Paper 2, Section II, 38C
The function satisfies the equation
Derive the dispersion relation, and sketch graphs of frequency, phase velocity and group velocity as functions of the wavenumber. In the case of a localised initial disturbance, will it be the shortest or the longest waves that are to be found at the front of a dispersing wave packet? Do the wave crests move faster or slower than the wave packet?
Give the solution to the initial-value problem for which at
and is real. Use the method of stationary phase to obtain an approximation for for fixed and large . If, in addition, , deduce an approximation for the sequence of times at which .
You are given that decreases like for large . Give a brief physical explanation why this rate of decay is slower than for . What can be said about for large if ? [Detailed calculation is not required in these cases.]
[You may assume that for