The real function $\phi(x, t)$ satisfies the Klein-Gordon equation

$$\frac{\partial^{2} \phi}{\partial t^{2}}=\frac{\partial^{2} \phi}{\partial x^{2}}-\phi, \quad-\infty<x<\infty, t \geqslant 0$$

Find the dispersion relation for disturbances of wavenumber $k$ and deduce their phase and group velocities.

Suppose that at $t=0$

$$\phi(x, 0)=0 \quad \text { and } \quad \frac{\partial \phi}{\partial t}(x, 0)=e^{-|x|}$$

Use Fourier transforms to find an integral expression for $\phi(x, t)$ when $t>0$.

Use the method of stationary phase to find $\phi(V t, t)$ for $t \rightarrow \infty$ for fixed $0<V<1$. What can be said if $V>1$ ?

[Hint: you may assume that

$$\left.\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}}, \quad \operatorname{Re}(a)>0 .\right]$$