Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 3 / e / ii / Solution

Write $k_{z+}=+i{q}_{+}$ above the interface and $k_{z-}=-i{q}_{-}$ below it, with $q_{\pm }=|k_{z\pm }|>0$, so both waves decay away from $z=0$. A common interface displacement ${\xi }_0$ givesPressure continuity therefore requires