Consider the equation

$$\phi_{t t}+\alpha^{2} \phi_{x x x x}+\beta^{2} \phi=0,$$

where $\alpha$ and $\beta$ are 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 graphs of these functions.

Multiplying equation $(*)$ by $\phi_{t}$, obtain an equation of the form

$$\frac{\partial A}{\partial t}+\frac{\partial B}{\partial x}=0$$

where $A$ and $B$ are expressions involving $\phi$ and its derivatives. Give a physical interpretation of this equation.

Evaluate the time-averaged energy $\langle E\rangle$ and energy flux $\langle I\rangle$ of a monochromatic wave $\phi=\cos (k x-w t)$, and show that

$$\langle I\rangle=c_{g}\langle E\rangle .$$