Sound transmission through annular cylindrical apertures using rapidly converging expansion series

The problem of sound transmission through an annular aperture with perfectly rigid walls and located in a hard infinite baffle was solved analytically. The three separated regions with suitable boundary conditions were considered. There are two regions for the upper and the lower half spaces and one for the aperture. The solutions to the Helmholtz equation in each region and the continuity conditions were satisfied. The rigorous formulations describing the acoustic pressure and the transmitted and scattered acoustic powers were obtained. The obtained solutions resulted in the derivation of the modal coupling coefficients in the form of infinite integrals. The spectral mapping technique based on the Zernike circle polynomials was adopted for the annular geometry. The modal coupling impedances were presented in the form of a rapidly convergent expansion series. Finally, high accuracy and time efficiency were achieved. The problem of sound scattering and transmission through a circular aperture could be generalized for the annular aperture.