- #1
deuteron
- 64
- 14
- TL;DR
- does the dispersion relation hold for electromagnetic waves in all media, or just in non-dispersive media?
In my lectures, we have derived the dispersion relation
$$ |\vec k|^2 = \frac {n^2 \omega^2}{c^2}$$
by substituting in a plane wave solution for the electromagnetic wave, into the wave equation derived from the Maxwell equations
$$\Delta\vec E= \mu_0\epsilon_0 \frac {\partial^2 \vec E}{\partial t^2}$$
But we also derived the same wave equation for classical waves
$$\Delta f = \frac 1 {v^2} \frac {\partial^2 f}{\partial t^2}$$
from observing a wave propagating with velocity ##v##, and then saying that the wave would be constant in a coordinate system that was moving with the wave with the speed ##v##. From that requirement, we got that the most general form of the wave solution is
$$f(x,t)= A_1 h(x-vt) + A_2 g(x+vt)$$
and by twice partial differentiating, we got the above wave equation.
But this derivation from classical physics requires the assumption of non-dispersion, since we say that the wave does not change its shape for a coordinate system moving with the speed of the wave.
However, we get the same wave equation from the Maxwell equations, even though electromagnetic waves are dispersed in some media
How is that possible that 1. we derive an equation by making an assumption on non-dispersion, 2. we derive the same equation from Maxwell equations, 3. but the solution of the wave equations describing EM waves are allowed to be dispersed?
And if the wave equation holds for dispersed waves too, does that mean that the dispersion relation we derive by substituting in a plane wave solution, hold for both dispersed and non-dispersed plane waves, thus essentially for all waves?
Or is the above dispersion relation only valid for non-dispersive media? (which would make me more confused, since we can substitute in a solution for the wave equation that is dispersed, into the wave equation; since by requirement all waves are supposed to fulfill the wave equation, and there are dispersed waves)
$$ |\vec k|^2 = \frac {n^2 \omega^2}{c^2}$$
by substituting in a plane wave solution for the electromagnetic wave, into the wave equation derived from the Maxwell equations
$$\Delta\vec E= \mu_0\epsilon_0 \frac {\partial^2 \vec E}{\partial t^2}$$
But we also derived the same wave equation for classical waves
$$\Delta f = \frac 1 {v^2} \frac {\partial^2 f}{\partial t^2}$$
from observing a wave propagating with velocity ##v##, and then saying that the wave would be constant in a coordinate system that was moving with the wave with the speed ##v##. From that requirement, we got that the most general form of the wave solution is
$$f(x,t)= A_1 h(x-vt) + A_2 g(x+vt)$$
and by twice partial differentiating, we got the above wave equation.
But this derivation from classical physics requires the assumption of non-dispersion, since we say that the wave does not change its shape for a coordinate system moving with the speed of the wave.
However, we get the same wave equation from the Maxwell equations, even though electromagnetic waves are dispersed in some media
How is that possible that 1. we derive an equation by making an assumption on non-dispersion, 2. we derive the same equation from Maxwell equations, 3. but the solution of the wave equations describing EM waves are allowed to be dispersed?
And if the wave equation holds for dispersed waves too, does that mean that the dispersion relation we derive by substituting in a plane wave solution, hold for both dispersed and non-dispersed plane waves, thus essentially for all waves?
Or is the above dispersion relation only valid for non-dispersive media? (which would make me more confused, since we can substitute in a solution for the wave equation that is dispersed, into the wave equation; since by requirement all waves are supposed to fulfill the wave equation, and there are dispersed waves)