Department of Mechanical Engineering
Application of the Multi-Modal Integral Method (MMIM) to Sound Wave Scattering in an Acoustic Waveguide.
Alexei Zinoviev
Thesis submitted for the Degree of Doctor of Philosophy October 1999
Alexei Zinoviev The Department of Mechanical Engineering, The University of Adelaide. 2 of 158
1 Table of Contents.
1 TABLE OF CONTENTS. 2
2 ABSTRACT. 5
3 STATEMENT OF ORIGINALITY. 7
4 ACKNOWLEDGMENTS. 8
5 NOTATIONS. 9
5.1 LATIN LETTERS. 9
5.2 GREEK LETTERS. 12
6 INTRODUCTION 14
7 LITERATURE REVIEW16
7.1 INTRODUCTION 16
7.2 CIRCUMFERENTIAL WAVES AROUND SOLID CYLINDERS AND
SPHERES. 16
7.3 CIRCUMFERENTIAL WAVES AROUND CYLINDRICAL AND
SPHERICAL SHELLS. 19
8 THEORETICAL FOUNDATIONS 22
8.1 INTRODUCTION 22
8.2 STATEMENT OF THE PROBLEM 22
8.2.1 Waveguide Layout 22
8.2.2 Equations and Boundary Conditions 23 8.3 SOLUTION OF THE BOUNDARY VALUE PROBLEM 24
8.3.1 Representation of the Pressure Field and Elastic
Potentials 24
8.3.2 System of Fredholm Integral Equations. 26 8.3.3 Improving Convergence of the Green’s function. 28
8.3.4 System of Linear Equations. 31
8.4 NUMERICAL SOLUTION. 33
8.5 CONCLUSIONS. 34
9 HOMOGENEOUS CYLINDERS IN ASYMMETRIC
WAVEGUIDE. 35
9.1 INTRODUCTION. 35
9.2 PARAMETERS OF THE SYSTEM. 35
9.3 AMPLITUDE-FREQUENCY CHARACTERISTICS (AFC) 35 9.4 PHASE-FREQUENCY CHARACTERISTICS (PFC). 36 9.5 THE SPATIAL DISTRIBUTION OF THE ABSOLUTE VALUE OF
PRESSURE AMPLITUDE. 41
9.6 SOURCE DENSITY ON THE CYLINDER SURFACE. 45
9.7 CONCLUSIONS. 46
10 POWER FLOW OF THE SCATTERED ACOUSTIC WAVE IN A
SYMMETRIC WAVEGUIDE. 48
10.1 INTRODUCTION 48 10.2 CALCULATION OF ENERGY STREAMLINES. 48
10.3 PARAMETERS OF THE SYSTEM. 49
10.4 ANALYSIS OF THE PICTURES. 49
10.5 CONCLUSION. 51
11 OSCILLATIONS OF A GAS FILLED ELASTIC SHELL AND
REFLECTION OF SOUND IN A LAYER OF FLUID. 53
11.1 INTRODUCTION. 53
11.2 DIFFERENCES IN THE THEORETICAL DESCRIPTION OF
SCATTERING BY HOMOGENEOUS CYLINDERS AND SHELLS. 53
11.3 PARAMETERS OF THE SYSTEM. 54
11.4 ANALYSIS. 55
11.5 CONCLUSIONS. 58
12 THE ACOUSTIC FIELD IN A WAVEGUIDE NEAR AN
OSCILLATING ELASTIC SHELL. 61
12.1 INTRODUCTION. 61
12.2 ACOUSTIC FIELD STRUCTURE IN THE LOW FREQUENCY
RANGE. 61
12.2.1 Acoustic Field in One-Mode Regime. 62 12.2.2 Acoustic Field in Multi-Mode Regime. 70 12.3 ACOUSTIC FIELD IN THE MIDDLE FREQUENCY RANGE. 76
12.3.1 Comparison of Resonant and Non-Resonant
Fields 76
12.3.2 Acoustic Field in the Case of the Shell Located
near the Waveguide Bottom. 100
12.4 CONCLUSION. 106
13 TARGET STRENGTH AND ITS APPLICATION TO
WAVEGUIDE SYSTEMS. 107
14 ACCURACY OF THE MULTI-MODAL INTEGRAL METHOD. 108
14.1 INTRODUCTION. 108
14.2 MATRIX OF THE SYSTEM OF LINEAR EQUATIONS (THE
SCATTERING MATRIX). 108
14.2.1 Truncation of the Scattering Matrix and the
Green’s function. 109
14.2.2 Determining of the Matrix Dimension in Numerical
Experiments. 109
14.2.3 Examples of the Scattering Matrix. 109 14.2.4 Balance of Power Flows in the Waveguide. 123
14.3 CONCLUSIONS. 128
15 SUMMARY. 129
16 APPENDIX A. DERIVATION OF THE MODIFIED GREEN’S
FUNCTION. 132
16.1 COMMON FORM OF GREEN’S FUNCTION 132 16.2 IMPROVING CONVERGENCE OF THE GREEN’S FUNCTION 132
Alexei Zinoviev The Department of Mechanical Engineering, The University of Adelaide. 4 of 158 16.3 IMPROVING CONVERGENCE OF THE GREEN’S FUNCTION
DERIVATIVE. 134
17 APPENDIX B. CALCULATION OF THE SINGULAR INTEGRAL.138 17.1 SINGULARITY IN THE GREEN’S FUNCTION. 138 17.2 SINGULARITY IN THE DERIVATIVE OF THE GREEN’S FUNCTION. 142 18 APPENDIX C. MODELLING OF SCATTERING BY A GAS-
FILLED SHELL. 146
