ON THE ACOUSTICS OF LOW MACH NUMBER BULGED, THROATED AND BAFFLED 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 (Section 2), by using as variables either the potential or the velocity (Section 2.1);the ray approximation, that holds only for wavelengths which are short compared with the length scales of the variation of the cross-section and mean flow velocity, is used as a factor to reduce the wave equation to a Schrödinger form (Section 2.2). It is shown that the latter, reduced form, for the potential, is the most convenient for studying the acoustics of catenoidal (Section 2.3) and sinusoidal (Section 3.1) 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 (Section 3.3), whereas for the Gaussian nozzle, the sound field can be expressed in terms of Hermite functions, by using a semi-reduced form (Section 3.2) of the wave equations. The exact solutions of the nozzle wave equation (Section 3), for the four families of ducts, are plotted (Section 4) as amplitude and phase versus distance, for several combinations of frequency and low Mach number. The nozzle families considered include the catenoidal in (Figures 1, 2, 6–9) in Section 4.1, the sinusoidal (Figures 3, 10, 11) and exponential (Figures 5, 12, 13) in Section 4.2, and the first six eigenfunctions of Gaussian nozzles (Figures 4, 14–17) in Section 4.3. The various cases presented here are applications of a technique which derives the acoustics of low Mach number nozzles from that of the corresponding horns: i.e., ducts of the same cross-section but without mean flow.