Sound propagation in parallel sheared flows in ducts: the mode estimation problem

Abstract

We consider the inviscid, linear propagation of discrete frequency sound in an infinitely long, two-dimensional duct, possibly with soft walls, carrying a steady parallel sheared flow. Completely explicit expressions are obtained for the amplitudes of the sound modes associated with the discrete wavenumber spectrum generated upstream and downstream of a specified source distribution. By comparison with a less explicit solution for this problem given by Swinbanks (1975) we obtain an explicit expression for the group velocity of a mode in the case of rigid duct walls. Explicit expressions for the mode amplitudes can also be obtained in the case that the pressure, axial pressure gradient and transverse velocity are specified at one axial station. In the case of the problem of expanding a pressure profile in terms of modes entirely of the downstream or upstream family it is shown that in general this results in the need to solve an infinite set of linear simultaneous equations. Two such sets can be developed, suggesting th at in fact non-trivial constraints exist on the kinds of initial pressure profiles that are amenable to expansion in such a partial set of eigenfunctions. A necessary condition for the ‘optimum impedance’ problem is also derived. The entire analysis is based on a new approach to non-classical eigenvalue problems that combines the elements of residue theory and Sturm-Liouville theory and that is first illustrated by consideration of a non-classical vibrating string problem.