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My plot seems wrong so I am not sure what the problem is: (a) mistake in sol (b) mistake in coding.

A clamped, uniform string under tension \(T\) has length \(\ell\). The string is struck in the middle, giving an initial tranverse velocity distribution

\[

\dot{u}(x, 0) = \delta(x - 1/2).

\]

So the solution I obtained:

\[

u(x, t) = \frac{2}{\pi}\sum_{n=1}^{\infty}\frac{1}{n} \sin\left(\frac{n \pi}{2}\right) \sin(n\pi x)\sin(n\pi t)

\]

Is the solution correct? If so, is the plot correct?

Also, how do I find the energy of each mode?

A clamped, uniform string under tension \(T\) has length \(\ell\). The string is struck in the middle, giving an initial tranverse velocity distribution

\[

\dot{u}(x, 0) = \delta(x - 1/2).

\]

So the solution I obtained:

\[

u(x, t) = \frac{2}{\pi}\sum_{n=1}^{\infty}\frac{1}{n} \sin\left(\frac{n \pi}{2}\right) \sin(n\pi x)\sin(n\pi t)

\]

Is the solution correct? If so, is the plot correct?

Also, how do I find the energy of each mode?

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