On the effect of wall undulations on the acoustics of ducts with flow

The acoustic wave equation for quasi-one-dimensional propagation is obtained, along a duct with a small wall sinusoidal perturbation and containing a low Mach number mean flow. The motivation is the study of the effect of wall roughness on the propagation of sound in a duct, and also of the effect of repeated reflections at periodic changes in cross-sectional area. The ray approximation, which 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 Schrِdinger form. The exact solutions are obtained, without restriction, as power series expansions around the middle of the duct; since this solution fails to converge at the two ends of the duct it is matched to the other solutions there. In this way it is possible to calculate everywhere reduced potential, (unreduced) potential, velocity and pressure perturbations. These are plotted as a function of the longitudinal co-ordinates along the duct for several values of the three dimensionless parameters in the problem, viz., (1) the relative height of wall corrugations, (2) the Mach number of the mean flow at the central section and (3) the wavenumber made dimensionless multiplying by the periodicity of corrugations.