On the wave propagation in isotropic fractal media

Abstract

In this paper, we explore the wave propagation phenomenon in three-dimensional (3D) isotropic fractal media through analytical and computational means. We present the governing scalar wave equation, perform its eigenvalue decomposition, and discuss its corresponding modal solutions. The homogenization through which this fractal wave equation is derived makes its mathematical analysis and consequently the formulation of exact solutions possible if treated in the spherical coordinate system. From the computational perspective, we consider the finite element method and derive the corresponding weak formulation which can be implemented in the numerical scheme. The Newmark time-marching method solves the resulting elastodynamic system and captures the transient response. Two solvers capable of handling problems of arbitrary initial and boundary conditions for arbitrary domains are developed. They are validated in space and time, with particular problems considered on spherical shell domains. The first solver is elementary; it handles problems of purely radial dependence, effectively, 1D. However, the second one deals with general advanced 3D problems of arbitrary spatial dependence.