Fourier transform solution of three-dimensional wave equation

One of the PDE books I'm studying says that the 3D wave equation can be solved via the Fourier transform, but doesn't give any details. I'd like to try to work the details out for myself, but I'm having trouble getting started - in particular, what variable should I make the transformation with respect to? I have one time variable and three space variables, and I can't use the time variable because the Fourier transform won't damp it out.

If I make the transformation with respect to one of the spatial variables, the differentiations with respect to time and the other two spatial variables become parameters and get pulled outside the transform. But it looks like then I'm still left with a PDE, but reduced by one independent variable. Where do I go from here? Thanks.

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1 Answer

You use the Fourier transform in all three space variables. The wave equation $\frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2}\right)$ becomes $\frac{\partial^2 U}{\partial t^2} = - c^2 (p_1^2 + p_2^2 + p_3^2) U$.

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Thanks, seems straightforward enough! – Bitrex Aug 29 '11 at 5:38