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# Resonant Cavities

We will consider a resonant cavity to be a waveguide of length with caps at both ends. As before, we must satisfy TE or TM boundary conditions on the cap surfaces, either Dirichlet in or Neumann in . In between, we expect to find harmonic standing waves instead of travelling waves.

Elementary arguments for presumed standing wave -dependence of:

 (10.82)

such that the solution has nodes or antinodes at both ends lead one to conclude that only:
 (10.83)

for are supported by the cavity. For TM modes must vanish on the caps because the nonzero field must be the only E field component sustained, hence:
 (10.84)

For TE modes must vanish as the only permitted field component is a non-zero , hence:

 (10.85)

Given these forms and the relations already derived for e.g. a rectangular cavity, one can easily find the formulae for the permitted transverse fields, e.g.:

 (10.86) (10.87)

for TM fields and
 (10.88) (10.89)

for TE fields, with determined as before for cavities.

However, now is doubly determined as a function of both and and as a function of and . The only frequencies that lead to acceptable solutions are ones where the two match, where the resonant in the direction corresponds to a permitted associated with a waveguide mode.