
propose several interesting and possible works to be done in the future:
1.
Multiple
linear
acoustic
waves
interact
with
irregular
rigid/elastic/bubble inclusion within an infinite/semi-infinite fluid by
the vector superposition of the linear incident waves.
2.
The analysis on the nonlinear effects for the bubble cavitation by
the
transformation
of
nonlinear
equation
into
linear
homogeneous/non-homogeneous equation with small parameter in
the spherical coordinates.
3.
Numerical calculation for the nonlinear acoustic equations by the
finite differential scheme for the model with two or three dimensional
condition.
70
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79
Appendix A
Partial Coding for Linear Acoustic Wave Propagation Relate
to Conformal Mapping Method
Code 1:
clear;
T=1.0;
R=1.0; % 0.7
r=1.0; % 1.3
q=2;
p=1/2;
b=5;
a=1/b;
d=6;
c=1/d;
liu=0;
for sita=0:0.01:2*pi;
liu=liu+1
S(liu)=sita/2/pi*360;
w(liu)=abs(R*(r*exp(i*sita)+p/(r*exp(i*sita))^q));
M(liu)=real(R*(r*exp(i*sita)+p/(r*exp(i*sita))^q));
N(liu)=imag(R*(r*exp(i*sita)+p/(r*exp(i*sita))^q));
w1(liu)=abs(R*(r*exp(i*sita)+a/(r*exp(i*sita))^b));
M1(liu)=real(R*(r*exp(i*sita)+a/(r*exp(i*sita))^b));
N1(liu)=imag(R*(r*exp(i*sita)+a/(r*exp(i*sita))^b));
w2(liu)=abs(R*(r*exp(i*sita)+c/(r*exp(i*sita))^d));
M2(liu)=real(R*(r*exp(i*sita)+c/(r*exp(i*sita))^d));
N2(liu)=imag(R*(r*exp(i*sita)+c/(r*exp(i*sita))^d));
end
Code 2:
clear;
r=1.0;
c1=1/2; % 1/9 , 1/5, 1/3, 1/2
80
a1=1/3; % 1/3, 1/6 ,1/9
a2=1/15; % 1/15, 2/15, 7/45
liu=0;
for sita=0:0.01:2*pi;
liu=liu+1
S(liu)=sita/2/pi*360;
w(liu)=abs((r*exp(i*sita))^(-1)+c1*(r*exp(i*sita))^2);
M(liu)=real((r*exp(i*sita))^(-1)+c1*(r*exp(i*sita))^2);
N(liu)=imag((r*exp(i*sita))^(-1)+c1*(r*exp(i*sita))^2);
w1(liu)=abs((r*exp(i*sita))^(-1)+a1*(r*exp(i*sita))^2+a2*(r*exp(i*sita))^5);
M1(liu)=real((r*exp(i*sita))^(-1)+a1*(r*exp(i*sita))^2+a2*(r*exp(i*sita))^5);
N1(liu)=imag((r*exp(i*sita))^(-1)+a1*(r*exp(i*sita))^2+a2*(r*exp(i*sita))^5);
w2(liu)=abs((r*exp(i*sita))^(1)+c1*(r*exp(i*sita))^(-2));
M2(liu)=real((r*exp(i*sita))^(1)+c1*(r*exp(i*sita))^(-2));
N2(liu)=imag((r*exp(i*sita))^(1)+c1*(r*exp(i*sita))^(-2));
w3(liu)=abs((r*exp(i*sita))^(1)+a1*(r*exp(i*sita))^(-2)+a2*(r*exp(i*sita))^(-5));
M3(liu)=real((r*exp(i*sita))^(1)+a1*(r*exp(i*sita))^(-2)+a2*(r*exp(i*sita))^(-5));
N3(liu)=imag((r*exp(i*sita))^(1)+a1*(r*exp(i*sita))^(-2)+a2*(r*exp(i*sita))^(-5));
end
Code 3:
clear;
fpi=0; % sita angle
mk=300; % the division of the angle
hh=2*pi/mk; % the step
w0=1; % pressure amplitude
mu=0.0; % damping constant
rou=1.0; % stands for the density of the fluid
nn=2.0; %%%%%%%%%%%%%%%%% incident wave number
