Integral equation methods for scattering by infinite rough surfaces
Integral equation methods for scattering by infinite rough surfaces
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DOI：
10.1002/mma.361
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发表时间：
2003-04
影响因子：
2.9
通讯作者：
Bo Zhang;S. Chandler-Wilde
中科院分区：
文献类型：
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作者：
Bo Zhang;S. Chandler-Wilde
本文考虑了非局部扰动半平面中Helmholtz方程的Dirichlet边值问题和阻抗边值问题。这些边值问题出现在研究入射场的时谐声散射时，入射场被一个全场为零的无限大的粗糙表面（狄利克雷问题）或被一个全场满足齐次阻抗条件的无限大的阻抗粗糙表面（阻抗问题）散射。我们提出了一个新的边界积分方程的Dirichlet问题，利用组合的双层和单层的潜力和Dirichlet半平面绿色的功能。对于阻抗问题，我们提出了两个边界积分方程公式，都使用半平面阻抗绿色函数，第一个来自绿色表示定理，第二个来自寻求作为单层势的解决方案。我们证明了所有的积分方程是唯一可解的空间中的有界和连续的函数的所有波数。作为一个重要的推论，我们证明了，对于各种入射场，包括入射平面波，散射场的阻抗边值问题有一个唯一的解决方案，在一定的约束条件下的边界阻抗。版权所有© 2003年约翰威利父子有限公司。
In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.