On sound in an inverse sinusoidal nozzle with low Mach number mean flow

Abstract

The quasi-one-dimensional propagation of sound, in a throated convergent–divergent nozzle, with entry and exit baffle at finite distance, containing a low Mach number mean flow, is studied in four steps. The low Mach number nozzle wave equation, is obtained, both for the acoustic potential and velocity, using two methods, viz. (a) elimination between the equations of motion and (b) an acoustic variational principle. For the case of inverse sinusoidal ducts, exact solutions are obtained, in terms of elementary functions, for the acoustic potential and velocity, in horns and nozzles. The reduced acoustic velocity, i.e., correction to ray approximation, is reconsidered as an exact, closed series solution, of a modified Mathieu equation, with imaginary coefficients. The plots of amplitude and phase versus distance, for various low Mach numbers, and increasing wave numbers show a number of effects, which may be interpreted as follows: (i) acoustic energy is focused in the converging duct, leading to a peak half-way between baffle and throat; (ii) a part of the sound field is reflected before the throat, leaving a weaker propagating acoustic field after. Thus although the duct is symmetric relative to the throat, the sound field is unsymmetric, due to the initial one-directional propagation, this effect being most marked for half-wavelength comparable to the length of the duct.