kd=nn; % nn*pi; % K of the function
oumiga=kd*1480; % incident frequency
a0=3*pi/4; %%%%%%%%%%%%%%%%% incident angle pi, 3*pi/4, pi/2
R=1.0; % 0.75;
q=2; % 2; 3
p=1/q;
% 1/q; % canonical ellipse R=0.75 p=1/3;
r=1.0; % radius of the circle 1.0 or 1.3
a=100;
nnn1=7; %terms of the series
81
for n=1-nnn1:nnn1-1;
for m=1-nnn1:nnn1-1;
z=r*exp(i*fpi);
wz=R*(z+p/z^q);
sz=R*(1-p*q/z^(q+1));
xx11=kd*abs(wz);
t=0;
t1=(besselh(m-1,xx11)*(wz/abs(wz))^(m-1)*exp(i*fpi)*sz/abs(sz)-besselh(m+1,xx11)*(wz/abs(
wz))^(m+1)*exp(-i*fpi)*conj(sz)/abs(sz))*exp(-i*n*fpi)/(2*pi);
for k=1:mk;
ct=fpi+k*hh;
z=r*exp(i*ct);
wz=R*(z+p/z^q);
sz=R*(1-p*q/z^(q+1));
xx11=kd*abs(wz);
t2=(besselh(m-1,xx11)*(wz/abs(wz))^(m-1)*exp(i*ct)*sz/abs(sz)-besselh(m+1,xx11)*(wz/abs(
wz))^(m+1)*exp(-i*ct)*conj(sz)/abs(sz))*exp(-i*n*ct)/(2*pi);
tt=(t1+t2)/2;
t=t+tt*hh;
t1=t2;
end;
slb(n+nnn1,m+nnn1)=t;
end
end %
for n=1-nnn1:nnn1-1;
z=r*exp(i*fpi);
wz=R*(z+p/z^q);
sz=R*(1-p*q/z^(q+1));
t=0;
t1=-i*w0*exp(i*kd/2*(wz*exp(-i*a0)+conj(wz)*exp(i*a0)))*(exp(i*fpi)*sz/abs(sz)*exp(-i*a0)+exp(
-i*fpi)*conj(sz)/abs(sz)*exp(i*a0))*exp(-i*n*fpi)/(2*pi);%
for k=1:mk;
ct=fpi+k*hh;
z=r*exp(i*ct);
wz=R*(z+p/z^q);
sz=R*(1-p*q/z^(q+1));
t2=-i*w0*exp(i*kd/2*(wz*exp(-i*a0)+conj(wz)*exp(i*a0)))*(exp(i*ct)*sz/abs(sz)*exp(-i*a0)+exp(i*ct)*conj(sz)/abs(sz)*exp(i*a0))*exp(-i*n*ct)/(2*pi);%
tt=(t1+t2)/2;
82
t=t+tt*hh;
t1=t2;
end;
srb(n+nnn1)=-t;
end
bn=slb\srb';
%%%%%%%% below is the velocity and pressure inside the fluid
Q=0;
for ppii=0:0.1:360; % the azimuthal along the solid surface
Q=Q+1
ppi=ppii/180*pi;
liu(Q)=ppii;
gang(Q)=ppi;
f=a*exp(i*ppi);
wf=R*(f+p/f^q);
sf=R*(1-p*q/f^(q+1));
xx22=kd*abs(wf);
usitai=-w0/2*kd*exp(i*kd/2*(wf*exp(-i*a0)+conj(wf)*exp(i*a0)))*(exp(i*ppi)*sf/abs(sf)*exp(-i*a0)
-exp(-i*ppi)*conj(sf)/abs(sf)*exp(i*a0));
pai=rou*(i*oumiga+mu)*w0*exp(i*kd/2*(wf*exp(-i*a0)+conj(wf)*exp(i*a0)));
faii=w0*exp(i*kd/2*(wf*exp(-i*a0)+conj(wf)*exp(i*a0)));
usitas=0;
pas=0;
fpas=0;
fais=0;
for m=1-nnn1:nnn1-1;
usitas=usitas+i*kd/2*bn(m+nnn1).*(besselh(m-1,xx22)*(wf/abs(wf))^(m-1)*exp(i*ppi)*sf/abs(sf)
+besselh(m+1,xx22)*(wf/abs(wf))^(m+1)*exp(-i*ppi)*conj(sf)/abs(sf));
pas=pas+rou*(i*oumiga+mu)*bn(m+nnn1).*besselh(m,xx22)*(wf/abs(wf))^m;
fpas=fpas+rou*(i*oumiga+mu)*bn(m+nnn1).*(-i)^m*(2/(pi*kd*abs(wf)))^(1/2)*exp(i*(kd*abs(wf)
-pi/4));
fais=fais+bn(m+nnn1).*besselh(m,xx22)*(wf/abs(wf))^m;
end
usita(Q)=abs((usitai+usitas)/kd/w0);
pa(Q)=abs(pai+pas);
coeff(Q)=abs(pas/pai);
fpa(Q)=abs(fpas);
fai(Q)=abs(faii+fais);
end
83
Code 4
clear;
fpi=0;
mk=300;
hh=2*pi/mk;
w0=1;
mu=0.0;
rou=1.0;
nn=2.0;
kd=nn;
oumiga=kd*1480;
a0=pi/2; %%%%%%%%%%%%%%%incident angle pi, 3*pi/4, pi/2
