In a cavity of size $L$, the wave must give zero amplitude at the wall, means wave equation has zero amplitude. Why? Answer from hyperphysics "since a non-zero value would dissipate energy and violate our supposition of equilibrium. To form a standing wave, the reflection path around the cavity must produce a closed path."

  1. what is supposition of equilibrium?

  2. in 3D, how do we know $$E=E_0\sin\frac{n_1\pi x}{L}\sin\frac{n_2\pi y}{L}\sin\frac{n_3\pi z}{L}\sin\frac{2\pi ct}{\lambda}$$

from link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/rayj.html


In a cavity of dimension L, the wave must give zero amplitude at the wall, means wave equation has zero amplitude. Why?

The cavity is supposed to be made of metal with low resistance. For frequencies low enough, the in-plane component of the electric field near the wall will be very small, because the metal surface carries electric currents that cancel the external electric field. Metals behave this way - for large electric currents in the metal only weak electric field is necessary, and in the limit of perfect conductor the electric field necessary to drive the currents vanishes.

what is supposition of equilibrium?

The EM field inside the cavity is in mutual thermodynamic equilibrium with small piece of matter in the cavity. The two have same temperature and their macroscopic properties do not change in time.

in 3D, how do we know E = ...

The formula is written incorrectly. One has to state how the cavity looks like and which component is meant. For example, the $x$ component of electric field of mode $n_x,n_y,n_z$ in cubic metallic cavity $xyz\in (0;L)\times(0;L)\times(0;L)$  is

$$ E_x(x,y,z,t) = E_0 \cos \frac{\pi n_x x}{L} \sin \frac{\pi n_y y}{L} \sin \frac{\pi n_z z}{L} \cos \omega t. $$ The $y$ component: $$ E_y(x,y,z,t) = E_0 \sin \frac{\pi n_x x}{L} \cos \frac{\pi n_y y}{L} \sin \frac{\pi n_z z}{L} \cos \omega t. $$ $$ \omega = \frac{\pi c}{L}\sqrt{n_x^2 + n_y^2 + n_z^2}. $$ One can show that this satisfies above boundary conditions (in-plane component of $E$ is zero at the wall) and Maxwell's equations.

For beginner, I recommend studying some good textbooks of physics instead of websites first, there are some good ones, but most of them perpetrate various misconceptions and contain mistakes.


In a cavity, you have only standing waves no? This is the only way to satisfy all the boundary conditions at the same time.

The assumption of equilibrium refers more to the fact that all the modes compatible with the dispersion relation are allowed in principle.

Most of the reasoning is essentially the same as that for for an ideal gas treated in the quantum case.

For the energy, I guess a simple idea is to make an analogy with acoustic standing sound waves in a cavity that is used for the specific heat of crystals and imagine that as for sound waves, there is $k_B T$ per mode and the rest follows.


The boundary condition is not exactly that the wave amplitude goes to zero at the boundary. It's that the component of the electric field parallel to the surface goes to zero. You are definitely allowed to have a perpendicular component.

It's not so hard to see how to make this work in two dimensions, which also gives you solutions for a three-dimensional waveguide. It's quite a bit harder to see how it works for a three-dimensional cavity. I struggled with this question for quite a while until I came up with this picture:enter image description here

(The green lines are electric fields and the purple lines are magnetic fields.) What was even harder for me was to figure out how this picture reconciled with the equation quoted above by the OP. I think I finally worked it out.

I discuss this question on in this blogpost:http://marty-green.blogspot.ca/2013/01/in-which-i-figure-out-how-to-count-modes.html


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