How Do Waves Propagate in an Anisotropic Dielectric Medium?

[sachi]

we have a non magnetic but anisotropic dielectric medium which has the following relationships between D and E

Dx = k1*Ex, Dy = k2*Ey, Dz = k3*Ez

we have to show that waves propogate in the z-dir'n at one speed only.

I can't get the wave eq'n to fall out. Usually you just use Maxwell's equations and the vector identity
curl(curl(E)) = grad(divE)- grad squared E
and set divE = 0. You can usually say this because in a dielectric there is no free charge therefore divD = 0 and in an LIH medium E is proportional to D therefore divE = 0 . Instead now I have to include the grad(divE) expression and the algebra doesn't even resemble the wave eq'n. I'd appreciate any hints on where I've gone wrong.

 

[dextercioby]

Well, you'll have to see what impact does

\left(\begin{array}{c} D_{x}\\D_{y}\\D_{z}\end{array}\right)=\left(\begin{array}{ccc} k_{1} & 0 & 0 \\ 0 & k_{2} & 0\\ 0 & 0 & k_{3} \end{array}\right) \left(\begin{array}{c} E_{x}\\E_{y}\\E_{z}\end{array}\right)

have upon the wave equations. I assume the Permittivity matrix has constant elements, if not, you'd have to be more careful with the calculus.

Daniel.

 

[Meir Achuz]

If k1=k2=k, then there is a relatively simple solution.
curl(curl B)=-del^2 B=d_t curl{[k]E}. (d_t is the partial wrt t, etc.)
For the z component in Cartesian coords:
curl{[k]E}_z=d_x(k2 E_y)-d_y(k1 E_x)=k (curl E)_x, if k1=k2=k.
Then, -del^2 B_z=-k(d_t)^2 B_z, and B_z has a wave solution with
velocity=1/\sqrt{k} (All with c=1)