c1=1/3; % 1/5, 1/3, 1/2, 1/3, 1/6, 1/9
c2=1/15;
c3=0;
%
0, 0,
0, 1/15, 2/15, 7/45
%0
r=1.0;
a=100;
nnn1=7;
for n=1-nnn1:nnn1-1;
for m=1-nnn1:nnn1-1;
z=r*exp(i*fpi);
wz=z+c1*z^(-2)+c2*z^(-5)+c3*z^(-8);
sz=1-2*c1*z^(-3)-5*c2*z^(-6)-8*c3*z^(-9);
xx11=kd*abs(wz);
t=0;
t1=(besselh(m-1,xx11)*(wz/abs(wz))^(m-1)*exp(i*fpi)*sz/abs(sz)-besselh(m+1,xx11)*(wz/abs(
wz))^(m+1)*exp(-i*fpi)*conj(sz)/abs(sz))*exp(-i*n*fpi)/(2*pi);
for k=1:mk;
ct=fpi+k*hh;
z=r*exp(i*ct);
wz=z+c1*z^(-2)+c2*z^(-5)+c3*z^(-8);
sz=1-2*c1*z^(-3)-5*c2*z^(-6)-8*c3*z^(-9);
xx11=kd*abs(wz);
t2=(besselh(m-1,xx11)*(wz/abs(wz))^(m-1)*exp(i*ct)*sz/abs(sz)-besselh(m+1,xx11)*(wz/abs(
wz))^(m+1)*exp(-i*ct)*conj(sz)/abs(sz))*exp(-i*n*ct)/(2*pi);
tt=(t1+t2)/2;
t=t+tt*hh;
t1=t2;
end;
slb(n+nnn1,m+nnn1)=t;
end
end
84
for n=1-nnn1:nnn1-1;
z=r*exp(i*fpi);
wz=z+c1*z^(-2)+c2*z^(-5)+c3*z^(-8);
sz=1-2*c1*z^(-3)-5*c2*z^(-6)-8*c3*z^(-9);
t=0;
t1=-i*w0*exp(i*kd/2*(wz*exp(-i*a0)+conj(wz)*exp(i*a0)))*(exp(i*fpi)*sz/abs(sz)*exp(-i*a0)+exp(
-i*fpi)*conj(sz)/abs(sz)*exp(i*a0))*exp(-i*n*fpi)/(2*pi);
for k=1:mk;
ct=fpi+k*hh;
z=r*exp(i*ct);
wz=z+c1*z^(-2)+c2*z^(-5)+c3*z^(-8);
sz=1-2*c1*z^(-3)-5*c2*z^(-6)-8*c3*z^(-9);
t2=-i*w0*exp(i*kd/2*(wz*exp(-i*a0)+conj(wz)*exp(i*a0)))*(exp(i*ct)*sz/abs(sz)*exp(-i*a0)+exp(i*ct)*conj(sz)/abs(sz)*exp(i*a0))*exp(-i*n*ct)/(2*pi);
tt=(t1+t2)/2;
t=t+tt*hh;
t1=t2;
end;
srb(n+nnn1)=-t;
end
bn=slb\srb';
%%%%%%%%%%%%%%%% below is the velocity and pressure inside the fluid
Q=0;
for ppii=0:0.1:360;
Q=Q+1
ppi=ppii/180*pi;
liu(Q)=ppii;
gang(Q)=ppi;
f=a*exp(i*ppi);
wf=f+c1*f^(-2)+c2*f^(-5)+c3*f^(-8);
sf=1-2*c1*f^(-3)-5*c2*f^(-6)-8*c3*f^(-9);
xx22=kd*abs(wf);
usitai=-w0/2*kd*exp(i*kd/2*(wf*exp(-i*a0)+conj(wf)*exp(i*a0)))*(exp(i*ppi)*sf/abs(sf)*exp(-i*a0)
-exp(-i*ppi)*conj(sf)/abs(sf)*exp(i*a0));
pai=rou*(i*oumiga+mu)*w0*exp(i*kd/2*(wf*exp(-i*a0)+conj(wf)*exp(i*a0)));
faii=w0*exp(i*kd/2*(wf*exp(-i*a0)+conj(wf)*exp(i*a0))); usitas=0;
pas=0;
fpas=0;
fais=0;
for m=1-nnn1:nnn1-1;
85
usitas=usitas+i*kd/2*bn(m+nnn1).*(besselh(m-1,xx22)*(wf/abs(wf))^(m-1)*exp(i*ppi)*sf/abs(sf)
+besselh(m+1,xx22)*(wf/abs(wf))^(m+1)*exp(-i*ppi)*conj(sf)/abs(sf));
pas=pas+rou*(i*oumiga+mu)*bn(m+nnn1).*besselh(m,xx22)*(wf/abs(wf))^m;
fpas=fpas+rou*(i*oumiga+mu)*bn(m+nnn1).*(-i)^m*(2/(pi*kd*abs(wf)))^(1/2)*exp(i*(kd*abs(wf)
-pi/4));
fais=fais+bn(m+nnn1).*besselh(m,xx22)*(wf/abs(wf))^m;
end
usita(Q)=abs((usitai+usitas)/kd/w0);
pa(Q)=abs(pai+pas);
coeff(Q)=abs(pas/pai);
fpa(Q)=abs(fpas);
fai(Q)=abs(faii+fais);
end
86
Appendix B
Neglecting the terms of third or higher orders, we can obtain the approximate formulate
of Westervelt equation as below:
2 p1 p2
2
1 p1 p2
2 p12
0.
