Find the Fourier sine series representation on the interval $0 \leqslant x \leqslant l$ of the function

$$f(x)= \begin{cases}0, & 0 \leqslant x<a \\ 1, & a \leqslant x \leqslant b \\ 0, & b<x \leqslant l\end{cases}$$

The motion of a struck string is governed by the equation

$$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}, \quad \text { for } \quad 0 \leqslant x \leqslant l \quad \text { and } \quad t \geqslant 0$$

subject to boundary conditions $y=0$ at $x=0$ and $x=l$ for $t \geqslant 0$, and to the initial conditions $y=0$ and $\frac{\partial y}{\partial t}=\delta\left(x-\frac{1}{4} l\right)$ at $t=0$.

Obtain the solution $y(x, t)$ for this motion. Evaluate $y(x, t)$ for $t=\frac{1}{2} l / c$, and sketch it clearly.