Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 38A / b / Solution

The cutoff occurs at $k=0$:For $k>0$, the phase velocity and group velocity areTheir product is $c_{p,n}{c}_{g,n}=c_S^2$. On a plot against positive $k$, $c_{p,n}$ decreases from $+\infty$ to $c_S$, while $c_{g,n}$ increases from zero to $c_S$; larger mode number shifts both transitions to larger wavenumber.