t 2
c2
c4 t 2
(B1)
Here, we can set two separated governing equations for the first linear solution of p1 and
following inhomogeneous equation with second order correction p2 :
2 p1
1 2 p1
0,
c2 t 2
(B2)
2 p2
1 2 p2
2 p12
.
c4 t 2
c2 t 2
(B3)
Here,
we
can
take
the
harmonic
problem
into
account
and
assume p1 P1 exp it , p2 P2 exp 2it , so we can obtain the reduced governing
equation as below:
2 P1 k 2 P1 0,
2 P2 4k 2 P2
(B4)
4k 2 2
P .
c2 1
(B5)
In summary, we need to obtain the solutions of the above Eq. (B5) and get the detail
expressions
for
P1
and
P2 ,
and
the
total
pressure
will
be
shown
as p P1 exp it P2 exp 2it .
87
Analytical Solution for Plane wave
The simplest and best studies case is the plane wave propagation. Various explicit solutions
can be found in the literatures. For the condition of one dimensional problem in Cartesian
coordinates, the governing equation can be reduced into:
d 2 P1
k 2 P1 0,
dx 2
(B6)
d 2 P2
4k 2 2
2
4
k
P
P .
2
dx 2
c2 1
(B7)
For the harmonic excitation of amplitude P0 , the acoustic pressure is assumed to be an
addition of two terms. The analytical solutions which have the form were obtained:
P1 P0 e ikx ,
P2
P02
1 2ikx e 2ikx .
2 c2
(B8)
(B9)
Analytical Solution for Cylindrical Wave
The acoustic wave propagation expansion in the polar coordinate with one-dimensional will
make the governing equation as below:
d 2 P1 1 dP1
k 2 P1 0,
dr 2 r dr
(B10)
d 2 P2 1 dP2
4k 2 2
2
4
k
P
P .
2
dr 2 r dr
c2 1
(B11)
It is easy to find that Eq.(B10) is the zero order Bessel function with the variable as kr , and
the solution is that:
88
P1 P0 J 0 kr or P0 H 0 kr .
(B12)
According to this certain problem, we can assume the particular solution basing on the time
term as exp it :
P1 P0 H 0 2 kr ,
(B13)
when r , the asymptotic expression of Eq.(B13) will turn to be:
P1 P0
2
i kr 4
e
.
kr
(B14)
Substituting Eq.(B14) into Eq.(B11), we can obtain:
8k P02 e 2i kr
d 2 P2 1 dP2
2
4
k
P
2
dr 2 r dr
c2
r
4
,
(B15)
Subsequently, we need to get the analytical solution of the second-order correction as shown
in Eq.(B15). Here, it is straight forward for us to obtain the analytical solution as below:
P2
4i P02 i 2 2ikr
e e .
c2
(B16)
Analytical Solution for Spherical Wave
For the condition of one dimensional problem in spherical coordinates, the governing
equation can be reduced into:
d 2 P1 2 dP1
k 2 P1 0,
2
dr
r dr
(B17)
d 2 P2 2 dP2
4k 2 2
2
4k P2
P .
dr 2 r dr
c2 1
(B18)
89
The solution of Eq.(B17) in spherical coordinate can be shown as below:
P1
P0 ikr
e .
r
(B19)
Where P0 is the amplitude of the incident pressure. It is also easy to observe that when
r , the pressure P1 gradually decreased into infinitesimal.
