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Wave Equation

dwsmith

Well-known member
Feb 1, 2012
1,673
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?

Screenshot from 2013-10-19 00:41:09.png
 
Last edited:

dwsmith

Well-known member
Feb 1, 2012
1,673
Here is the plot t = 0:0.01:0.11.

We can see that the amplitude grows as time grows. Shouldn't the amplitude spike since it is a delt function then decay?

Screenshot from 2013-10-19 11:43:29.png
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Does the wording under tension \(T\) something to the equation I am missing?