Total reflection and tunneling of transient waves at a layer

Within the acoustic approximation, the paper provides the transient reflected and transmitted waves produced by a plane wave, obliquely incident on the plane interface of a layer which is sandwiched between two homogeneous half spaces. First, the unknown waves are investigated in the Fourier transform domain and Snell's law is applied. Next, attention is restricted to incidence beyond the critical angle and the solution to the reflection-transmission problem is shown to be unique. The reflected and transmitted waves and the waves in the layer are eventually obtained in the time domain. The transient waves turn out to be expressed as convolutions of the incident wave with appropriate solutions to the wave equation. Such solutions, which are parameterized by the mass density, the speed and the width of the layer, express the pertinent effects of multiple reflections within the layer.