Paper 2, Section II, 38C

Waves
Part II, 2014

The function ϕ(x,t)\phi(x, t) satisfies the equation

2ϕt22ϕx2=4ϕx2t2\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}=\frac{\partial^{4} \phi}{\partial x^{2} \partial t^{2}}

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 t=0t=0

ϕ=A(k)eikxdk and ϕt=0\phi=\int_{-\infty}^{\infty} A(k) e^{i k x} d k \quad \text { and } \quad \frac{\partial \phi}{\partial t}=0

and ϕ(x,0)\phi(x, 0) is real. Use the method of stationary phase to obtain an approximation for ϕ(Vt,t)\phi(V t, t) for fixed 0<V<10<V<1 and large tt. If, in addition, ϕ(x,0)=ϕ(x,0)\phi(x, 0)=\phi(-x, 0), deduce an approximation for the sequence of times at which ϕ(Vt,t)=0\phi(V t, t)=0.

You are given that ϕ(t,t)\phi(t, t) decreases like t1/4t^{-1 / 4} for large tt. Give a brief physical explanation why this rate of decay is slower than for 0<V<10<V<1. What can be said about ϕ(Vt,t)\phi(V t, t) for large tt if V>1V>1 ? [Detailed calculation is not required in these cases.]

[You may assume that eau2du=πa\int_{-\infty}^{\infty} e^{-a u^{2}} d u=\sqrt{\frac{\pi}{a}} \quad for Re(a)0,a0.]\left.\operatorname{Re}(a) \geqslant 0, a \neq 0 .\right]