The transverse displacement $y(x, t)$ of a stretched string clamped at its ends $x=0, l$ satisfies the equation

$$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}-2 k \frac{\partial y}{\partial t}, \quad y(x, 0)=0, \frac{\partial y}{\partial t}(x, 0)=\delta(x-a)$$

where $c>0$ is the wave velocity, and $k>0$ is the damping coefficient. The initial conditions correspond to a sharp blow at $x=a$ at time $t=0$.

(a) Show that the subsequent motion of the string is given by

$$y(x, t)=\frac{1}{\sqrt{\alpha_{n}^{2}-k^{2}}} \sum_{n} 2 e^{-k t} \sin \frac{\alpha_{n} a}{c} \sin \frac{\alpha_{n} x}{c} \sin /\left(\sqrt{\alpha_{n}^{2}-k^{2}} t\right)$$

where $\alpha_{n}=\pi c n / l$.

(b) Describe what happens in the limits of small and large damping. What critical parameter separates the two cases?