High-contrast approximation for penetrable wedge diffraction

Abstract

The important open canonical problem of wave diffraction by a penetrable wedge is considered in the high-contrast limit. Mathematically, this means that the contrast parameter, the ratio of a specific material property of the host and the wedge scatterer, is assumed small. The relevant material property depends on the physical context and is different for acoustic and electromagnetic waves for example. Based on this assumption, a new asymptotic iterative scheme is constructed. The solution to the penetrable wedge is written in terms of infinitely many solutions to (possibly inhomogeneous) impenetrable wedge problems. Each impenetrable problem is solved using a combination of the Sommerfeld–Malyuzhinets and Wiener–Hopf techniques. The resulting approximated solution to the penetrable wedge involves a large number of nested complex integrals and is hence difficult to evaluate numerically. In order to address this issue, a subtle method (combining asymptotics, interpolation and complex analysis) is developed and implemented, leading to a fast and efficient numerical evaluation. This asymptotic scheme is shown to have excellent convergent properties and leads to a clear improvement on extant approaches.

Diffraction by a Right-Angled No-Contrast Penetrable Wedge Revisited: A Double Wiener-Hopf Approach

“…Let the domains PW and R 2 \ PW's material properties be described by the densities ρ 1,2 in the acoustic setting, and by the magnetic permeabilities µ 1,2 or electric permittivities 1,2 in the electromagnetic setting (E-polarised and H-polarised field respectively, c.f. [28], [19], or [25]). Crucial to the present work (c.f.…”

“…Equations (2.10) and (2.11) are the sought edge conditions. It should be noted that this particular expression is only valid since we have chosen λ = 1, see [25] for the general case.…”

“…Due to the wavenumbers' positive imaginary part > 0, this is automatically satisfied and by the limiting absorption principle, the radiation condition also holds in the limit → 0. See [25] for more information on the radiation condition for penetrable wedges.…”

“…Other innovative approaches suitable for high contrast penetrable wedge problems were given by Lyalinov [21] and, more recently, by Nethercote et al [25]. In [21] Lyalinov uses the Sommerfeld-Malyuzhinets technique to obtain a system of two coupled Malyuzhinets equations which were solved approximately, giving the leading order far-field behaviour, whereas in [25] Nethercote et al combine the Wiener-Hopf and Sommerfeld-Malyuzhinets method. In [25] a solution to the penetrable wedge is given as an infinite series of impenetrable wedge problems.…”

“…In [21] Lyalinov uses the Sommerfeld-Malyuzhinets technique to obtain a system of two coupled Malyuzhinets equations which were solved approximately, giving the leading order far-field behaviour, whereas in [25] Nethercote et al combine the Wiener-Hopf and Sommerfeld-Malyuzhinets method. In [25] a solution to the penetrable wedge is given as an infinite series of impenetrable wedge problems. Each of these impenetrable wedge problems is solved exactly, and the resulting infinite series for the penetrable wedge can be evaluated rapidly and efficiently using asymptotic and numerical methods.…”