Substituting Eq.(B19) into the non-homogeneous term, the following wave equation for the
second order correction P2 is obtained:
4 k 2 P02 2ikr
d 2 P2 2 dP2
2
4k P2
e ,
dr 2 r dr
c2 r 2
(B20)
The following solution was proposed to satisfy Eq.(B20):
ln r n
n
P2
n2 r
r
2ikr
e ,
(B21)
By substituting Eq.(B21) into Eq.(B20), we can obtain the approximate solutions of the
unknown coefficients and n can be:
i kP02
,
c2
(B22)
1 n 2 ! ,
n
n 1
4ik
n
(B23)
So we can obtain the approximated analytical solution for the second order Eq.(B20) as
below:
n
i kP02 ln r 1 n 2 ! 2ikr
P2
e .
c2 r n 2 4ik n 1 r n
(B24)
90
Actually, this solution is asymptotic one, and when n as well as the increase of the r
will make the solution to be more accurate.
91
Appendix C
Analytical Solution for Plane Wave
For the one dimensional plane wave, the governing equations are (the higher order small
terms will be omitted subsequently):
2 *0 2 *0
0,
r*2
t*2
(C1)
2
*0 2 *0
2 *1 2 *1 1 *0 *0 2 *0
n
1
.
2 t* r*
r*2
t*2
r* r*t*
t* t*2
(C2)
Here, we would like to point out that r* can be directly insteaded by the Cartesian
coordinate x* for the plane wave. We propose the one dimensionless solution for the linear
acoustic wave in Eq.(C1) as below:
*0 A* cos k*r* ei t A* cos *r* ei t
**
**
(C3)
where A* , k* , * are the dimensionless amplitude, wave number, frequency, respectively,
and defined as:
A*
R
A
U
, k* Rs k , * s , r*
r , t* t.
RsU
Rs
Rs
U
(C4)
It is evident that Eq.(C3) is a solution of Eq.(C1). By substituting Eq.(C3) into Eq.(C2) and
assuming that *1 *1 r* e 2i*t* , we obtain the simplified second-order non-homogeneous
differential equation as:
2*1
4*2*1 2iA*2*3 sin 2 *r* i n 1 A*2*3 cos 2 *r* .
2
r*
(C5)
92
The general analytical solution for this second-order non-homogeneous Eq.(C5) is:
*1 C1 cos 2*r* e2i t C2 sin 2*r* e2i t
**
**
i* A*2
n 1
cos 2*r* n 3 e2i*t* .
n 1 *r* sin 2*r*
8
4
(C6)
where, C1 and C2 are unknown coefficients to be determined by the imposed boundary
conditions. For simplicity in the discussion below, the coefficients for the second-order
nonlinear terms can be reasonably assumed as C1 A* , C2 0 without
loss of
generality.
Analytical Solution for Cylindrical Wave
For the one dimension cylindrical wave, the governing equations are:
2 *0 1 *0 2 *0
0,
r* r*
r*2
t*2
(C7)
2
*0 2 *0
2 *1 1 *1 2 *1 1 *0 *0 2 *0
.
n 1
r* r*
2 t* r*
r*2
t*2
r* r*t*
t* t*2
(C8)
The proposed solution for the linear acoustic wave (second kind of Hankel function with zero
order) is assumed as:
*0 A* H 0 2 *r* ei t
(C9)
**
By substituting Eq.(C9) into Eq.(C8) and assuming *1 *1 r* e 2i*t* , we obtain the
simplified second-order non-homogeneous differential equation as:
2
2
2*1 1 *1
2
2
2
2 3
2 3
4
2
iA
H
r
i
n
1
A
H
r
,
* *1
* *
1
**
* *
0
**
r*2 r* r*
(C10)
and the analytical solution is (the softwave named MAPLE will be used here):
93
*1 C1 J0 2*r* e2i t C2 Y0 2*r* e2i t
**
**
J 2 r n 1 H 2 r 2 2 H 2 r 2 Y 2 r r dr
* * 0 * *
1 * *
0
* * * *
(C11)
i A 0
e2i*t* .
2
2
2
2
2
Y0 2*r* n 1 H0 *r* 2 H1 *r* J0 2*r* r*dr*
3 2
* *
Here, J 0 is the Bessel function of order zero, Y0 is the second kind of Bessel
2
function of order zero, and H 0 J 0 iY0 is second kind Hankel function with
order zero. For ease of discussion and without loss of generality, one can reasonably assumed
that C1 A* and C2 iA* .
