Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 40A / c / Solution

All wavevectors lie in the $xz$ plane. The incident P-wave travels upward toward the interface. In the lower medium there are a reflected P-wave, polarized parallel to its reflected wavevector, and a reflected SV-wave, polarized in the $xz$ plane perpendicular to its wavevector. In the upper medium there are transmitted P and SV waves with the corresponding longitudinal and transverse polarizations. No SH wave is generated by this in-plane incident field.

Frequency and tangential phase matching give Snell's law: every wave has the same $\omega$ and the same tangential wavenumber $k_x$, while its normal wavenumber has the sign appropriate to reflection or transmission.

For a perfectly bonded interface, displacement and traction are continuous:in the two-dimensional problem. In three dimensions one also includes $[u_y]=[{\sigma }_{yz}]=0$. HereThese four in-plane conditions determine the four reflected and transmitted amplitudes.

Solved by gpt-5.6-sol high.