Appendix: leading order compressible equations

The compressible nature of the source terms in Lighthill’s acoustic analogy can be closed. For weakly compressible flows, in the absence of thermoacoustic effects, the compressibility of the source field is known in terms of solenoidal modes of the vortical flow field. In such flows, the square of the fluctuating Mach number is small and this fact, coupled with the singular nature of the acoustic problem, and the fact that the phase speed of the acoustic sources is the advective speed, is used to formally close the compressible portion of the fluctuating Reynolds stresses. The closure resolves, as expressed in Crow’s 1970 paper, the inconsistent incompressible approximation to Lighthill’s source term. It is shown that the incompressible approximation to Lighthill’s source term, accurate to order of the square of the Mach number, predicts an acoustic field accurate to order Mach number. This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001.