18.1 THE IMPEDANCES OF AN ELASTIC GAS-FILLED SHELL. 146 18.2 DEFINITION OF THE REFLECTION COEFFICIENT. 150 18.3 VELOCITY AMPLITUDE ON THE SHELL BOUNDARY. 151
19 BIBLIOGRAPHY. 153
20 PUBLICATIONS ORIGINATING FROM THE THESIS. 157
20.1 MAIN PUBLICATIONS. 157
20.1.1 Refereed Journal Publication. 157
20.1.2 Conference Proceedings. 157
20.2 OTHER PUBLICATIONS. 157
20.2.1 Refereed Journal Publications. 157
20.2.2 Conference Proceedings. 158
2 Abstract.
The current work is devoted to the problem of sound wave scattering by elastic cylindrical objects in a plain acoustic waveguide. The Multi- Modal Integral Method (MMIM) is proposed, which is based on non- standard representation of the Green’s function. It combines advantages of integral equation and eigenfunction methods and provides a quickly converging and highly accurate solution, taking into account all the waveguide modes up to infinite order. As illustrations of application of this method, acoustic diffraction is calculated from a system of several parallel homogeneous cylinders and from an air-filled cylindrical elastic shell.
Numerical solutions are found for various versions of the system of elastic cylinders in a fluid layer with perfectly soft and rigid boundaries. Phase - frequency and amplitude - frequency characteristics are found for modal coefficients of the scattered field.
Sharp increase of their amplitudes is found near resonance frequencies of the waveguide and the scattering cylinders. Pictures of the source density on the surface of the cylinders show that the nature of their distribution strongly depends on the frequency and the mutual location of the cylinders in the waveguide. Field structure near the cylinders reveals that higher-order waveguide modes play a significant role in the scattering process.
Spatial distribution of the acoustic power flow near the scattering object is calculated for several frequencies and two sets of elastic properties of the cylinder. It is shown that at the critical frequencies of the waveguide as well as at the internal resonances of the cylinder the acoustic energy flows in closed paths in some regions of the waveguide. Near the internal resonances of the cylinder the closed paths are located in the near vicinity of the scattering object and partially go through its interior. It is suggested that re-radiation of the energy stored in the vortices may contribute to the echo phenomenon.
The integral reflection coefficient is calculated for a system waveguide/shell for different values of wall thickness and distance between the shell and the waveguide bottom. Maxima and minima in the reflection coefficient associated with cut-on frequencies of the waveguide modes and structural resonances of the shell are identified. The calculations show that the conventional definition of target strength in a shallow waveguide is inappropriate.
Different kinds of resonances are identified in frequency and angular dependencies of the velocity amplitude of the shell surface. These resonances belong to the following groups: a) critical frequencies of the waveguide modes, b) structural resonances of the elastic shell, c) resonance oscillations of the gas-filled interior of the shell, d) resonance oscillations of the coupled fluid-shell.
Alexei Zinoviev The Department of Mechanical Engineering, The University of Adelaide. 6 of 158 Temporal sequences of pictures showing the spatial structure of the total and scattered fields in the near and far field zones are obtained.
