Dept. of Mech. Eng., Univ. of Houston, Houston, TX 77204-4792
In J. Acoust. Soc. Am. 36, 830--832 (1964): ``The sound radiation from a slightly compressible [compact] flow can be calculated by estimating the pressure that would exist at an enclosing [spherical] surface just as if the fluid were incompressible, this [hydrodynamic] pressure then being inserted in the appropriate point-multipole expression for the sound radiation in the [surrounding] compressible fluid...the [resultant] far field sound pressure is the same as for the theory of vortex sound,'' for which the quadrupole strength contribution is the volume integral of y[sub x] ((zeta)xu)[sub x][sup ''], where u=velocity, (zeta)=(cursive beta)xu=vorticity, y=coordinate in flow, x=far-field point, ( )[sub x](identically equal to)(vector)(centered dot)x/x, and ( )'(identically equal to)(cursive beta)/(cursive beta)t( ). Now work in terms of induced velocity instead of pressure, starting with the so-called Biot--Savart law. The result is the integral of (1/2)(yx(zeta)'')[sub x] for the dipole contribution and of (1/3)y[sub x](yx(zeta)''')[sub x] for quadrupoles. These involve only the vorticity and not the velocity. These final results were first given by Mohring [J. Fluid Mech. 85, 685--691 (1978)] and later by Kambe [J. Fluid Mech. 173, 643--681 (1986)], both using more complex methods.