# PDE: Nontrivial solution to the wave equation

1. Feb 20, 2016

### RJLiberator

1. The problem statement, all variables and given/known data
Consider the wave equation:

$u_{tt} - c^2u_{xx} = f(x,t), \hspace{1cm} for \hspace{1cm} 0 < x < l \\ u(0,t) = 0 = u(l,t) \\ u(x,0) = g(x), u_t(x,0) = f(x) \\$
Find a nontrivial solution.
2. Relevant equations

3. The attempt at a solution

Here's what I did, but I have little understanding of it other than I know that I am using boundary conditions and some previous material to get here:

We form a series solution:

$u(x,t) = \sum_{n=1}^{\infty} u_n(x,t) = \sum_{n=1}^{\infty} sin{\frac{nπx} {l}}u_n(t) \\ g(x) = \sum_{n=1}^{\infty}b_nsin{\frac{nπx}{l}} \\ f(x) = \sum_{n=1}^{\infty} \frac{cnπb_n*}{l} sin{\frac{nπx}{l}} \\ \\ b_n = \frac{2}{l} \int_{0}^{l} g(x)sin{\frac{nπx}{l}}dx \\ b_n* = \frac{2}{l} \int_{0}^{l} f(x)sin{\frac{nπx}{l}}dx \\$

All together:

$u(x,t) = \sum_{n=1}^{\infty} sin{\frac{nπx} {l}} [b_n cos(\frac{cnπt}{l}) + b_n*sin(\frac{cnπt}{l})]$

From my understanding, inputting b_n and b_n* would give us the nontrivial solution. Yes?

2. Feb 20, 2016

### Orodruin

Staff Emeritus
Yes, although you should consider using the notation $b_n^*$ instead, it would be less confusing.

If you would like a better understanding of what is going on, I suggest reading up on function spaces and orthogonal functions.

3. Feb 20, 2016

### RJLiberator

Thank you kindly for the confirmation and suggestion.