Acoustic Wave Propagation in Anisotropic Functionally Graded Materials: A Simulation Study

The use of functionally graded materials (FGMs) is experiencing rapid growth. One notable application of FGMs is noise reduction, which is intricately linked to the mechanics of acoustic wave propagation. Acoustic wave propagation is typically modeled through the Helmholtz equation. This study focuses on transient two-dimensional (2D) problems governed by the Helmholtz equation, where the coefficients vary with time and space. To address this complexity, a mathematical transformation, along with the Laplace transform, is employed to convert the space-time-dependent coefficient equation into one with constant coefficients. This constant coefficient equation can subsequently be expressed as a boundary-only integral equation. We solve this integral equation using the standard boundary element method (BEM) to derive numerical solutions for the relevant problems. The obtained numerical solutions are then subjected to an inverse Laplace transformation utilizing the Stehfest formula. Various problems with distinct classes of time-space-dependent coefficients were explored, and the numerical solutions obtained were highly accurate.