Acoustic energy balances for sound radiated from duct exit with mean flow

An old model problem for the exchange of energy between sound field and mean flow by vortex shedding has been worked out in numerical detail. The analytically exact solution of the problem of reflection, diffraction and radiation of acoustic modes in a semi-infinite annular duct with uniform subsonic mean flow, including shedding of unsteady vorticity from the duct exit, allows a precise formulation of Myers’ energy for perturbations of an inviscid mean flow. The transmitted power (Formula presented.) in the duct and the radiated power (Formula presented.) in the far field differ by the amounts of hydrodynamic far field powers (Formula presented.) inside and (Formula presented.) outside the wake (vortex sheet) emanating from the duct edge, plus the power (Formula presented.) that disappears into the vortex sheet. This last component represents the source term in Myers’ energy equation. This is evidence of the non-conserved character of acoustic energy in mean flow, owing to the coupling of the acoustic field with the mean flow. (Formula presented.), (Formula presented.) and (Formula presented.) are always positive. This is normally the case too for (Formula presented.) and (Formula presented.). But for not too high frequencies or other circumstances where shed vorticity produces more sound than was necessary for its creation, (Formula presented.) and even (Formula presented.) may also be negative.