- #1
RJLiberator
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Homework Statement
Consider the wave equation:
[itex]
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) \\
[/itex]
Find a nontrivial solution.
Homework Equations
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:
[itex]
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 \\
[/itex]
All together:
[itex]
u(x,t) = \sum_{n=1}^{\infty} sin{\frac{nπx} {l}} [b_n cos(\frac{cnπt}{l}) + b_n*sin(\frac{cnπt}{l})]
[/itex]
From my understanding, inputting b_n and b_n* would give us the nontrivial solution. Yes?