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I am trying to work through this question about the wave equation and I just don't know what to do.

Solve the wave equation in the rectangle $R=\{(x,y):0, with homogenous Dirichlet conditions on the boundary and initial conditions $u(x,y,0) \equiv xy(3-y)(1-x)\;$ and
$u_t(x,y,0) \equiv 0$.

Thank you in advance for the help.

edit for more general example:

Solve the wave equation in the rectangle $R=\{(x,y):0, with homogenous Dirichlet conditions on the boundary, and the initial conditions $u(x,y,0) \equiv xy(b-y)(a-x)\;$ and $u_t(x,y,0) \equiv 0$.

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    Does the PDE be $u_{tt}=u_{xx}+u_{yy}$ ?2012-10-24

1 Answers 1

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Do you have a set of modes of the rectangle? These are solutions to the wave equation without regard to the initial conditions and are of the form $f(x,y)e^{i\omega t}$. If not, do you have a set for the square?

If you do, you need to expand your $u(x,y,0)$ in these modes. They tend to be numbered $u_{mn}(x,y)$ where $m$ is the number of half waves in the $x$ direction and $n$ is the number in the $y$ direction. You are looking for a set of coefficients $a_{mn}$ so the solution $u(x,y,t)=a_{mn}u_{mn}e^{i \omega_{mn}t}$There is probably a proof that the modes are orthogonal, in which case you can just do like a Fourrier expansion. $a_{mn}=\int_{rectangle}u(x,y,0)u_{mn}(x,y)dxdy$ This only works if the modes are orthogonal.

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    I worked something out I think is correct, thank you for all your help.2012-05-07