Analytical Solution for Spherical Wave
For the one dimension spherical wave, the governing equations are:
2 *0 2 *0 2 *0
0,
r* r*
r*2
t*2
(C12)
2
*0 2 *0
2 *1 2 *1 2 *1 1 *0 *0 2 *0
.
n 1
r* r*
2 t* r*
r*2
t*2
r* r*t*
t* t*2
(C13)
Again, the solution for the linear acoustic wave is assumed as:
*0
A*
cos *r* ei*t* .
r*
(C14)
By substituting Eq.(C14) into Eq.(C13) and assuming *1 *1 r* e 2i*t* , we have:
2
2*1 2 *1
4*2*1 2iA*2* r*2 cos*r* r*1* sin *r* i n 1 A*2*3r*2 cos2 *r* , (C15)
2
r* r* r*
In which the analytical solution is:
94
*1
C1
C
cos 2*r* e 2i*t* 2 sin 2*r* e 2i*t*
r*
r*
(C16)
i* A*2 *r* n 1 Ci 4*r* 2Ci 2*r* ln r* sin 2*r*
2 i*t*
e
.
8r*2 *r* n 1 Si 4*r* 2 Si 2*r* cos 2*r* 8cos 2 *r*
where Ci and Si stand for the Cosine Integral and Sine Integral function,
respectively. As before, C1 and C2 are unknown coefficients to be solved based on
imposed boundary conditions. For simplicity, we shall assume C1 A* and C2 0 .
Since we have assumed r
p p U 2 p* , A*
Rs
r* , t
Rs
t* , RsU * , h U 2 h* , c c c* ,
U
R
A
U
U
t , r* r ,
, * s , t*
, k
, the
c
c
RsU
Rs
Rs
U
waves above can be transformed accordingly to the one dimensional plane wave as:
0 A cos kr eit
1 Acos 2kr e2it
(C17)
i A2
n 1
n 1 kr sin 2kr
cos 2kr n 3 e2it ,
2
8U
4
(C18)
the one dimensional cylindrical wave as
0 A H 0 2 kr eit
1 A H0 2 2kr e2it
(C19)
J 2kr n 1 H 2 kr 2 2 H 2 kr 2 Y 2kr rdr
0
0
1
2it
i A 0
e ,
2
2
2U 4
2
2
Y0 2kr n 1 H0 kr 2 H1 kr J0 2kr rdr
2
3 2
(C20)
and the one dimensional spherical wave as
0
ARs
cos kr eit ,
r
(C21)
95
ARs
cos 2kr e2it
r
r
2it
i A2 Rs2 kr n 1 Ci 4kr 2Ci 2kr ln sin 2kr
Rs
2 2 2
e .
8U r
2
kr n 1 Si 4kr 2Si 2kr cos 2kr 8cos kr
1
(C22)
The total velocity potential is then 0 2 1 O 4 .
96
Appendix D
Partial Coding for the Nonlinear Acoustic Wave Propagation
Code 1:
clear;
p0=4*10^5;
c=1500;
f=15000;
w=2*pi*f;
k=w/c;
rou=1000;
bt=3.5;
t=0;
liu=0;
for x=1.01:0.1:5.01;
liu=liu+1
gang(liu)=x;
pp1(liu)=abs(p0*exp(-i*k*x)*exp(i*w*t));
pp2(liu)=abs(bt*p0^2/2/rou/c^2*(1+2*i*k*x)*exp(-2*i*k*x)*exp(2*i*w*t));
pp(liu)=abs(p0*exp(-i*k*x)*exp(i*w*t)+bt*p0^2/2/rou/c^2*(1+2*i*k*x)*exp(-2*i*k*x)*exp(2*i*w*t))
;
Rapp(liu)=abs(pp2/pp1);
ppc1(liu)=abs(p0*besselh(0,2,k*x)*exp(i*w*t)); % k*(x-1.0)
ppc2(liu)=abs(2*i*bt*k*x/rou/c^2*(p0*besselh(0,2,k*x)*exp(i*w*t))^2);
ppc(liu)=abs(p0*besselh(0,2,k*x)*exp(i*w*t)+2*i*bt*k*x*p0^2/rou/c^2*(besselh(0,2,k*x))^2*exp(
2*i*w*t));
Rappc(liu)=abs(ppc2/ppc1);
s=0;
for n=2:1:100;
s=s+(-1)^n*gamma(n-1)/(4*i*k)^(n-1)/x^n;
end
LL(liu)=s;
ps1(liu)=abs(p0/x*exp(-i*k*x)*exp(i*w*t));
ps2(liu)=abs(i*bt*k*p0^2/rou/c^2*(log(x)/x+LL(liu))*exp(-2*i*k*x)*exp(2*i*w*t));
ps(liu)=abs(p0/x*exp(-i*k*x)*exp(i*w*t)+i*bt*k*p0^2/rou/c^2*(log(x)/x+LL(liu))*exp(-2*i*k*x)*exp
97
(2*i*w*t));
Raps(liu)=abs(ps2/ps1);
end
Code 2:
clear;
p0=4*10^5;
c=1500;
f=15000;
w=2*pi*f;
k=w/c;
rou=1000;
bt=3.5;
t=0;
liu=0;
for x=1.01:0.1:5.01;
liu=liu+1
gang(liu)=x;
pp1(liu)=abs(p0*exp(-i*k*x)*exp(i*w*t));
pp2(liu)=abs(bt*p0^2/2/rou/c^2*(1+2*i*k*x)*exp(-2*i*k*x)*exp(2*i*w*t));
pp(liu)=abs(p0*exp(-i*k*x)*exp(i*w*t)+bt*p0^2/2/rou/c^2*(1+2*i*k*x)*exp(-2*i*k*x)*exp(2*i*w*t))
;
Rapp(liu)=abs(pp2/pp1);
ppc1(liu)=abs(p0*besselh(0,2,k*x)*exp(i*w*t)); % k*(x-1.0)
ppc2(liu)=abs(2*i*bt*k*x/rou/c^2*(p0*besselh(0,2,k*x)*exp(i*w*t))^2);
ppc(liu)=abs(p0*besselh(0,2,k*x)*exp(i*w*t)+2*i*bt*k*x*p0^2/rou/c^2*(besselh(0,2,k*x))^2*exp(
2*i*w*t));
