Consider the equation

$$\frac{\partial^{2} \phi}{\partial t^{2}}+\alpha^{2} \frac{\partial^{4} \phi}{\partial x^{4}}+\beta^{2} \phi=0$$

with $\alpha$ and $\beta$ real constants. Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Find the phase velocity $c(k)$ and the group velocity $c_{g}(k)$, and sketch the graphs of these functions.

By multiplying $(*)$ by $\partial \phi / \partial t$, obtain an energy equation in the form

$$\frac{\partial E}{\partial t}+\frac{\partial F}{\partial x}=0$$

where $E$ represents the energy density and $F$ the energy flux.

Now let $\phi(x, t)=A \cos (k x-\omega t)$, where $A$ is a real constant. Evaluate the average values of $E$ and $F$ over a period of the wave to show that

$$\langle F\rangle=c_{g}\langle E\rangle$$

Comment on the physical meaning of this result.