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

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

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 $t=0$

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

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

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

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