Abstract
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.
Highlights
Waves diffraction by edges, wedges and ledges has been studied in numerous physical contexts by many distinguished scientists
Its rigorous mathematical treatment was established in the late 19th century thanks to Sommerfeld who solved the famous half-plane problem (Sommerfeld, 1896) and later conceived the first solution that models impenetrable wedge diffraction using the method of images (Sommerfeld, 1901)
Relying on previous papers on Rayleigh wave scattering by elastic wedges (Budaev & Bogy, 1995, 1996, 1998), they designed the problem symmetrically with an antisymmetric incident field so that they could formulate it on a half-space domain
Summary
Wedges and ledges has been studied in numerous physical contexts (acoustics, elasticity, electromagnetism) by many distinguished scientists. Relying on previous papers on Rayleigh wave scattering by elastic wedges (Budaev & Bogy, 1995, 1996, 1998), they designed the problem symmetrically with an antisymmetric incident field so that they could formulate it on a half-space domain They derived a system of difference equations that were solved via singular integral operators and a Neumann series using the assumptions that the host wedge region is very thin and the ratio of densities is close to unity. This is done by a subtle combination of interpolation, asymptotic analysis and analytic continuation via functional difference equations that leads to an efficient approximation of the host and scatterer wave fields for sufficiently high contrast. The appendices discuss the derivation of an important complex mapping that is used to link the wave fields in spectral space as well as some asymptotic analysis of the integral solutions
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