On localized trapped modes in a pipe-cavity system (Mathematical Aspects and Applications of Nonlinear Wave Phenomena)

Abstract

The work is concerned with an analytical study of acoustic trapped modes in a cyhndrical expansion chamber, placed in between two semi-infinite pipes (waveguides). 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 wavenumber k, correspond to trapped modes. In the case of a shallow cavity, in the sense that the cavity radius is only slightly larger than the pipe radius, asymptotic approximations for the coeficients of the determinant can be apphed. The determinant then reduces to a simple form with four-rowed minors placed on a diagonal, enabling analytical evaluation and a proof of existence of trapped modes. We consider here circumferential mode numbers mgeq 1. For a shallow cavity and for low values of the circumferential mode number there is just one trapped mode in the allowable wave number domain k_{min}<k<k_{mathrm{m}mathrm{w}}, where k_{min} is the cutoff frequency for acoustic waves in the cavity and k_{max} is the corresponding cutoff frequency in the pipes. This mode is symmetric about a radial axis in the center of the cavity.