Acoustic Scattering in Prolate Spheroidal Geometry via Vekua Tranformation -- Theory and Numerical Results
L.N. Gergidis, D. Kourounis, S. Mavratzas and A. Charalambopoulos

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 21, No. 2, pp. 157-176, 2007
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Keywords Prolate Spheroid; Acoustic Scattering; Vekua Transformation; Arbitrary Precision;$L^2$-norm Minimization; Collocation; Mathematical Modeling; Special Functions; Scientific Computing.
Abstract A new complete set of scattering eigensolutions of Helmholtz equation in spheroidal geometry is constructed in this paper. It is based on the extension to exterior boundary value problems of the well known Vekua transformation pair, which connects the kernels of Laplace and Helmholtz operators. The derivation of this set is purely analytic. It avoids the implication of the spheroidal wave functions along with their accompanying numerical deficiencies. Using this novel set of eigensolutions, we solve the acoustic scattering problem from a soft acoustic spheroidal scatterer, by expanding the scattered field in terms of it. Two approaches concerning the determination of the expansion coefficients are extensively studied in terms of their numerical and convergence properties. The first one minimizes the$L^2$-norm of a suitably constructed error function and the second one relies on collocation techniques. The robustness of these approaches is established via the adoption of arbitrary precision arithmetic.
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