Consider wave equation in three dimensions $$\square f=0$$ Consider first cartesian coordinates. With separation of variables one can find an infinite number of indipendent solutions
$$f_{\omega,\alpha,\beta,\gamma}(x,y,z,t)=f^{0}_{\omega,\alpha,\beta,\gamma}e^{i(\alpha x+\beta y+ \gamma z-\omega t)}\tag{1}$$
To get the general solution it is necessary to write the linear combination
$$f(x,y,z,t)=\sum_{\omega,\alpha,\beta,\gamma}f^{0}_{\omega,\alpha,\beta,\gamma}e^{i(\alpha x+\beta y+ \gamma z-\omega t)}\tag{2}$$
On textbook is it claimed that $(2)$ is a Fourier series, as long as $f$ is periodic. It is not explicit but I suppose that values of $\omega,\alpha,\beta,\gamma$ considered in $(2)$ must be a multiple of a foundamental value $\omega_0,\alpha_0,\beta_0,\gamma_0$. Is that correct?
If $f$ is not periodic, then it is necessary to use an integral. What is the form of the integral rapresentation of $f$ in this case and what is its Fourier trasform? My guess is:
$$f(x,y,z,t)=\frac{1}{(2\pi)^{\frac{4}{2}}}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}F(\omega,\alpha,\beta,\gamma)e^{i(\alpha x+\beta y+ \gamma z-\omega t)} d\omega \,\,d\alpha \,\, d\beta \,\,d \gamma$$
$$F(\omega,\alpha,\beta,\gamma)=\frac{1}{(2\pi)^{\frac{4}{2}}}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y,z,t)e^{-i(\alpha x+\beta y+ \gamma z-\omega t)} dx\,\,dy \,\, dz \,\,d t$$
Are these the correct expressions?
If I go in cylindrical coordinates for istance, the solution is
$$f(r,\phi , z,t) =\sum_{\omega,n,h} R^{0}_{\omega, n, h } H_n\Bigg(r \sqrt{\frac{\omega^2}{c^2}-h^2}\Bigg) e^{i(n\phi +hz-\omega t)} \tag{3}$$
Is $(3)$ still a Fourier series? There is an Hankel function $H$ that contains $r$ but also $\omega$ and $h$ (and its order is $n$), so it should not be a Fourier series but then what "kind" of series is $(3)$?
In particular supposing $f(r,\phi , z,t) =R(r) \Phi(\phi) Z(z) \chi (t)$ can I factorize $(2)$ as a Fourier series for $\Phi(\phi) Z(z) \chi (t)$ and a Fourier-Bessel series for $R(r)$?
Finally if $f$ is not periodic and I'm using cylindrical coordinates, is $(3)$ subsituted by something else (on textbook there are no explicit assumption for $(3)$, i.e. periodicity)?
(I have similar doubts for solution in spherical coordinates)