Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 785–810

EXISTENCE, UNIQUENESS AND EXPLICIT BOUNDS FOR ACOUSTIC SCATTERING BY AN INFINITE LIPSCHITZ BOUNDARY WITH AN IMPEDANCE CONDITION

Thomas Baden-Riess

thomas_baden_riess 'at' yahoo.co.uk

Abstract. We study a boundary value problem for the Helmholtz equation with an impedance boundary condition, in two and three dimensions, modelling the scattering of time harmonic acoustic waves by an unbounded rough surface. Via analysis of an equivalent variational formulation we prove this problem to be well-posed when: i) the boundary has the strong local Lipschitz property and the frequency is small; ii) the rough surface is the graph of a bounded Lipschitz function (with arbitrary frequency). An attractive feature of our results is that the bounds we derive, on the inf-sup constants of the sesquilinear forms, are explicit in terms of the wavenumber k, the geometry of the scatterer and the parameters describing the surface impedance.

2010 Mathematics Subject Classification: Primary 78A45; Secondary 65J05, 45B05, 28A80.

Key words: Helmholtz equation, reduced wave equation, rough surface scattering, Rellich identity.

Reference to this article: T. Baden-Riess: Existence, uniqueness and explicit bounds for acoustic scattering by an infinite Lipschitz boundary with an impedance condition. Ann. Acad. Sci. Fenn. Math. 45 (2020), 785–810.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4540