The goal of this paper is to investigate the characteristics of high-frequency whispering-gallery modes in embedded elastic spheres, that is, surrounded by an infinite elastic matrix. Due to several modeling difficulties, the high-frequency regime of embedded spheres has remained unexplored in elasticity. Our approach consists of formulating a specific finite-element method in spherical coordinates. The basic idea is to discretize only the radial coordinate while describing analytically the angular distribution of elastodynamic fields. Then, we also introduce a radial perfectly matched layer to cope with the unbounded nature of the external medium. Our approach yields a linear matrix eigensystem, simple and costless to solve. In order to identify general trends, both stiff and soft configurations are considered, corresponding to a sphere stiffer and softer than the external medium, respectively. Including material loss, our results highlight the behavior of leaky elastic whispering-gallery modes in the high-frequency regime. This work is motivated by the well-known behavior of whispering-gallery modes in optical resonators, reaching high $Q$ factors as the frequency increases. Identifying high-$Q$-factor whispering-gallery modes in elastic spheres could find promising applications for sensing the mechanical properties of external media.

中文翻译：

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嵌入弹性球体中的高频泄漏回音壁模式

本文的目的是研究嵌入弹性球体中高频回音壁模式的特性，即被无限弹性矩阵包围。由于几个建模困难，嵌入球体的高频区域在弹性方面仍未得到探索。我们的方法包括在球坐标系中制定特定的有限元方法。基本思想是在分析描述弹力场的角度分布时仅离散径向坐标。然后，我们还引入了一个径向完美匹配层来应对外部介质的无界性质。我们的方法产生了一个线性矩阵特征系统，求解简单且成本低。为了确定总体趋势，同时考虑刚性和柔软配置，分别对应于比外部介质更硬和更软的球体。包括材料损失在内，我们的结果突出了高频区域中泄漏弹性回音壁模式的行为。这项工作的动机是众所周知的光谐振器中回音壁模式的行为，达到高$问$频率增加的因素。识别高$问$弹性球体中的因子回音壁模式可以在传感外部介质的机械性能方面找到有前景的应用。