Is it possible to find an analytic solution for the modes of vibration of a hollow cylinder, assuming azimuthal symmetry? That is can the following PDE be evaluated:
$$\frac{1}{\rho}\frac{\partial}{\partial\rho}\left[\rho\frac{\partial f}{\partial\rho}\right] + \frac{\partial^2 f}{\partial z^2} = \frac{1}{v^2}\frac{\partial^2f}{\partial t^2}$$
With boundary conditions (I believe):
$$f(0, \rho, t) = 0$$ $$f(H, \rho, t) = H$$ $$f(z, r, t) = z$$ $$f(z, R, t) = z$$
Where $H$ is the height of the cylinder, and $r$ and $R$ are respectively the inner and outer radii of the cylinder.