Macquarie University
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Potential and acoustic diffraction problems for arbitrarily shaped rotationally symmetric doubly-connected screens

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posted on 2022-12-14, 00:05 authored by Martin Sagradian

This thesis develops a rigorous mathematical approach to the analysis of acoustic wave scattering and electrostatic problems for three-dimensional structures of revolution. An object can be formed by rotation of an arbitrarily shaped generating curve and has coaxial apertures on both ends which topologically corresponds to a doubly-connected surface. Such flexibility allows to consider various conductors and scatterers that are widely used in practice. Additionally, objects complementary to doubly-connected structures (an example of which would be a parallel disc capacitor) are analysed.

Analysis of open structures by purely numerical techniques results in Fredholm integral equation of the first kind which is ill-posed in its numerical implementation and may result in a solution that does not correspond to a physical situation. Due to the presence of the edges, the accuracy of such a solution can be difficult to improve. The Method of Analytical Regularisation (MAR), a semi-analytical semi-numerical technique, is used in this thesis to transform the initial problem to an infinite system of linear algebraic equations of the second kind. It has been previously used to solve electrostatic and wave scattering problems for canonical structures and cavities with one aperture. The method is based on the theory of triple series equations with the Jacobi polynomials and the Abel integral transform. Such a system can be solved with a predefined accuracy via a truncation technique. For the numerical investigation, object-oriented software that supports a wide range of structures and different problem parameters has been built. 

The shape variability of the developed approach stimulated a study of various practical problems. Calculated capacitance and potential distribution for widely used conductors such as spheroidal barrels, finite open cylinders, truncated cones and parallel disk capacitors demonstrate excellent agreement with the known results from the literature. The research is also conducted on modifications of these structures that shows how electrostatics characteristics change with the shape modification. 

Finally, acoustic wave scattering from soft objects is investigated. The MAR approach allows to obtain the full spectral portrait of the scatterer, that is, to calculate the complex eigenvalues of the cavity. Scattering characteristics, such as sonar and bistatic cross sections, and field distributions are calculated and analysed for a wide range of objects.  


Table of Contents

1 Introduction -- 2 Geometry of the considered structures -- 3 The Dirichlet mixed boundary value problem for the Helmholtz equation. Method of Analytical Regularisation -- 4 Advanced theory of the Laplace equation and electrostatic applications -- 5 Spectral studies: complex oscillations in open rotationally symmetric cavities -- 6 Resonance scattering of a plane wave from cavities with two coaxial apertures -- 7 Conclusions -- A Special functions used -- B Remainder calculation -- C Series convergence improvement -- Bibliography

Awarding Institution

Macquarie University


A thesis submitted to Macquarie University for the degree of Doctor of Philosophy

Department, Centre or School

Department of Mathematics and Statistics

Year of Award


Principal Supervisor

Elena Vynogradova

Additional Supervisor 1

Stuart Hawkins

Additional Supervisor 2

Paul Smith


Copyright: The Author Copyright disclaimer:




165 pages