Given an empty cilinder with a $\backslash$leq r $\backslash$leq b and hight L$.$ Assume the cilinder is starts initially ouf of balance and set free. Assyme initially that there are no external sources, but a frictionforce that is in proportion to the speed $(-\backslash$alpha$\backslash$partial$\_$t u$($x$,$t$).$ The cilinder is uniform and has an initial speed c$.$ Vind the resulting vibrations with the following boundary conditions:

$$ the topflat and the inside of the mantle are absorbing

$$the bottomflat and the outside of the mantle are reflecting.

a$)$ translate this information into mathematical equations and boundary conditions

b$)$ solve the equation

c$)$ assume the intial displacement is given as cos$(\backslash$pi z$/2$L$)$ how does the solution look like now?

d$)$ give a schematics of how to solve the equation if there was an externam source