The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory

Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283 and Potthast, R. (2002) The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 458 (2028). pp. 2967-3001. ISSN 1471-2946 doi: 10.1098/rspa.2002.0999

Abstract/Summary

We investigate Fréchet differentiability of the scattered field with respect to variation in the boundary in the case of time–harmonic acoustic scattering by an unbounded, sound–soft, one–dimensional rough surface. We rigorously prove the differentiability of the scattered field and derive a characterization of the Fréchet derivative as the solution to a Dirichlet boundary value problem. As an application of these results we give rigorous error estimates for first–order perturbation theory, justifying small perturbation methods that have a long history in the engineering literature. As an application of our rigorous estimates we show that a plane acoustic wave incident on a sound–soft rough surface can produce an unbounded scattered field.