Residue Enhanced Embedding Formulae

For certain scattering geometries, embedding formulae [1] can be used to efficiently compute the far-field pattern induced by a large range of incident angles. In principle, only a relatively small subset of far-field patterns, induced by canonical angles are required as inputs to each embedding formula. Despite the embedding formulae being exact in theory, in [2] it was shown that they are very sensitive to numerical errors in the canonical far-fields. This Matlab package is a numerically stable implementation of the embedding formulae of [1]. Numerical stability is achieved by adding residue contributions to the embedding formulae, hence the name Residue Enhanced Embedding Formulae.

The user must provide M, and p, which are defined as in [1], the wavenumber kwave, and M canonical far-field patterns induced by incident angles in a vector alphas. The far-field patterns must be defined as cell array of function handles:

val = D{m}[obs]

where m is the index of the far-field induced by alphas(m); obs and vals are Nx1 vectors of observation angles. Then the code

E = Reef(D,alphas,kwave,p);
Eout = E.getFarField(obs_test,inc_test);

will efficiently compute the cross-section for a large number of incident angles inc_test at the observation angles obs_test.

A full example is provided in EG1.m, which produces the following cross-section with wavenumber 15 on a unit square:

[1] Biggs, N.R.T., 2006. A new family of embedding formulae for diffraction by wedges and polygons. Wave Motion, 43(7), pp.517-528.
[2] Gibbs, A., Langdon, S. and Moiola, A., 2018. Numerically stable computation of embedding formulae for scattering by polygons. arXiv preprint arXiv:1805.08988.