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.

中文翻译：

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可穿透楔形衍射的高对比度近似

在高对比度范围内考虑了一个重要的开放规范问题，即由可穿透的楔形波衍射。从数学上讲，这意味着对比度参数（主体和楔形散射体的特定材料属性的比率）被假定为较小。相关的材料属性取决于物理环境，例如对于声波和电磁波而言是不同的。基于此假设，构造了一个新的渐近迭代方案。针对（可能不均匀）不可渗透的楔形问题的许多解决方案来写可渗透楔形的解决方案。使用Sommerfeld–Malyuzhinets和Wiener–Hopf技术的组合可以解决每个难以解决的问题。所得的可穿透楔形的近似解涉及大量嵌套的复杂积分，因此难以进行数值评估。为了解决这个问题，开发并实施了一种微妙的方法（结合渐近，内插和复杂分析），从而实现了快速有效的数值评估。该渐近方案显示具有出色的收敛性，并导致对现有方法的明显改进。