An investigation is conducted on the acoustic resonance that develops in rectangular and cylindrical cavities with rigid walls. An analytical model for the small amplitude acoustic perturbations inside an enclosure with rigid walls is developed from classical linerized acoustics. The method is first applied to a rectangular geometry and the normalized mode shapes and frequencies of the first six standing wave modes are given. The results are used to diagnose whether coupling is likely to occur between the first two Rossiter modes and the acoustic standing waves, which may lead to a reinforcement of the flow instability. At the selected test conditions, the method indicates that the second Rossiter mode can couple with the first longitudinal acoustic mode. The acoustic resonant mode predictions for a cylindrical cavity of length to depth ratios 0.71 and 2.5, tested at near-incompressible speeds, suggest that the acoustic resonant modes and the main fluid dynamic instability in the enclosure are sufficiently apart in frequency not to strongly interact with one another. The parametrized analytical solutions developed in this study enable the aero-acoustic engineer to diagnose whether coupling between a given fluid dynamic instability and acoustic resonance is likely to affect a rectangular or cylindrical cavity component.