Characterization of acoustic disturbances in linearly sheared flows
Inviscid fluctuations in a compressible linearly sheared, but otherwise homogeneous, parallel two-dimensional flow are considered. The equation describing the plane wave propagation (PWP), the stability, or the rectangular duct mode characteristics in such a flow is shown to be reducible to Whittaker's equation. The exact solutions are applied to problems of PWP and stability in linearly sheared flows as limiting cases in which the speed of sound goes to infinity (incompressible limit) or the shear layer thickness, or wave number, goes to zero (vortex sheet limit). With respect to the PWP it is shown that the shear layer possesses no resonances and no Brewster angles, while with regard to the problem of the stability of a finite thickness shear layer with a linear velocity profile, it is shown that the thin layer is unstable to long wavelength disturbances for all Mach numbers.