Consider the equation

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

where $\alpha$ is a positive constant. Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Sketch graphs of the phase velocity $c(k)$ and the group velocity $c_{g}(k)$.

A disturbance localized near $x=0$ at $t=0$ evolves into a dispersing wave packet. Will the wavelength and frequency of the waves passing a stationary observer located at a large positive value of $x$ increase or decrease for $t>0$ ? In which direction do the crests pass the observer?

Write down the solution $\phi(x, t)$ with initial value

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

What can be said about $A(-k)$ if $\phi$ is real?

Use the method of stationary phase to obtain an approximation for $\phi(V t, t)$ for fixed $V>0$ and large $t$. What can be said about the solution at $x=-V t$ for large $t$ ?

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