Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 3 / e / Solution

Multiplyby $\hat{w}$ and integrate from the rigid bottom to an upper endpoint $z$ at which $\hat{w}{\hat{w}}_z=0$. The bottom term also vanishes because $\hat{w}(0)=0$. Integration by parts givesTherefore the horizontal wavenumber has the Rayleigh quotientFor a trapped wave the upper endpoint may be taken to infinity because the eigenfunction decays.