Analytical and numerical study of anti-plane elastic wave scattering in a structured quadrant subjected to a boundary point load

Abstract

We verify the recently developed method by Kisil (Kisil 2024 SIAM J. Appl. Math. 84 , 464–476 (doi: 10.1137/23M1562445 )) through its application in the study of the anti-plane dynamic response of an elastic square-cell lattice quadrant having a free lateral boundary that is subjected to a sinusoidal point load. The vertical boundary of the quadrant is assumed to be either fixed or free. The approach adopted utilizes the discrete Fourier transform in both principal lattice directions to reduce the governing equations to a functional equation linking both the bulk and boundary response of the system. For both problems, this functional equation is shown to reduce to scalar Wiener–Hopf equation whose solution can be exploited to determine the full response of the lattice system. The analytical solutions based on the Wiener–Hopf method are also implemented in numerical investigations that demonstrate the lattice response to various positions and frequencies of the load. In particular, this study includes dynamic regimes induced by the load that possess strong anisotropy and shows how the associated waveforms interact with the boundaries of the quadrant.