Rappc(liu)=abs(ppc2/ppc1);
s=0;
for n=2:1:100;
s=s+(-1)^n*gamma(n-1)/(4*i*k)^(n-1)/x^n;
end
LL(liu)=s;
ps1(liu)=abs(p0/x*exp(-i*k*x)*exp(i*w*t));
ps2(liu)=abs(i*bt*k*p0^2/rou/c^2*(log(x)/x+LL(liu))*exp(-2*i*k*x)*exp(2*i*w*t));
ps(liu)=abs(p0/x*exp(-i*k*x)*exp(i*w*t)+i*bt*k*p0^2/rou/c^2*(log(x)/x+LL(liu))*exp(-2*i*k*x)*exp
(2*i*w*t));
Raps(liu)=abs(ps2/ps1);
98
end
Code 3:
clear;
A=1.0;
Rs=1.0;
k=2.0;
U=450;
c=1500;
e=U/c;
w=k*c;
t=0;
n=7.15;
%%%%%%%%%%%%%%%%%%%%%%%% the integrations for the cylindrical wave
x=1.02;
tt=0;
gg=0;
tt1=((n-1)*besselh(0,2,k*x)^2-2*besselh(1,2,k*x)^2)*bessely(0,2*k*x)*x;
for x=1.03:0.01:20.01;
tt2=((n-1)*besselh(0,2,k*x)^2-2*besselh(1,2,k*x)^2)*bessely(0,2*k*x)*x;
ttt=(tt1+tt2)/2;
tt=tt+ttt*0.01;
tt1=tt2;
end
gg1=((n-1)*besselh(0,2,k*x)^2-2*besselh(1,2,k*x)^2)*besselj(0,2*k*x)*x;
for x=1.03:0.01:20.01;
gg2=((n-1)*besselh(0,2,k*x)^2-2*besselh(1,2,k*x)^2)*besselj(0,2*k*x)*x;
ggg=(gg1+gg2)/2;
gg=gg+ggg*0.01;
gg1=gg2;
end
%%%%%%%%%%%%%%%%%%%%%%%%
liu=0;
for x=1.02:0.01:20.01;
% the distance away from the original point
liu=liu+1
gang(liu)=x;
% plane wave
99
fai0(liu)=abs(A*cos(k*x)*exp(i*w*t));
fai1(liu)=e^2*abs(A*cos(2*k*x)*exp(2*i*w*t)-i*w*A^2/8/U^2*((n+1)*k*x*sin(2*k*x)+(n+1)/4*cos(
2*k*x)+n-3)*exp(2*i*w*t));
fai(liu)=fai0(liu)+fai1(liu);
Ratiofai(liu)=fai1(liu)/fai0(liu);
% cylindrical wave
cai0(liu)=abs(A*besselh(0,2,k*x)*exp(i*w*t));
cai1(liu)=e^2*abs(A*besselh(0,2,2*k*x)*exp(2*i*w*t)+i*pi*A^2*w^3*e^2/2/U^4*(besselj(0,2*k*x)
*tt-bessely(0,2*k*x)*gg)*exp(2*i*w*t));
cai(liu)=cai0(liu)+cai1(liu);
Ratiocai(liu)=cai1(liu)/cai0(liu);
% spherical wave
pai0(liu)=abs(A*Rs/e/x*cos(k*x)*exp(i*w*t));
pai1(liu)=e^2*abs(A*Rs/e/x*cos(2*k*x)*exp(2*i*w*t)-i*w*A^2*Rs^2/8/U^2/e^2/x^2*(k*x*(n+1)*(c
osint(4*k*x)+2*cosint(2*k*x)+log(e*x/Rs))*sin(2*k*x)-k*x*(n+1)*(sinint(4*k*x)+2*sinint(2*k*x))*c
os(2*k*x)-8*cos(k*x)^2)*exp(2*i*w*t));
pai(liu)=pai0(liu)+pai1(liu);
Ratiopai(liu)=pai1(liu)/pai0(liu);
end
100
Appendix E
The calculated results from Pillai et al. 1982:
101
[...]... Linear Acoustic Wave Scattering by Two Dimensional Scatterer with Irregular Shape in an Ideal Fluid 3.1 Governing equations of linear acoustic wave The propagation of linear sound waves in a fluid can be modeled by the equation of motion (conservation of momentum) and the continuity equation (conservation of mass) With some simplifications by taking the fluid as homogeneous, inviscid, and irrotational, acoustic. .. present work are presented There is a brief presentation on the background of linear /nonlinear acoustic wave propagation, in which attention is centered on using conformal mapping method for linear acoustic wave scattering by the inclusion with irregular across section and perturbation method for the nonlinear acoustic wave propagation In Chapter 2, we outline the mathematical background for the conformal... linear acoustic wave equation, the nonlinear counterpart can handle waves with large finite amplitudes, and allow accurate modeling of nonlinear constitutive models in the fluid Interesting phenomena unknown in linear acoustics can be observed, for example, waveform distortion, formation of shock waves, increased absorption, nonlinear interaction (as opposed to superposition) when two sound waves are... y x Solid Incident Acoustic wave Infinite fluid Figure 3.1: The model for scattering of acoustic wave by rigid