The use of microperforated plates to introduce damping in a closed cavity is examined. By placing a microperforated plate well inside the cavity instead of near a wall as traditionally done in room acoustics, high attenuation can be obtained for specific acoustic modes, compared with the lower attenuation that can be obtained in a broad frequency range with the conventional position of the plate. An analytical method for predicting the attenuation is presented. The method involves finding complex eigenvalues and eigenfunctions for the modified cavity and makes it possible to predict Green's functions. The results, which are validated experimentally, show that a microperforated plate can provide substantial attenuation of modes in a cavity. One possible application of these findings is the treatment of boiler tones in heat-exchanger cavities.