On the acoustic of inverse sinusoidal and catenoidal nozzles

Abstract

The acoustic wave equation, for quasi-one-dimensional propagation, along a duct of varying cross section, containing a low Mach number mean flow, is obtained using as variables either the potential or the velocity; the ray approximation, which holds only for wavelength short compared with the lengthscales or variation of cross section and mean flow velocity, is used as a factor to reduce the wave equation to a Schrödinger form. It is shown that the latter, reduced from, for the potential, is the most convenient to study the acoustics of catenoidal and sinusoidal nozzles; it is found that these inherit respectively the filtering properties of catenoidal horns, and transparency properties of sinusoidal horns. This approach also applies to the exponential nozzle whereas, for the Gaussian nozzle, the sound field can be expressed in terms of Hermite functions, using a semi-reduced form of the wave equations. The exact solutions of the nozzle wave equation, for the four families of ducts, are plotted as amplitude and phase versus distance, for several combinations of frequency and low Mach number. The nozzle families considered include the catenoidal in the sinusoidal and exponential and the first six eigenfunctions of Gaussian nozzles. The acoustics of inverse sinusoidal and catenoidal nozzles can also be studied, without use of the preceding transformations, via the solution of modified Mathieu equation.