It is shown that the incident field produces waves of acoustic pressure propagating along the boundary of the scattering object, which, in turn, generate the scattered acoustic field. In the process of propagation, the waves may interact with each other via the fluid or the scattering object. This leads to significant changes of the structure of the acoustic field and of the amount of acoustic energy reflected from the scatterer. It is also shown that, in most cases, standing waves exist between the scatterer and the waveguide boundaries.
Accuracy of the Multi-Modal Integral Method is discussed. It is shown that the implementation of the method requires few computer resources for good accuracy of the solution.
3 Statement of Originality.
This thesis contains no material, which has been accepted for the award of any other degree or diploma in any university. To the best of my knowledge and belief, this thesis contains no material previously published or written by another person, except where due reference has been made in the text.
I give consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying.
Alexei Zinoviev 1 October 1999
Alexei Zinoviev The Department of Mechanical Engineering, The University of Adelaide. 8 of 158
4 Acknowledgments.
The author wishes to express his gratitude to his supervisors, Prof.
Colin H. Hansen and Dr. David A. Bies. Special appreciation goes to Prof. Hansen for providing resources and financial support, which made this work possible. The author is most grateful to Dr. Bies for his guidance and encouragement throughout the duration of this research.
Sincere thanks to Ms Brenda Hope and Ms Tonia Camporeale as well as other Department staff for their invaluable help in organising smooth research process.
Last but not least, the author thanks members of his family for their patience and understanding.
5 Notations.
5.1 Latin Letters.
A
amν amplitude of the m-th cylindrical mode of the incident field with respect to the cylindrical coordinate system with the origin in the centre of ν-th cylinder
An amplitude of the n-th normal mode of the incident field
Aνm amplitude of the m-th mode in the representation of the scalar elastic potential in the ν-th cylinder
B
bpν p-th Fourier coefficient in the representation of the function µν(ϕν) Bνm amplitude of the m-th mode in the representation of the z-component of
the vector elastic potential in the ν-th cylinder
C
c sound speed in fluid cs Stoneleywave speed cr Rayleigh wave speed cf Franz wave speed
cls speed of symmetric Lamb waves cla speed of asymmetric Lamb waves cwg Whispering Gallery wave speed cl longitudinal elastic wave speed ct transversal elastic wave speed
clν longitudinal elastic wave speed in ν-th cylinder ctν transversal elastic wave speed in ν-th cylinder cg sound speed in the gas-filled interior of the shell
D
D depth of the waveguide
E
E unity matrix
F
F(ν) z-component of the vector potential of displacement in ν-th cylinder
G
gn longitudinal wavenumber of the waveguide mode of order n Gν Green’s function associated with the ν-th cylinder
H
H the distance between the centre of a scattering object and the bottom of the waveguide
Alexei Zinoviev The Department of Mechanical Engineering, The University of Adelaide. 10 of 158
I
i imaginary unit (square root of –1) I acoustic intensity vector
∆I discrepancy in power flows on both sides of scattering objects
J
Jm Bessel’s function of the first kind of order m
K
k non-dimensional acoustic wavenumber
kg non-dimensional acoustic wavenumber in the gas filled shell
kA non-dimensional wavenumber (frequency) of the monopole resonance A
kB non-dimensional wavenumber (frequency) of the dipole resonance B kn non-dimensional critical wavenumber of the waveguide mode of order
n
kt non-dimensional wavenumber of transversal elastic waves
ktν non-dimensional wavenumber of transversal elastic waves in ν-th cylinder
kl non-dimensional wavenumber of longitudinal elastic waves
klν non-dimensional wavenumber of longitudinal elastic waves in ν-th cylinder
k dimensional acoustic wavenumber
Kνl acoustic field radiated by a point source on the surface of the l-th cylinder and observed on the surface of the ν-th cylinder
n
Kνl n-th term in the series Kνl n
Kνl approximation of the n-th term in the series Kνl
Kνl function describing the contribution of the evanescent modes to the Green’s function
L
Lν external boundary line of the ν-th cylinder
