A string vibrates according to: $\frac{1}{c^2} = \frac{d^2s}{dt^2} = \frac{d^2s}{dx^2}$, where c is constant while s,x,t are displacement, spatial position and time. One end of the string is fixed so that @x=0, s=0. Its other end @x=l is attached to a mass-less damper so that @x=l, $\frac{ds}{dx} = \frac{ds}{dt}$ with being another constant.
You can find the frequencies of vibration for such system by using the separation of variables technique. The frequencies are simply related to the eigen value associated with the variation in time. Derive the relation which these eigen values have to satisfy (this relation is also called characteristic equation). Are the eigen functions for x-variations orthogonal to each other in this problem according to Sturm-Liouville theory? Explain.
