线性非自伴算子下的散射：平面弹性波的情形
Scattering under Linear Non Self-Adjoint Operators: Case of in-Plane Elastic Waves
摘要：
本文研究了均匀介质与异向介质交界面上平面波的散射问题。在这两个区域的相关本征模计算通过解决最近描述的非自伴本征值问题，特别适合于散射研究。该方法有效地产生所有的传播和消逝模式符合应用的斯涅耳定律，是适用于非常一般的散射问题。在一个模型复合，我们阐明了出现了丰富的频谱特征值退化。这些简并性出现在波矢的复域和实域中。然而，由于这个问题是非自伴的，这些退化通常表示特征值和特征向量（例外点）的合并。通过对坡印廷矢量的显式计算，我们指出了一个有趣的现象：在一个例外点的两侧，波的折射角的符号总是出现突变。此外，这些简并的存在，在某些情况下，暗示在散射场的快速变化的入射角是由少量的变化。我们计算这些散射场通过一个新的应用贝蒂-瑞利互易定理。我们提出了几个数值例子，显示了丰富的散射光谱。在一个特别有趣的例子中，我们指出波的行为可能与共振捕获现象有关。我们还表明，有一个深刻的联系之间的能量通量守恒和非自伴问题的双正交关系。证明适用于一般类的散射问题，涉及弹性波（下自伴或非自伴运营商）。
英文摘要：
In this paper, we consider the problem of the scattering of in-plane waves at an interface between a homogeneous medium and a metamaterial. The relevant eigenmodes in the two regions are calculated by solving a recently described non self-adjoint eigenvalue problem particularly suited to scattering studies. The method efficiently produces all propagating and evanescent modes consistent with the application of Snell's law and is applicable to very general scattering problems. In a model composite, we elucidate the emergence of a rich spectrum of eigenvalue degeneracies. These degeneracies appear in both the complex and real domains of the wave-vector. However, since this problem is non self-adjoint, these degeneracies generally represent a coalescing of both the eigenvalues and eigenvectors (exceptional points). Through explicit calculations of Poynting vector, we point out an intriguing phenomenon: there always appears to be an abrupt change in the sign of the refraction angle of the wave on two sides of an exceptional point. Furthermore, the presence of these degeneracies, in some cases, hints at fast changes in the scattered field as the incident angle is changed by small amounts. We calculate these scattered fields through a novel application of the Betti-Rayleigh reciprocity theorem. We present several numerical examples showing a rich scattering spectrum. In one particularly intriguing example, we point out wave behavior which may be related to the phenomenon of resonance trapping. We also show that there exists a deep connection between energy flux conservation and the biorthogonality relationship of the non self-adjoint problem. The proof applies to the general class of scattering problems involving elastic waves (under self-adjoint or non self-adjoint operators).
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