Mode conversion at a planar elastic interface

An obliquely incident in-plane P- or SV-wave generally produces reflected and transmitted P- and SV-waves. Continuity of displacement and traction supplies four scalar conditions for their four amplitudes, while frequency and tangential wavenumber are shared.

Conversion of an incident S wave into a P wave at a material contrast. It provides a complementary receiver function constraint because its travel-time ordering and interference with reverberations differ from P-to-S conversion.

Conversion of an incident P wave into an S wave at a material contrast. The converted arrival's relative time and amplitude can constrain subsurface structure through a receiver function, subject to the velocity model and interface geometry.

For incidence from a solid with compressional/shear speeds $\alpha ,\beta$ into a fluid of sound speed ${\alpha }_f$, let $E={\cos }^{2}2{\theta }_S+({\beta }^2/{\alpha }^2)\sin 2{\theta }_P\sin 2{\theta }_S$. The displayed coefficient multiplies the transmitted displacement field potential relative to the incident P wave potential, using the same retarded-time waveform. Snell law for elastic and acoustic waves matches the tangential slowness. Continuity of normal displacement field, zero tangential traction, and continuity of normal traction give the coefficient. It is not a displacement-amplitude coefficient: longitudinal displacement field acquires an additional ratio $\alpha /{\alpha }_f$.

Phase matching at a planar interface preserves frequency and tangential wavenumber. Thus waves of speeds $c_j$ and angles ${\theta }_j$ from the normal satisfy