Wave propagation and profile reconstruction in inhomogeneous elastic media

In this work, an analytical solution is derived for a wave reflection-transmission problem that involves non-homogeneous materials with position-dependent elastic parameters. The Frobenius method is applied to analyse the propagation of a time-harmonic elastic plane wave in layered inhomogeneous elastic media in one dimension. Through a series of numerical examples, this approach is contrasted with a conventional collocation formulation and it is proved that an excellent solution accuracy is achieved. Finally, in the inverse problem regime, synthetic data obtained by collocation, along with the Frobenius direct problem solver, are utilized by a differential evolution genetic algorithm to reconstruct inhomogeneous profiles up to quadratic order.