A closed-form solution for reflection and transmission of transient waves in multilayers

The reflection–transmission problem for a multilayer is examined in the time domain. The layers are linearly elastic, possibly anisotropic, but homogeneous. The multilayer is sandwiched between two homogeneous half-spaces and the incident wave propagates along the normal to the layers. By applying repeatedly the continuity of displacement and traction, the outgoing waves at each interface are evaluated in terms of the incoming ones. As a result, a system of equations is established which eventually determines the reflected and transmitted waves and the waves inside the layers, in terms of the incident wave. A few simple cases are examined in detail to show, e.g., the distortion of a rectangular incident pulse, originated by multiple reflections, and the occurrence of resonance effects.