Other frequencies in a cavity This is a fairly basic question but is something that I've never properly understood. If you have a cavity with perfectly reflecting walls, I understand that there are obviously frequencies which generate standing waves but I'm not sure I understand why they are the only frequencies which need to considered. What happens to waves of other frequencies as they propagate in the cavity? The energy of the waves cannot just disappear, so by what mechanism are they suppressed? 
 A: Let's consider a 1D cavity with one wall at $x=0$ and the other wall at $x=L$. We know we have the wave equation for the electric potential $\phi$, $\nabla^2 \phi - \frac{1}{c^2}\partial^2_t \phi = 0$. There would be a similar one for $\vec{A}$ in three dimensions. We additionally have the boundary condition that the potential must be zero on the boundary. 
The usual standing wave solutions are like $\sin(\pi n x /L) \sin(\pi n c t /L)$. Now given any other instantaneous potential that goes to zero at the boundary, we can find its time evolution by doing a fourier decomposition to write it as a sum of plane waves. Since we know how each of these plane waves will evolve, we know the evolution of our initial potential profile by linearity. 
This applies just as well to the case of your suggestion where the initial potential profile is a localized wave packet with some frequency different from the standing mode frequency. What you will see is that the wave packet will spatially decohere as the different modes it is composed of oscillate at different rates. Eventually it will just look like a more or less random distribution of normal modes.
Notice this discussion works just as well in 3D. Since the potential must be zero at the surface. You can prove that the $\nabla^2$ operator is hermitian on the space of all instantaneous potential profiles satisfying the boundary condition. Then it can be "diagonalized"; i.e., there exists a complete set of eigenfunctions, which each have their different eigenfrequencies. As before, if you start with a localized wave packet, you will see it decohere and turn into a random-looking superposition of eigenmodes. 
Now there is the question of energy. Energy ought to be conserved because the total energy should be the energy of an eigenmode times the square of the amplitude of the mode, summed over all modes. This does not care about the relative phases of the modes. As the wave-packet decoheres, the amplitudes of the modes remain constant even though the relative phases change. Since the amplitudes of each mode remains constant, the total energy will be constant.
A: The wave touching normal to a reflector must either be an E-mode standing node or anti-node, depending upon whether the interface is a conductor or an insulator.  If you want it not to destructively add on reflection, how many half-wavelengths can fit into the gap - and the minimum number?
