# Polarisation directions in standing waves in cubical cavity

I was studying Rayleigh-Jean's formula. The author has assumed a cubical cavity of each side $L$ with perfectly reflecting surfaces. According to author, there are two perpendicular directions of polarization for a standing wave taken in arbitrary direction. I am stuck up here, my question is why two directions of polarization?

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It is possible because there are two linearly-independent solutions for wave-equation of the light traveling in the direction $\vec n$. Let $\vec n$ coincides with z axis then there are two polarizations for $\vec E$: x- and y-. Every wave of light traveling along z axis may be seen as superposition of two waves with x- and y- polarizations.

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I had a huge problem with this question last winter, and I absolutley could not come up a reasonable way of fitting in two directions of polarization. In fact, I could not fit a standing wave into a cubical cavity without having the polarization varying eveywhere. I came up with a pattern that seemed to work, but it gave me a multiplicity of 3 orientations. I am sure the pattern is right, but there is no factor of 3 in the Rayleigh-Jeans formula, so I just don't get it. Here is the pattern I came up with:

The green lines are the electric fields, which must be either perpendicular (two walls) or null (third wall) to the boundaries. The pink lines are the magnetic fields which can be parallel to the walls. You can set up this pattern, as far as I can see, in three different orientations: but in the formula for counting the modes, there is no factor of three:

I discuss this problem in more detail in my blogpost, "Counting Modes".

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