Three dimensional wave propagation in time-varying materials: A mathematical model based on the weak solutions of continuity in the moving property interface

Abstract

Abstract In this work, the elastic wave propagation through the moving property interface (MPI) in time-varying materials has been investigated. First, a general description of the MPI in time-varying materials has been given. Then, the elastic wave propagation through MPI has been systematically explored. In general, the wave propagation behaviors through MPI can be divided into two types depending on moving velocity of MPI and incident angle. For the first type, it has similar wave propagation behavior of that in static property interface, that is the reflection and refraction waves can be determined by the displacement and stress continuity in MPI. While, for the second type, it may have two transmitted waves, or only one transmitted wave or one reflected wave. Therefore, the transmitted waves or reflected wave cannot be solved by the commonly used continuous conditions, i.e. the displacement and stress continuities in MPI, due to the over constrained or under constrained problem. In order to deal with the wave propagation through MPI, a novel mathematical model is proposed based on the weak form of continuity in MPI. The results have shown that the wave propagation coefficients depend on not only the wave impedance radio and incident angle, but also the moving velocity of MPI. Besides, the transmitted and reflected wave frequencies and wavelengths are also significantly influenced by the moving velocity of MPI. When MPI catches up or exceeds the reflected or transmitted waves, the wavelength of the reflected or transmitted waves degenerates to zero, so that a shock wave is found in MPI and the displacements at the two sides of MPI are discontinuous. The wave propagation behaviors across MPI are further simulated via COMSOL Multiphysics software which shows very good agreement with the theoretical predictions. Moreover, the energy balance of SH waves in MPI is discussed. This work indicates that the novel mathematical model proposed in this paper is applicable to high dimensional wave phenomenon with moving boundaries, and is capable for studying complex wave equations, especially for wave equations with moving boundaries.