First kind BIEs and BEM for acoustic scattering

Simon Chandler-Wilde (University of Reading)

Abstract: We consider the so-called sound-soft problem in time-harmonic acoustic scattering, where the total field, a solution to the Helmholtz equation in the exterior of some obstacle, vanishes on the boundary of the obstacle.  Making an ansatz for the scattered field as a single-layer potential with some unknown density, a BIE for the density is obtained by applying this sound-soft boundary condition. This is an old formulation; it can be found, in the context of scattering by screens and apertures, already in 19th century texts [1]. In this talk we make a survey of previous results, and we explain in what sense this formulation applies when the obstacle is some arbitrary compact set. In the case when the scattering obstacle is a self-similar fractal of some fractal dimension d, we implement and establish convergence rates for a fully discrete Galerkin BEM where all integrals are with respect to d-dimensional Hausdorff measure [2,3].
[1] Lord Rayleigh, “Theory of Sound”, 2nd Ed., Vol. II, Macmillan, New York, 1896
[2] A. M. Caetano, S. N. Chandler-Wilde, A. Gibbs, D. P. Hewett and A. Moiola, A Hausdorff-measure boundary element method for acoustic scattering by fractal screens. Numer. Math., 156, 463-532 (2024).
[3] A. M. Caetano, S. N. Chandler-Wilde, X. Claeys, A. Gibbs, D. P. Hewett and A. Moiola, Integral equation methods for acoustic scattering by fractals. Proc. R. Soc. A.