On the acoustics of an exponential boundary layer

A brief derivation is given of the acoustic wave equation describing the propagation of sound in a unidirectional shear flow. This equation has been solved exactly in only one instance, namely a linear velocity profile; in the present paper a second exact solution is given, for the exponential velocity profile, which represents a boundary layer with weak suction at a high Reynolds number. The acoustic wave equation has a critical layer where the Doppler shifted frequency vanishes, and this corresponds to a regular singularity; another regular singularity corresponds to the free stream and the sound field consists either of propagating waves or of surface waves, showing that the critical; layer can act as an absorbing layer. Analytical continuation is used to cover the whole flow region, from the wall boundary layer to the free stream; the appropriate :boundary, radiation and stability conditions are discussed, and the acoustic pressure is plotted as a function of distance from the wall for several combinations of frequency, wavenumber parallel to the wall and low Mach number free-stream velocity; the combination of solutions appropriate to rigid and impedance walls is also plotted. The solutions are expressible in terms of Bessel functions only when the critical layer is in the free stream; when the critical layer is in the boundary layer, or when there is no critical layer, the solutions require an extension of the Gaussian hypergeometric equation, in which one of the three singularities is irregular; its solutions are extensions of Gaussian hypergeometric and Mathieu functions, whose general properties are discussed elsewhere.