Polarized wave in an anisotropic medium

[lholmes135]

Homework Statement

Calculate the wavelength for an Ex polarized wave

Relevant Equations

Unsure

Calculate the wavelength for an $E_x$ polarized wave traveling through an anisotropic material with $\overline{\overline{\epsilon}}=\epsilon_0diag({0.5, 2, 1})\text{ and }\overline{\overline{\mu}}=2\mu_0$ in:
a. the y direction
b. the z direction
Leave answers in terms of the free space wavelength.

All I've gotten so far is:
$$E(y, t)=E_0cos(k_yy-wt)$$
$$\lambda=\frac{2\pi}{k_y}$$
I don't know how to determine $k_y$ or $\overline{k}$. I'm basically totally stumped on this problem. Of course it needs to be in terms of:
$$\lambda_0=\frac{2\pi c}{\omega_0}$$

 

[jasonRF]

Do you know how to calculate the wavelength in an isotropic medium with (for example) $\epsilon = 2 \epsilon_0$ and $\mu=2\mu_0$?

jason

 

[lholmes135]

Sure. $\lambda=\frac{\lambda_0}{\sqrt{\epsilon_r\mu_r}}$, so in your example the wavelength would be half of that as in free space. The problem in anisotropic materials is that when $\epsilon$ and $\mu$ are tensors, I don't know what values to use.

 

[lholmes135]

I think I figured it out. Because the electric field is polarized in the x direction and the magnetic field in the z direction, I can just use the x component of permittivity and the z component of the magnetic field, so in this problem $\lambda=\lambda_0$ in both cases.