The exact solution, in infinite series form, is given to the problem of the transmission of an axially symmetric sound wave through an ideal stretched membrane in a rigid circular tube. Certain simple exact results are deduced, and the exact equations for the resonance and antiresonance frequencies are considered. It is shown that, for an incident plane wave, above a certain frequency there are no exact resonances or antiresonances. Below this frequency, the resonances and antiresonances differ from those predicted by approximate methods, and, contrary to the simple results, their frequencies depend on the density of the medium surrounding the membrane. The same analysis, which involves the inversion of an infinite matrix, applies also to the parallel‐plate guide and to transmission through a thin elastic plate. Numerical results are given for a particular case.