# Wave Equation

#### dwsmith

##### Well-known member
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?

Last edited:

#### dwsmith

##### Well-known member
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?

#### dwsmith

##### Well-known member
Does the wording under tension $$T$$ something to the equation I am missing?