A physical system permits one-dimensional wave propagation in the $x$-direction according to the equation

$$\frac{\partial^{2} \psi}{\partial t^{2}}-\alpha^{2} \frac{\partial^{6} \psi}{\partial x^{6}}=0$$

where $\alpha$ is a real positive constant. Derive the corresponding dispersion relation and sketch graphs of frequency, phase velocity and group velocity as functions of the wave number. Is it the shortest or the longest waves that are at the front of a dispersing wave train arising from a localised initial disturbance? Do the wave crests move faster or slower than a packet of waves?

Find the solution of the above equation for the initial disturbance given by

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

where $A(k)$ is real and $A(-k)=A(k)$.

Use the method of stationary phase to obtain a leading-order approximation to this solution for large $t$ when $V=x / t$ is held fixed.

[Note that

$$\int_{-\infty}^{\infty} e^{\pm i u^{2}} d u=\pi^{\frac{1}{2}} e^{\pm i \pi / 4}$$