Lνl radial derivative of the acoustic field of a point source on the surface of the l-th cylinder observed on the surface of ν-th cylinder
n
Lνl n-th term in the series Lνl n
Lνl approximation of the n-th term in the series Lνl
Lνl function describing the contribution of the evanescent modes to the derivative of the Green’s function
M
ν l
Mpm element of the matrix of the system of linear equations Mˆ scattering matrix
N
n order of current waveguide mode
N the number of cylinders in the waveguide
ν l
Npm element of the auxiliary matrix for calculating coefficients Bνm
Nint number of integration points on the interval
P
P total pressure in the acoustic wave in fluid Pe pressure in the incident (external) wave
P- pressure in the reflected (scattered back to the source) wave P+ pressure in the transmitted (scattered forward) wave
Psν pressure in the wave scattered by the ν-th cylinder Pg pressure in the gas-filled interior of the shell
Pn n-th eigenfunction of the boundary value problem in the free waveguide Pn n-th eigenfunction of the conjugate boundary value problem in the free
waveguide
ν
Ppm coefficients describing the right-hand part of the system of linear equations
Pm m-th cylindrical component of the acoustic pressure in the exterior fluid
g
Pm m-th cylindrical component of the acoustic pressure in the interior fluid
R
rν radial polar coordinate with the origin in the centre of the ν-th cylinder Rν radius of the ν-th cylinder
R radius of a cylindrical scattering object R1 outer radius of a cylindrical shell R2 inner radius of a cylindrical shell
∆rν displacement vector in the ν-th cylinder
S
S− reflected wave power flow through the waveguide cross-section S+ transmitted wave power flow through the waveguide cross-section Se incident wave power flow through the waveguide cross-section
T
t time
TS target strength
V
V particle velocity amplitude
X
x vertical Cartesian coordinate
xν x-coordinate of a point on the surface of the ν-th cylinder
xl0 x-coordinate of an integration point on the surface of the l-th cylinder Xν x-coordinate of the centre of the ν-th cylinder
Y
y horizontal Cartesian coordinate
Alexei Zinoviev The Department of Mechanical Engineering, The University of Adelaide. 12 of 158 yν y-coordinate of a point on the surface of the ν-th cylinder
yl0 y-coordinate of an integration point on the surface of the l-th cylinder Yν y-coordinate of the centre of the ν-th cylinder
Y0 y-coordinate of the centre of a scattering object
Z
z Cartesian coordinate normal to the (x,y) plane
ν
z1m dummy variable
ν
z2m dummy variable
zνm acoustical impedance of the m-th mode on the surface of the ν-th cylinder
5.2 Greek Letters.
α , Α
α0, α1 coefficients describing boundary conditions on the waveguide walls
β , Β
β0, β1 coefficients describing boundary conditions on the waveguide walls
δ , ∆
∆ Laplace operator δ delta-function δpm Kronecker’s symbol
ε , Ε
ε a small vicinity of a point of singularity
φ , Φ
Φ(ν) scalar potential of the displacement in the ν-th cylinder Φnm, auxiliary functions
γ , Γ
γ
n coefficient in the eigenfunctions of the conjugate boundary value problemη , Η
ηn transversal wavenumber of the waveguide mode of order n
ϕ , ϑ
ϕ angle polar coordinate of the observation point ϕ0 angle polar coordinate of the source point
κ , Κ
ν l
Κpm component of the scattering matrix, determining the acoustic pressure on the surface of the ν-th cylinder
λ , Λ
λν Lame coefficient of the ν-th cylinder
λ a) acoustic wavelength; b) Lame coefficient of the shell
ν l
Λpm component of the scattering matrix, determining the radial derivative of the acoustic pressure on the surface of the ν-th cylinder
µ , Μ
µν Lame coefficient of the ν-th cylinder
ν0
µ source field distribution on the surface of the ν-th cylinder
µν µν = Rνµν0, source field distribution on the surface of the ν-th cylinder related by the radius of the ν-th cylinder
µ Lame coefficient of the shell
ν , Ν
ν a) sequential number of a cylinder; b) complex mode order
π , Π
π
3.1415926…θ , Θ
θn phase of the waveguide mode of order n
ρ , Ρ
ρ density of the fluid
ρν density of the ν-th cylinder ρg density of the gas
ρs density of the shell
ω , Ω
ω angular frequency of the acoustic wave
ξ , Ξ
ξ0, ξ1 coefficients describing boundary conditions on the waveguide walls, related to α0,1 and β0,1 as ξ0,1=α0,1/β0,1
ψ , Ψ
Ψn coefficient in the representation of the incident field as a sum of cylindrical modes