If we have 1D wave equation:
$$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$$
we say that it's always possible to decompose the generic solution $f(x+ct)+g(x-ct)$ using Fourier Transforms.
But
$$\frac{1}{2\pi} \int_R f(\omega)e^{i\omega t}d\omega$$
and we have two variables, $x$ and $t$, how can we use the FT with
$$ \int_R f(\omega)e^{i(k(\omega)x\mp\omega t)}d\omega \; ?$$
We have two different types of evolution, $+ct$ and $-ct$... I have seen situations with no dispersion law, where we consider only the $x+ct$ wave and use the point $x=0$ to find the spectrum of waves $f(\omega)$, but can someone explain me the general method?
In 2D or 3D I haven't any idea about how I can do all this. Someone can explain me this too?
Thank you very much!
