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Currently I am trying to understand the physics of waves in cold plasmas. So in a cold, collison-less, unmagnetized plasma you can derive two different different dispersion relations: one for longitudinal oscillations ($\parallel$ to $\vec{k}$)

$\omega^2 = \omega^2_{pe}$

and one for transverse oscillations ($\perp$ to $\vec{k}$)

$\omega^2 = \omega_{pe}^2 + k^2c^2$

I understand mathematically how these are derived. My confusion comes from the fact that we end up with two different equations for $\omega(k)$.

If, for example, $\vec{k} = k\hat{z}$, and we had an electric field of the form

$\vec{E}(\vec{r},t) = (E_x\hat{x} + E_z\hat{z})e^{(kz - \omega t)}$

where $E_x$ and $E_z$ are constants, does this mean that the two components of the electric field would have to obey different dispersion relationships and not necessarily have the same frequency?

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