On longitudinal acoustic propagation in convergent and divergent nozzle flows

The longitudinal propagation of sound in quasi-one-dimensional low Mach number nozzle flow is considered in section 1. The solution in the ray approximation (section 2.1) is used to transform the wave equation into a Schrödinger form, which is studied for the family of power-law ducts, including, as a particular case, the conical nozzle. It is shown that the coincidence of flow sources/sinks with sound sources can lead to appearance of essential singularities (section 2.2), which can be removed by using a Riccati transformation (section 2.4). The exact solutions of the acoustic equations for the parabolic (Figure 1) and hyperbolic (Figure 2) nozzles are obtained in terms of Bessel functions (section 2.3), respectively of complex order and argument. The general formulas, together with limiting forms in the compactness, ray and asymptotic approximations (section 3.1), are used to establish properties of the acoustic velocity and pressure (section 3.2), kinetic and compression energies, and energy flux and wave action (section 3.3); for example, it is shown that the equipartition of energies for moderate variations in cross-section, gives way (Table 2) to a predominance of kinetic/compression energies respectively near blockages/openings. The effects of non-uniform mean flow (Table 1) are discussed by comparing horns with nozzles (section 3.4): e.g., it is shown that the duality principle, in three alternative forms, does not extend from horn to nozzles, and the acoustic equations have no elementary solutions for the latter, in contrast with the former.