I am trying to find the displacement $u(r,t)$ in a circular membrane of radius $5$, that is clamped down along the circumference. If the initial displacement is $f(r)$ and the membrane is released from rest.
$$a^2\left(\frac{d^2u}{dr^2}\right) + \frac{1}{r}\left(\frac{du}{dr}\right)=\frac{d^2u}{dt^2}$$
Where $\frac{du}{dt}(r,0)=0,\quad u(5,t)=0, \quad u(r,0)=f(r)$
Transforming the boundary conditions is what throws me off.
Thanks.