ON THE CONVECTION OF SOUND IN INVERSE CATENOIDAL NOZZLES

Abstract

A variational method is used to deduce the acoustic wave equation, satisfied by the potential, for quasi-one-dimensional propagation, in a duct of varying cross-section, containing a low Mach number mean flow; both wave equations, for the acoustic potential and velocity, are reduced to a ‘Schrödinger’ form, by using the ray approximation as a factor, in the exact solution. The latter is obtained, for the acoustic potential and velocity perturbations, both in horns (no flow) and low Mach number nozzles, of the inverse catenoidal family of ducts. The latter consists of (see Figure 1 top) the “bulged” divergent–convergent duct of profile sech, and (see Figure 1 bottom) the twin “baffles” of profile csch. The acoustic velocity perturbation, apart from one amplitude and one phase term, is common to the two cases, and is calculated in a second way, by solving a modified form of Mathieu's equation, with imaginary and hyperbolic, hence non-periodic, coefficients. These solutions are used to plot (see Figures 2–5), the amplitude (top) and phase (bottom) of the wavefields, versus distance, for several low Mach numbers, and a wide range of wavenumbers, with the compactness and the ray approximations as extremes, and intermediate values as well.