Methods | Part IB, 2001

Find the Fourier sine series representation on the interval 0xl0 \leqslant x \leqslant l of the function

f(x)={0,0x<a1,axb0,b<xlf(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

2yt2=c22yx2, for 0xl and t0\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=0y=0 at x=0x=0 and x=lx=l for t0t \geqslant 0, and to the initial conditions y=0y=0 and yt=δ(x14l)\frac{\partial y}{\partial t}=\delta\left(x-\frac{1}{4} l\right) at t=0t=0.

Obtain the solution y(x,t)y(x, t) for this motion. Evaluate y(x,t)y(x, t) for t=12l/ct=\frac{1}{2} l / c, and sketch it clearly.

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