Show that a plane wave satisfies: $µ \frac{∂^2y}{∂t^2} = T \frac{∂^2y}{∂x^2} − α \frac{∂^4y}{∂x^4}$ I am unsure how to go about this question and I would appreciate some guidance here.
 A: You want to show that a generic plane wave, $y(x,t)$, solves this equation:
$$ \mu \frac{\partial^{2}y}{\partial t^{2}} = T \frac{\partial^{2}y}{\partial x^{2}} − \alpha \frac{\partial^{4}y}{\partial x^{4}}  $$
Of course, since this is your homework problem I will only set you in the right direction, but won't solve it for you :)
A generic two dimensional plane wave (one time dimension and one spatial dimension) can be written as,
$$ y(x,t) = A_{o} cos(kx - \omega t + \phi)$$
where $A_{o}$ is the amplitude, $k$ is the wave number, $\omega$ is the angular frequency, and $\phi$ is the constant phase shift of the wave.
Now, what you want to do is show algebraically that when you plug in $y(x,t)$ into the given partial differential equation the two sides equal each other. So, it's simplest to calculate each derivative in the equation separately (to avoid mistakes), and then use those derivatives to construct the equation (multiply the correct derivatives with the coefficients $\mu, T, \alpha$), then show that the two sides of the equation equal the same thing (show that the equal sign is indeed true). Then you're done! Let us know if you get stuck.
