Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 4

For a dispersion relation $\omega (k)$, phase velocity is $v_p=\omega /k$ and group velocity is $v_g=d\omega /dk=v_p+kd{v}_p/dk$. Their local equality of phase and group velocity holds throughout an interval for a nondispersive branch $\omega =ck$. Equality at one particular $k$ requires only that $v_p$ have zero derivative there, so it does not imply a globally nondispersive medium.

Use $k=2\pi /\lambda$ in the prescribed gravity-capillary wave relation. Then ${\omega }^2=gk+(T/\rho )k^3$, and differentiation givesIn the gravity-dominated limit, $v_g=v_p/2$; in the surface-tension-dominated limit, $v_g=3v_p/2$. For $g,T,\rho >0$, the two are equal at $\lambda =2\pi \sqrt{T/(\rho g)}$, the minimum of the phase velocity for this dispersion relation.