Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 40B / b / iii / Solution

The normal acoustic energy flux isIn either half-space, the pressure fields above are proportional to $\cos (kx-\omega t)$, whereas the normal velocities are proportional to $\sin (kx-\omega t)$. They are therefore in temporal quadrature. Averaging over one period givesThus the field is evanescent and stores reactive acoustic energy flux near the membrane without transporting net acoustic energy away from it in the normal direction; the wave's mean energy transport is along the membrane.