Let's assume I have general wave function written like this:
$$ f(x,t) = \displaystyle\int_{-\infty}^{\infty}A(k)e^{i(kx-\omega(k) t)}dk $$
So it's composited from lots of plane waves and each of them is traveling with different speeds depending on their wavelengths.
There $x$ is a space coordinate, $t$ is time. $k$ is the angular wavenumber (radians/meter) (inversely proportional to the wavelength). $\omega$ is the angular frequency (radians/second). $A(k)$ is magnitude for a given component with a given wavenumber.
In more than 1 dimensions $k$ and $x$ is a vector and there is a dot product in the formula.
The frequency depends on the wavenumber (dispersion relation).
For sound waves and light there is a fixed "speed of sound" or "speed of light". So the relation for them is $\omega(k) = kc$, where $c$ is the speed of the wave propagation in this case the corresponding wave equation is the wave equation, whose shortest form is: $\square f = 0$. (The $\square$ is the D'Alembert operator).
For different dispersion relation there corresponds a different wave equation eg. for $-|k|^2$ the free Schrödinger equation. (Or there are other variations here)
Now the question: There are lots of kinds of waves that can be described using the integral above, but the difference is the dispersion relation. And apparently the dispersion relation and the corresponding partial differential equation is quite closely related. Is there a method that let's you obtain the corresponding equation for any differentiable $\omega(k)$ function?
First I considered asking this on Physics.SE but decided to ask here due to the math involved.