Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 40C / a / Solution

Use a velocity potential proportional to $e^{ik(x-Vt)+ik{\beta }_{1}y}$, where
${\beta }_1=\sqrt{(V/c_1{)}^2-1}$. The wall condition gives amplitude $-Vϵ/{\beta }_1$. With $p^{\prime }=-{\rho }_1{\partial }_t\varphi$, acoustic energy flux givesFor $V<c_1$ the vertical wave number is imaginary: the disturbance is evanescent and carries no mean vertical energy flux.