ON SOUND PROPAGATION IN A LINEAR SHEAR FLOW

It is shown that the acoustic wave equation in a linear shear flow always has a critical layer, where the Doppler shifted frequency vanishes; since this is the only singularity of the wave equation apart from the point at infinity, the power series solution about the critical layer has infinite radius of convergence. Two linearly independent solutions are even and odd functions of distance from the critical layer. Their plots show that acoustic oscillations are suppressed near the critical layer. A linear combination of these solutions specifies the general acoustic field; the constants of integration are determined from boundary conditions of which several examples are given. The total acoustic field is illustrated for rigid and impedance wall conditions. The cases illustrated include both real and complex wave fields. These wave fields are small amplitude perturbations of the acoustic wave equation in a linear shear flow; it is shown (in the Appendix) that the perturbations of vorticity and dilatation are coupled, and thus combine features of “acoustic” and “hydrodynamic” modes.