The sound field generated by a source distribution in a long duct carrying sheared flow

The sound field generated by a source distribution in a long duct carrying sheared flow is derived. For a source of prescribed frequency this field takes the form of a sum of modes, each with its characteristic wavenumber. Although for any given frequency these modes do not satisfy an orthogonality condition, it is still possible to obtain an explicit expression for the amplitude with which each is generated. The result is obtained by considering the Fourier transform of the solution, which amounts to solving the equation for a source distribution of fixed frequency and wavenumber. Under these circumstances, if one regards the phase velocity (ratio of frequency to wavenumber) as being the prescribed parameter, one does obtain an orthogonal set of modes. One can then express the Fourier transform as a sum of these modes, and it is found that individual terms in the expression become singular whenever the driving frequency corresponds to an eigenfrequency of the mode. When the Fourier transform is subsequently inverted, this singularity (which takes the form of a simple pole) results in the mode appearing in the solution. By evaluating the residue, the mode amplitude can easily be determined. However it is also found that the expression for the Fourier transform contains branch cuts and logarithmic singularities at values of the phase velocity equal to the local fluid velocity (the critical layer effect.) It is shown that these singularities can only affect the field in the downstream direction, and in fact correspond to hydrodynamic disturbances propagating with the fluid velocity. For most sources of a distributed nature these hydrodynamic effects tend to interfere destructively with distance from the source and do not affect the asymptotic far field. Furthermore it should be emphasized that in many practical problems one is concerned only with propagation in the upstream direction (e.g., the problem of aircraft compressor noise) and under these circumstances the field consists simply of a sum of compressible acoustic modes: the hydrodynamic effects do not arise. The results are applied to the case of a hard-walled duct with a velocity profile which is constant except for a thin boundary layer at the duct walls. First order expressions are obtained for the shape and amplitude of the modes under these circumstances. It is found that the expression for the amplitude of the lowest mode applies not only to the boundary layer problem, but is a general result which holds for low frequency wave propagation in a hard-walled duct with any velocity profile.