On the reflection and transmission of sound in a thick shear layer

Abstract

The propagation of sound across a shear layer of finite thickness is studied using exact solutions of the acoustic wave equation for a shear flow with hyperbolic-tangent velocity profile. The wave equation has up to four regular singularities: two corresponding to the upper and lower free streams; one corresponding to a critical layer, where the Doppler-shifted frequency vanishes if the free streams are supersonic; and a fourth singularity which is always outside the physical region of interest. In the absence of a critical layer the matching of the two solutions, around the upper and lower free streams, specifies exactly the acoustic field across the shear layer. For example, for a sound wave incident from below (i.e. upward propagation in the lower free stream), the reflected wave (i.e. downward propagating in the lower free stream) and the transmitted wave (i.e. upward propagating in upper free stream) are specified by the continuity of acoustic pressure and vertical displacement. Thus the reflection and transmission coefficients, which are generally complex, i.e. involve amplitude and phase changes, are plotted versus angle of incidence for several values of free stream Mach number, and ratio of thickness of the shear layer to the wavelength; the vortex sheet is the particular case when the latter parameter is zero. The modulus and phase of the total acoustic field are also plotted versus the coordinate transverse to the shear flow, for several values of angle of incidence, Mach number and shear layer thickness. The analysis and plots in the present paper demonstrate significant differences between sound scattering by a shear layer of finite thickness, and the limiting case of the vortex sheet.