Reflection and transmission of transient acoustic waves with oblique incidence

Abstract

The aim of the paper is to determine, within the time domain, the waves produced by an oblique incident wave at the interface between two homogeneous half-spaces. By following the acoustic approximation, the wave solutions for the Fourier transform of the displacement field in a viscous fluid are established in a form which generalizes the concept of plane wave. Next the reflection-transmission problem, associated with the interface between an inviscid fluid and a viscous one, is investigated. The incident wave is supposed to propagate in the inviscid fluid. The reflected and transmitted waves, in the time domain, are eventually determined in two particular cases, namely that of normal incidence on a viscous half-space and that of oblique incidence, beyond the critical angle, on an inviscid half-space. In the first case it follows that, provided an approximation of band-limited data holds for the incident wave, the reflected and the transmitted waves are given by linear combinations of the values of the incident wave and of its time derivative. In the second case, the reflected (transmitted) wave is shown to be the sum of a term proportional to the incident wave and another one, proportional to the Hilbert transform of (a convolution of) the incident wave.