Hello, I am at present reading about black-body radiation and the derivation of the Rayleigh-Jeans law. I have a few queries, and am hoping they can be answered somewhat simply.

What exactly is the mode of an oscillation?

Why do the waves in the box have to be standing, and what exactly are standing waves?

Why do the waves have to be zero at the boundaries of the black-body cavity, and what does that mean for the waves to be zero at the boundaries?

Strange, I just answered a similar question in another thread! A mode of oscillation is a fundamental way in which a molecular structure can vibrate. You can break down any possible state of vibration of a molecule down into its individual modes, provided the oscillations are not too large.

A standing wave is what you get when to add up two individual waves with the same frequency but opposite direction. So you tend to see standing waves in boxes because a wave goes from left to right then is reflected and goes back, right to left. Adding these two waves gives you a standing wave. Google standing wave and look for some animations.

The electromagnetic waves have to be zero at the boundaries because the fields have to be zero here. These are transverse waves, so we are talking about electric fields that point up/down if the wave goes right/left. If the electric field is not zero at the wall then there is some electric field pointing upwards, i.e. in the plane of the wall. There are charges in the wall and they will move under the influence of this force, and a part of the wave will be absorbed. So it's not that E has to be zero at the walls, it's just that if E is not zero then the wave will quickly decay because it's losing energy every time it hits the wall and causes some charges to move.

Once these waves gone, we are left with only standing waves with zero E field at the walls.

A simple way to model a standing wave is by moving a piece of string up/down or back/forth, making "S" patterns. One "S" is like one complete wavelength. If you were to tie the piece of string to something at one end, you would notice that you have to make certain whole, or half wavelengths to get the wave you are making to reflect back from the tied off end and thus to "oscillate". Otherwise your wave interferes with itself in irregular ways.

Each pattern is a "mode" - for example 1/2 wavelength, 1 wavelength, 1 1/2 wavelength represents three different patterns you could make on your string, and thus three different modes of oscillation.

The formula you mention is an idealised model which is dependent upon using these modes of oscillation.

All right. I have another question. I am reading about the Rayleigh-Jeans Law, given in this link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/rayj.html , and am wondering why the argument of the sine functions is $\frac{n_1 \pi x}{L}$. What does each variable represent, and why are they combined in such a way?

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