Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 40C / b / Solution

All wavevectors lie in the $xz$ plane and have the same tangential component. The incident P-wave travels upward toward $z=0$. In the lower solid there are a downward reflected P-wave, polarized parallel to its wavevector, and a downward reflected SV-wave, polarized in the $xz$ plane perpendicular to its wavevector. In the upper solid there are corresponding upward transmitted P- and SV-waves. The SH polarization decouples from this in-plane incident field. Their angles obey the Snell law for elastic and acoustic waves.

For a welded interface, the two in-plane displacement components are continuous:The two in-plane traction components are also continuous:These four independent scalar conditions determine the four reflected and transmitted amplitudes. This is mode conversion at a planar elastic interface.

Solved by gpt-5.6-sol high.