Consider the elastic scattering of a plane or point incident wave by an unbounded and rigid rough surface. The angular spectrum representation (ASR) for the time-harmonic Navier equation is derived in three dimensions. The ASR is utilized as a radiation condition to the elastic rough surface scattering problem. The uniqueness is proved through a Rellich-type identity for surfaces given by uniformly Lipschitz functions. In the case of flat surfaces with a local perturbation, we deduce an equivalent variational formulation in a truncated bounded domain and show the existence results for general incoming waves. The main ingredient of the proof is the radiating behavior of the Green tensor to the first boundary value problem of the Navier equation in a half space.

中文翻译：

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三维粗糙表面的弹性散射

考虑平面或点入射波被无界且刚性粗糙表面的弹性散射。时谐纳维方程的角谱表示法 (ASR) 是从三个维度推导出来的。ASR 被用作弹性粗糙表面散射问题的辐射条件。唯一性是通过一致 Lipschitz 函数给出的表面的 Rellich 型恒等式来证明的。在具有局部扰动的平坦表面的情况下，我们在截断有界域中推导出等效变分公式，并显示一般入射波的存在结果。证明的主要成分是绿色张量对半空间中纳维方程的第一边值问题的辐射行为。