Sound radiation by a vibrating circular plate located at the bottom of a non-rigid flanged circular cylindrical tube

Abstract

The sound transmission through a non-rigid circular tube has been investigated in this study. For the same, the Neumann–Robin boundary value problem has been solved. The problem deals with sound radiation by considering a complex system consisting of a fluid filled in the circular cylindrical tube having a finite impedance and the half-space. The outlet of the tube is embedded into a rigid infinite flat screen. The fluid inside the tube is disturbed by a vibrating circular plate embedded at the bottom of the tube. A coupled system of partial differential equations governing the vibration of the plate and the fluid has been solved. For this purpose, the superposition of the two solutions has been applied: one is the Neumann boundary value problem in the form of the Dini series and the other is the correction for the impedance boundary in the form of a Fourier series. The continuity conditions at the tube outlet have been expressed in terms of the modal coupling acoustic impedance coefficients. The use of the radial polynomials together with Cerjan’s expansion greatly improve the numerical analysis. Application of the additional Fourier series helps to avoid finding the complex eigenvalues. Truncation conditions in the Dini series and in the Fourier series have been applied to assure valid and accurate numerical results for frequencies up to 8kHz, a tube radius from 30 to 70 mm, and a tube length from 3 to 7 times the tube radius with an arbitrary excitation of the plate. The selected tube materials range from being acoustically soft, such as mineral wool, to acoustically hard, such as a porous material. The other advantage of using the Dini and the Fourier series is that they require only real eigenvalues, and thus it is not necessary to find the cumbersome complex eigenvalues of the tube. One more advantage of the rigorous solution is that it automatically satisfies the radiation condition in the half-space, whereas the finite elements method always satisfies it approximately. The findings of this study can be summarized as follows: The main computational cost is associated with a relatively significant number of the Fourier expansion terms. As a result, for frequencies smaller than 1.2kHz, the first result can be obtained rapidly using the finite elements method. In addition, it has been shown that using the radial polynomials to calculate the coupling integrals enables faster calculations by at least 120 times.