A uniform stretched string of length $L$, density per unit length $\mu$ and tension $T=\mu c^{2}$ is fixed at both ends. Its transverse displacement is given by $y(x, t)$ for $0 \leqslant x \leqslant L$. The motion of the string is resisted by the surrounding medium with a resistive force per unit length of $-2 k \mu \frac{\partial y}{\partial t}$.

(i) Show that the equation of motion of the string is

$$\frac{\partial^{2} y}{\partial t^{2}}+2 k \frac{\partial y}{\partial t}-c^{2} \frac{\partial^{2} y}{\partial x^{2}}=0$$

provided that the transverse motion can be regarded as small.

(ii) Suppose now that $k=\frac{\pi c}{L}$. Find the displacement of the string for $t \geqslant 0$ given the initial conditions

$$y(x, 0)=A \sin \left(\frac{\pi x}{L}\right) \quad \text { and } \quad \frac{\partial y}{\partial t}(x, 0)=0$$

(iii) Sketch the transverse displacement at $x=\frac{L}{2}$ as a function of time for $t \geqslant 0$.