Finite Fourier Transform on a 3d wave

[brandy]

Finite Fourier Transform on a 2d wave

How does the finite Fourier transform work exactly?

The transform of f(x) is
\widetilde{f}(\lambda_{n}) =\int^{L}_{0} f(x) X_{n} dx

If I had a 3d wave equation pde and I applied Finite Fourier transform on the pde for
z(x,y,t)=X(x)Y(y)T(t)

f(x)=z_{xx}
\tilde{z_{xx}(\lambda_{n})} =\int^{L}_{0} z_{xx} X_{n}(x) dx

f(x)=z_{yy}
\tilde{z_{yy}(\lambda_{n})} =\int^{L}_{0} z_{yy} Y_{n}(y) dy

etc for T

or

is the transform the same for everything, namely:
f(x)=z_{xx}
\tilde{z_{xx}(\lambda_{n})} =\int^{L}_{0} z_{xx} X_{n}(x) Y_{n}(y) dx

or is it a double integral?
or do you apply finite Fourier twice with respect to y and x.

 

[I like Serena]

Hi brandy!

You can take any transform you want, including a 3D Fourier transform, which would be a triple integral.
And also a transform of for instance $z_{xx}$ with respect to y.

In all cases you can apply the Fourier's theorems.
It's just a matter of what you want to achieve.

 

[jasonRF]

Note that you should have a different "frequency" variable for each dimension $x$ and $y$.