A string of uniform density $\rho$ is stretched under tension along the $x$-axis and undergoes small transverse oscillations in the $(x, y)$ plane with amplitude $y(x, t)$. Given that waves in the string travel at velocity $c$, write down the equation of motion satisfied by $y(x, t)$.

The string is now fixed at $x=0$ and $x=L$. Derive the general separable solution for the amplitude $y(x, t)$.

For $t<0$ the string is at rest. At time $t=0$ the string is struck by a hammer in the interval $[l-a / 2, l+a / 2]$, distance being measured from one end. The effect of the hammer is to impart a constant velocity $v$ to the string inside the interval and zero velocity outside it. Calculate the proportion of the total energy given to the string in each mode.

If $l=L / 3$ and $a=L / 10$, find all the modes of the string which are not excited in the motion.