Abstract Acoustic exterior problems are in this paper solved numerically using the Astley-Leis infinite element method (IFEM). Normal modes can be determined thanks to the frequency independence of the system matrices. Convergence properties of the harmonic sound pressure solution as well as of normal mode eigenvalues are investigated for two-dimensional elliptical computational domains to estimate the essential requirements of accuracy. A relationship between the half-axis ratio of the ellipses and the eigenvalues is identified. By solving half-space problems, symmetry of the computational domains is utilized, which is shown for the first time for frequency-independent normal modes in exterior acoustics. This paper discusses applicability of the modes in the example of sonic crystals—periodic arrays of scatterers, in this context denoted as acoustic meta-atoms—that have recently attracted attention for their possible use as noise barriers. It can be shown that the sound-insulating effect of finite sonic crystals and individual meta-atoms in the free field can be related to normal modes in exterior acoustics. With the help of this approach, the absorption by Helmholtz resonators and due to a boundary admittance inside are studied. This work provides a new point of view and physical insights into the effects and underlying physics of sound insulation by finite sonic crystals and acoustic meta-atoms in free field. Although it is not the intention of this article to optimize the arrays, the method of normal modes in exterior acoustics is presented as an appropriate and novel tool for their dimensioning and design.

中文翻译：

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具有无限元和法向模式的自由场中有限声波晶体的散射分析

摘要 本文采用Astley-Leis 无限元法(IFEM) 数值求解声学外部问题。由于系统矩阵的频率无关性，可以确定正常模式。研究了二维椭圆计算域的谐波声压解以及正常模式特征值的收敛特性，以估计精度的基本要求。确定椭圆的半轴比和特征值之间的关系。通过解决半空间问题，利用了计算域的对称性，这是首次在外部声学中与频率无关的正常模式中显示出来。本文以声波晶体——散射体的周期性阵列为例，讨论了这些模式的适用性，在这种情况下，称为声学元原子——最近因其可能用作隔音屏障而引起了人们的注意。可以证明，自由场中有限声波晶体和单个元原子的隔音效果可能与外部声学中的正常模式有关。在这种方法的帮助下，研究了亥姆霍兹谐振器的吸收和由于内部边界导纳。这项工作为自由场中有限声波晶体和声学元原子对隔音的影响和基础物理学提供了新的观点和物理见解。尽管本文的目的不是优化阵列，但将外部声学中的法向模式方法作为其尺寸和设计的合适且新颖的工具提出。