Acoustic X-wave reflection and transmission at a planar interface: Spectral analysis

The spectral structure of a three-dimensional X-wave pulse incident on a planar surface of discontinuity is examined. Introducing a novel superposition of azimuthally dependent pulsed plane waves, it is shown for oblique incidence that the reflected pulse has a localized wave structure. On the other hand, the transmitted field maintains its localization up to a certain distance from the interface, beyond which it starts disintegrating. An estimate of the localization range of the transmitted pulse is established; also, the parameters affecting the localization range are identified. The reflected and transmitted fields are deduced for X-waves incident from either a slower medium or a faster one. For the former case the evanescent fields in the second medium are calculated and their explicit time dependence is deduced for a normally incident X-wave. Furthermore, at near-critical incidence the transmitted pulse exhibits significant pulse compression and focusing.