  We consider a one-dimensional membrane-in-the-middle model for a cavity that
consists of two fixed, perfect mirrors and a mobile dielectric membrane between
them that has a constant electric susceptibility. We present a sequence of
exact cavity angular frequencies that we call {\it structural angular
frequencies} and that have the remarkable property that they are independent of
the position of the membrane inside the cavity. Furthermore, the case of a thin
membrane is considered and simple, approximate, and accurate formulae for the
angular frequencies and for the modes of the cavity are obtained. Finally, the
cavity electromagnetic potential is numerically calculated and it is found that
a multiple scales, analytic solution is an accurate approximation.
