On the acoustic trapped modes and their symmetry properties in a circular cylindrical waveguide with a cavity

The paper is concerned with a partly analytical, partly numerical study of acoustic trapped modes in a cylindrical cavity (expansion chamber), placed in between two semi-infinite pipes acting as a waveguide. Trapped mode solutions are expressed in terms of Fourier–Bessel series, with the expansion coefficients determined from a determinant condition. The roots of the determinant, expressed in terms of the real wavenumber k, correspond to trapped modes. For a shallow cavity and for low values of the circumferential mode number it is found that there is just one trapped mode in the allowable wave number domain, and this mode is symmetric about a radial axis in the center of the cavity. As the circumferential mode number is increased, more and more trapped modes, placed between two cutoff frequencies, come into play, and they alternate between symmetric and antisymmetric modes. An analytical explanation of the mechanism behind the mode increasing and mode alternation is given via asymptotic expressions of the determinant condition. Numerical computations are done for verification of the analytical results and for consideration of less shallow cavities. Also for these cases, similar phenomena of an increasing number of trapped modes, and alternation between symmetric and antisymmetric modes, are found.