Past exam of the mathematical methods II course of the University of Oxford / 2022 / 8 / Solution

For positive wavenumber, the condition gives $\omega =C{k}^{-n/2}$ on the positive-frequency branch. The group velocity and phase velocity are consequently related byThe required positive ratio $v_g/v_p=2$ gives $\boxed{n=-4}$. Isotropy makes these velocities parallel to the wave vector when this coefficient is positive. Identifying group velocity with energy transport assumes the usual lossless narrowband wave packet regime.