 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2022/8/solution

For positive wavenumber, the condition gives

ω = C k^{- n /2}

on the positive-frequency branch. The group velocity and phase velocity are consequently related by

v_{g} = \frac{d ω}{d k} = - \frac{n}{2} \frac{ω}{k} = - \frac{n}{2} v_{p} .

The required positive ratio

v_{g} / v_{p} = 2

gives

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.

 New to topics? Read the docs here!