Past exam of the physics A course of the University of Cambridge / 2019 / Paper 2 / Section B / B9 / b / Solution

For the trial displacement $z=\sin (qy)e^{i(kx-\omega t)}$, direct differentiation givesThe two-dimensional wave equation is therefore satisfied ifHere $k$ is the longitudinal wavenumber, $q$ the transverse wavenumber, and $c_m$ the speed of an unconstrained membrane wave. The physical displacement is the real part, multiplied by an arbitrary displacement amplitude. $T$ has units $\mathrm{N}{\mathrm{m}}^{-1}$ and $\sigma$ is surface mass density in $\mathrm{kg}{\mathrm{m}}^{-2}$, so $T/\sigma$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-2}$ as required.