Embedding formulae for wave diffraction by a circular arc

C.A.J. Moran,N.R.T. Biggs,P.G. Chamberlain

doi:10.1016/j.wavemoti.2016.07.003

Abstract

For certain wave diffraction problems, embedding formulae can be derived, which represent the solution (or far-field behaviour of the solution) for all plane wave incident angles in terms of solutions of a (typically small) set of other auxiliary problems. Thus a complete characterisation of the scattering properties of an obstacle can be determined by only determining the solutions of the auxiliary problems, and then implementing the embedding formula. The class of scatterers for which embedding formulae can be derived has previously been limited to obstacles with piecewise linear boundaries; here this class is extended to include a simple curved obstacle, consisting of a thin circular arc. Approximate numerical calculations demonstrate the accuracy of the new embedding formulae.

Highlights

To fully characterise the wave scattering properties of an obstacle, solutions may be required for a range of plane wave incident angles
Following [2], subsequent extensions [3,4,5,6] required the boundary-value problem to be formulated as an integral equation; the derivation of the embedding formulae exploited the structure of the integral equation, and expressed the solution for arbitrary plane wave incident angle in terms of solutions corresponding to other plane wave incident angles
Derivation of embedding formulae initially follows one of two distinct but related routes: either a particular differential operator is used, which commutes with the Helmholtz operator, boundary conditions, and radiation condition, and annihilates the incident wave; or else the boundary-value problem is reformulated as an integral equation, and its structure is exploited

Summary

Introduction

To fully characterise the wave scattering properties of an obstacle, solutions may be required for a range of plane wave incident angles. Following [2], subsequent extensions [3,4,5,6] required the boundary-value problem to be formulated as an integral equation; the derivation of the embedding formulae exploited the structure of the integral equation, and expressed the solution for arbitrary plane wave incident angle in terms of solutions corresponding to other plane wave incident angles This approach was generalised in [7] in which a generalised integral equation problem, divorced from a particular wave diffraction interpretation, was addressed. The papers [8,9] instead derived embedding formulae directly from the boundaryvalue problem, without recourse to an integral equation formulation, and expressed the far-field of the solution for arbitrary plane wave incident angle in terms of the far-field of solutions corresponding to particular multipole forcing at the corners of the scatterers.

The boundary-value problem

Embedding formulae

From the boundary-value problem

Embedding formulae for the modal problem

From an integral equation formulation

Embedding formulae for the integral equation

Numerical implementation

Numerical approximation

Results

Conclusions and future directions