Equation of electromagnetic waves solution, what are the parametres represent? I am trying to understand the solution of the electromagnetic waves equation dervied from maxwell's equations.
I have been told that every function that is $f(\vec{k} \cdot \vec{r}
+\omega t)$ is the solution, and I can understand that, but I do not understand what each variable in the function is represent.
 A: A function is of the form:
$$ f(\vec r, t) $$
so the $\vec r$ is a position in 3D space, and $t$ is a time coordinate. Any function of the form:
$$ f(\vec r, t) = f(\vec k \cdot \vec r - \omega t) $$
is a solution of the wave equation. (Here I choose a minus sign instead of a plus sign in front of the $\omega t$ term. Both work, but this form provides a more common definition of $\omega$).
When you plug that form into:
$$ \nabla^2 f - \frac 1 {c^2} \frac{\partial^2 f}{\partial t^2}=0$$
you get an algebraic condition:
$$ k^2 - \frac{c^2}{\omega^2} = 0$$
so that:
$$ \omega = kc$$
provides a relation between the scales in time and space.
Since any function can be decomposed in to its Fourier components, choose:
$$ f(\vec k \cdot \vec r - \omega t) = e^{i(\vec k \cdot \vec r - \omega t)}$$
it's clear that $\omega$ is the angular frequency and $\vec k$ is the wave vector defining the wavelength and propagation direction (at speed $c$).
The combination:
$$ \phi \equiv \vec k \cdot \vec r - \omega t $$
is the phase, while $\omega = kc$ is the dispersion relation defining dispersion less propagation.