inclusion with irregular across section Consequently, the corresponding governing equation (3.9) in , plane takes on the following form: w w 2 2 k 0 4 (3.10) Equation (3.10) is a general expression for the spatial linear acoustic wave in the , plane... Mitri 2005) On the other hand, the theoretical aspect of acoustic study on inclusion with arbitrary cross sections in fluids are far fewer Our proposed method is an attempt to meet the need for various geometries and extend the classical conformal mapping within the framework of complex variable methods for the acoustic wave scattering problem in fluids Incorporation of the mapping technique into the... model, conformal mapping was applied to solve the in-plane elastic wave propagation through the infinite domain with irregular-shaped cavity and dynamic stress concentration (Liu et al 1982), the anti-plane shear wave propagation via mapping into the Cartesian coordinates (Han & Liu 1997; Liu & Han 1991) and the anti-plane shear wave propagation via mapping of the inner/outer domain into polar coordinates... 23 if we set 0 , the wave number k will be the same as the harmonic wave k c0 In this paper, the outgoing scattered wave will be combined with Hankel function of the first kind and the time term e i t 3.2 Conformal transformations of Helmholtz equation and corresponding physical vector For the model of acoustic wave scattering by three dimensional inclusion with arbitrary geometry embedded... acoustic beams, cavitation and sonoluminescence (Crocker 1998) As far as we are aware, there are various models to simulate the nonlinear characteristic of the acoustic wave propagating through the fluid For instance, the one-dimensional Burgers equation has been found to be an excellent approximation of the conservation equations for plane progressive waves of finite amplitude in a thermoviscous fluid. .. possibly one of the first few to calculate the linear acoustic wave scattering of noncircular cylinders with the use of conformal mapping within the context of the complex variables method in the fluid The results obtained are validated against some special cases available in the literature, and then the effect of different geometries of the solid inclusion with sharp corners is studied (It may also be remarked... Problem Definition, Motivation and Scope of Present Work A better understanding of the physics of linear /nonlinear acoustic wave interact with inclusion is important for a wide range of applications including underwater detection, biomedical and chemical processes On the aspect of linear acoustic wave, considerable work has been done on the scattering by objects having regular cross section For instance, ... Linear Acoustic Wave Scattering by Two Dimensional Scatterer with Irregular Shape in an Ideal Fluid 3.1 Governing equations of linear acoustic wave The propagation of linear sound waves in a fluid. .. conditions of acoustic wave scattering in fluids, e.g irregular elastic inclusion within fluid with viscosity, etc Our calculated results have shown that the angle and frequency of the incident waves... background of linear /nonlinear acoustic wave propagation, in which attention is centered on using conformal mapping method for linear acoustic wave scattering by the inclusion with irregular across
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