On the fundamental acoustic mode in variable area,low Mach number nozzles

We consider the propagation of sound in nozzles of substantially varying cross-section, taking into account reflections from the tapering walls and non-uniform convection by the axially accelerated or decelerated mean flow. Exact solutions of the acoustic equations are obtained for the simplest case of the fundamental longitudinal mode in quasi-one-dimensional, low Mach number nozzle flow. The effects of non-uniform mean flow on sound are demonstrated by comparing horns with nozzles, and 15 properties (stated in italics and labelled P1, P2 ... P15 in the text) are proved and discussed, viz. concerning (1) relations between acoustic pressure and velocity; (2) equipartition or biasing of kinetic and compression energies; (3) conservation or evolution of wave action; (4) existence of elementary or special solutions and (5) cut-off frequencies, and amplitude and phase laws. We conclude with a simple formula for the approximate calculation of the acoustic fields; the formula can be made more accurate by using a correction factor for amplitude and phase, which is plotted against area ratio for propagation upstream or downstream in diffusers and convergent nozzles.