阶梯状表面波浪辐射的数学分析
Mathematical analysis of wave radiation by a step-like surface
摘要：
本文首次提出了波在具有理想导电阶梯表面的整体扰动半平面中传播的精确索末菲辐射条件（SRC），给出了远场方向图的解析公式，并给出了高精度数值解。我们采用Wiener-Hopf方法来计算裂纹半平面的格林函数，这是扰动半平面的背景。我们严格证明了格林函数渐近满足一个普适方向SRC（uSRC），并在无穷远处辐射纯出射。这有助于提出一个隐式的透明边界条件的散射波，无论是一个圆柱形的入射波由于线源或平面入射波。然后，通过相关的变分公式建立适定性理论。该理论表明，散射波，后减去一个已知的波场，满足相同的uSRC，使其远场图案是理论上可访问的。对于一个平面波入射，渐近分析表明，仅仅减去反射平面波，由于非均匀高度的阶梯状表面在无穷远，从散射波在各自的区域产生一个不连续的波满足uSRC以及。数值上，我们采用一个先前开发的完全匹配层（PML）边界积分方程方法来解决这个问题。数值结果表明，PML截断误差随PML厚度或吸收功率的增加呈指数衰减，其收敛性依赖于PML中的格林函数的指数衰减.
英文摘要：
This paper proposes, for wave propagating in a globally perturbed half plane with a perfectly conducting step-like surface, a sharp Sommerfeld radiation condition (SRC) for the first time, an analytic formula of the far-field pattern, and a high-accuracy numerical solver. We adopt the Wiener-Hopf method to compute the Green function for a cracked half plane, a background for the perturbed half plane. We rigorously show that the Green function asymptotically satisfies a universal-direction SRC (uSRC) and radiates purely outgoing at infinity. This helps to propose an implicit transparent boundary condition for the scattered wave, by either a cylindrical incident wave due to a line source or a plane incident wave. Then, a well-posedness theory is established via an associated variational formulation. The theory reveals that the scattered wave, post-subtracting a known wave field, satisfies the same uSRC so that its far-field pattern is accessible theoretically. For a plane-wave incidence, asymptotic analysis shows that merely subtracting reflected plane waves, due to non-uniform heights of the step-like surface at infinity, from the scattered wave in respective regions produces a discontinuous wave satisfying the uSRC as well. Numerically, we adopt a previously developed perfectly-matched-layer (PML) boundary-integral-equation method to solve the problem. Numerical results demonstrate that the PML truncation error decays exponentially fast as thickness or absorbing power of the PML increases, of which the convergence relies heavily on the Green function exponentially decaying in the PML.
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