I'm working on my final project in this independent study I'm in. I worked out this problem, but I want to verify that I'm right.

A string of length L with both endpoints fixed, such as the string of a piano, is struck at its midportion, the impact thus imparting to the string an initial velocity $\displaystyle \psi(x)$ which is defined by:

$\displaystyle 0, 0<= x<\frac{L-a}{2}$

$\displaystyle 1, \frac{L-a}{2}<x<\frac{L+a}{2}$

$\displaystyle 0, \frac{L+a}{2}<x<=L $

Find the motion of the string.

I set up the following boundary value problem:

$\displaystyle y_{tt}=a^2y_{xx}$

$\displaystyle y(0,t)=0$

$\displaystyle y(L,t)=0$

$\displaystyle y(x,0)=0$

$\displaystyle y_{t}(x,0)=\psi(x)$

and I got the following Fourier Series Solution:

$\displaystyle y(x,t)=\frac{4L}{\pi^2a}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^2}sin{\left[\frac{(2n-1)\pi a}{2L}\right]}sin{\left[\frac{n\pi a}{L}t\right]}sin{\left[\frac{n\pi}{L}x\right]}$

Does this look right? I would appreciate any help!!!