# Fixed String Oscillation

1. Dec 10, 2012

### saturnsalien

This isn't actually a homework or coursework problem, but the style of the question is similar so I'm posting it here. Anyways, here goes. Consider a string of length L clamped at both ends, with one end at x=0 and the other at x=L. The displacement of the oscillating string can be described by the following equation:
$\frac{\partial^2 \Psi}{\partial t^2}=\frac{\partial^2 \psi}{\partial x^2}$

$\textrm{Given that at t=0:}\\ \>\>\>\> \psi(x,0)=\frac{2xh}{L},0\leq x\leq \frac{L}{2}\\ \>\>\>\> \psi(x,0)=\frac{2xh}{L},(L-x),\frac{L}{2}\leq x\leq L\\ \textrm{Show:}\\ \>\>\>\> \psi(x,t)=\sum_{m=1}^\infty\sin\left(\frac{m\pi x}{L}\right)\cdot\cos\omega_mt\cdot\left(\frac{8h}{\pi^2m^2}\right)\cdot\sin\left(\frac{\pi m}{2}\right)$

So, how do we go about doing that?

Last edited: Dec 10, 2012
2. Dec 10, 2012

### haruspex

Do you know the method of separation of variables?

3. Dec 10, 2012

### saturnsalien

Yeah, that's just rearranging the equation so that different variables occur on opposite sides of the equation. You can also do this by defining a variable as some expression, substitute, and the separate them.

4. Dec 10, 2012

### haruspex

So what does that give you for generic solutions of the PDE?