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

$$\left(1-2 \frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{4}}{\partial x^{4}}\right) \frac{\partial^{2} \varphi}{\partial t^{2}}+\frac{\partial^{4} \varphi}{\partial x^{4}}=0$$

Derive the corresponding dispersion relation and sketch graphs of frequency, phase velocity and group velocity as functions of the wavenumber. Waves of what wavenumber are at the front of a dispersing wave train arising from a localised initial disturbance? For waves of what wavenumbers do wave crests move faster or slower than a packet of waves?

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

$$\varphi(x, 0)=\int_{-\infty}^{\infty} 2 A(k) e^{i k x} d k, \quad \frac{\partial \varphi}{\partial t}(x, 0)=0$$

where $A^{*}(-k)=A(k)$, and $A^{*}$ is the complex conjugate of $A$. Let $V=x / t$ be held fixed. Use the method of stationary phase to obtain a leading-order approximation to this solution for large $t$ when $0<V<V_{m}=(3 \sqrt{3}) / 8$, where the solutions for the stationary points should be left in implicit form.

Very briefly discuss the nature of the solutions for $-V_{m}<V<0$ and $|V|>V_{m}$.

[Hint: You may quote the result that the large time behaviour of

$$\Phi(x, t)=\int_{-\infty}^{\infty} A(k) e^{i k x-i \omega(k) t} d k$$

due to a stationary point $k=\alpha$, is given by

$$\Phi(x, t) \sim\left(\frac{2 \pi}{\left|\omega^{\prime \prime}(\alpha)\right| t}\right)^{\frac{1}{2}} A(\alpha) e^{i \alpha x-i \omega(\alpha) t+i \sigma \pi / 4}$$

where $\left.\sigma=-\operatorname{sgn}\left(\omega^{\prime \prime}(\alpha)\right) .\right]$