Refraction of acoustic duct waveguide modes by exhaust jets

Abstract

The refraction of acoustic duct waveguide modes emitted from the open end of a semi-infinite rectangular duct by a jet-like exhaust flow is studied theoretically. The problem is formulated as a Wiener-Hopf problem and is ultimately solved by an approximate method due to Carrier and Koiter. Continuity of transverse acoustic particle displacement and of acoustic pressure is assumed at the jet/still-air interface. The solution exhibits several features of the acoustics of moving media such as a source convection effect, zones of relative silence, simple refraction, etc. Plots of far-field directivity patterns are presented for several cases and show refraction effects to be important even at modest exhaust Mach numbers of order 0.3. Only subsonic exhaust Mach numbers are considered. In view of the problem’s technological interest, the solution for the far-field directivity is written out in full detail in the appendices. In the low-frequency limit when only one duct waveguide mode (the plane wave mode) propagates, we also examine the reflection coefficient. It is found that this reflection coefficient, in general, considerably exceeds either the no-flow value or the value for the case with uniform flow both inside and outside the waveguide (i.e., in the whole space). Apparently the acoustic medium mismatch enhances the geometric mismatch in the jet flow case, thus producing a higher reflection coefficient.