Alan Powell
Dept. of Mech. Eng., Univ. of Houston, TX 77204-4792
In the contiguous method [J. Acoust. Soc. Am. 36, 830--832 (1964)] an incompressible inviscid flow causes pressure or velocity perturbations, p[sub inc] or u, on geometrically distant surface (cylinder) that drives the contiguous external acoustic field. Solid surfaces are replaced by some appropriate vortex image system. The far field sound pressures are found to be p(x)=[1/[radical (8(pi))[radical ] [((rho)(omega)[sup 1/2])/(x[sup 1/2]c[sup 1/2])]I[sub D][sup (star)] for dipole sound, with c[sup -3/2]I[sub Q][sup (star)] for quadrupoles, (rho)=density, c=sound speed, I=sinusoidal source term (or its Fourier component or transform) of frequency (omega), f(t)[sup (star)](identically equal to)f(t-x/c). In terms of vortex force, -(rho)((zeta)(conjunction)u), where (zeta)=(cursive beta)(conjunction)u=vorticity, (del)[sup 2](p[sub inc]+(rho)u[sup 2]/2)=-(rho)(cursive beta)(centered dot)((zeta)(conjunction)u) leads to I[sub D]=-(integral)((zeta)(conjunction)u)[sub x][sup '] dS(y), with f(t)'(identically equal to)(cursive beta)f/(cursive beta)t and I[sub Q]=-(integral)y[sub x]((zeta)(conjunction)u)[sub x][sup ''] dS(y). For vorticity alone, use u=(cursive beta)(conjunction)B, where (del)[sup 2]B=-(zeta), to get I[sub D]=(integral)(y(conjunction)(zeta),'')[sub x] dS(y) and I[sub Q]=(integral)(1/2)y[sub x](y(conjunction)(zeta)''') [sub x] dS(y). (In 3D, constants in the integrals are 1, 1 and 1/2, 1/3, respectively.) The implications of these all being integral results are discussed; while the vortex force -(rho)((zeta)(conjunction)u) is a local flow property, (y(conjunction)(zeta)) is not.