A wave disturbance satisfies the equation

$$\frac{\partial^{2} \psi}{\partial t^{2}}-c^{2} \frac{\partial^{2} \psi}{\partial x^{2}}+c^{2} \psi=0$$

where $c$ is a positive constant. Find the dispersion relation, and write down the solution to the initial-value problem for which $\partial \psi / \partial t(x, 0)=0$ for all $x$, and $\psi(x, 0)$ is given in the form

$$\psi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k$$

where $A(k)$ is a real function with $A(k)=A(-k)$, so that $\psi(x, 0)$ is real and even.

Use the method of stationary phase to obtain an approximation to $\psi(x, t)$ for large $t$, with $x / t$ taking the constant value $V$, and $0 \leqslant V<c$. Explain briefly why your answer is inappropriate if $V>c$.

[You are given that

$$\left.\int_{-\infty}^{\infty} \exp \left(i u^{2}\right) d u=\pi^{1 / 2} e^{i \pi / 4} .\right]$$