Birefringent materials and Maxwell equation

Question

Maxwell's equations define the speed of light in a given medium at a given point through the equation:
$$frac{partial^2E}{partial t^2}=muepsilonnabla^2E$$
so according to it, the speed of light in a given medium should be constant in all directions. Then why do birefringent materials, which have different speed of light in different directions, even exist?

Dave · Accepted Answer

In anisotropic materials there is a permittivity tensor instead of just a scalar.
Then the wave equation is
$$
nabla^2 vec{E} = -omega^2 mu_0 vec{D}
$$
With
$$
vec D = bar epsilon vec E
$$
where $bar epsilon$ is a (rank 2) tensor, c.f.
https://courses.cit.cornell.edu/ece303/Lectures/Lectures.htm, Lecture 17

Birefringent materials and Maxwell equation

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