This is quite an interesting problem in astrophysics so I thought it would be a good idea to ask here so we can archive the solution for future reference.

Consider a pulsar that emits pulses of frequencies $\omega_1$ and $\omega_2$. Due to the interstellar space, these pulses will each take a different time to arrive to Earth, thus we will observe a delay between signals.

If we know the dispersion relation $\epsilon(\omega)$, how can one find the delay between these two signals? For example, let's consider the most simple dispersion relation:


where $\omega_p$ is the plasma frequency (for simplicity, let's take it as constant).

My idea is to start from the wave equation for a wave packet in a dispersive media and take an arbitrary distance $L$ for the wave to travel, and calculate the times $t_1$ and $t_2$ for the wave to take using each frequency, and somehow introduce the dispersion relation somewhere on the wave equation. However, while I know the evolution for a wave packet $\psi(x,t)$, I don't know how to find the time it takes to travel a distance $L$.

Edit: I found the solution using a less complicated method. We know that the wave number $k$ follows the relation,

$$k^2=\epsilon \mu\frac{\omega^2}{c^2}$$

Substituting the dispersion relation and considering non-magnetic media ($\mu=1$),


Thus we get,


Taking the derivative and remembering the definition of group velocity,


Substituting $k$ in the previous expression, we have:


If we consider that the wave travel a distance $L$, each pulse takes a time:



And thus the time difference is a function of the distance and frequencies,

$$\Delta t=\frac{L}{c}\left ( \frac{\omega_1}{\sqrt{\omega_1^2-\omega_p^2}} - \frac{\omega_2}{\sqrt{\omega_2^2-\omega_p^2}} \right )$$

If anyone can tell me it's correct so I can post it as an answer to my question, and thus close this thread.


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