## 5aEA5. Vortex sound: Equivalence of ``vortex-force'' and ``vorticity-alone'' forms.

Alan Powell

Dept. of Mech. Eng., Univ. of Houston, TX 77204-4792

A direct general proof is offered of the equivalence of the source integrals of the ``vortex-force'' and ``vorticity-alone'' formulations of aerodynamic dipole and quadrupole sound in both two and three dimensions as given in the previous paper. For the dipole (with solid surfaces replaced by an image system), take the expansion of (cursive beta)(a(centered dot)b), a=y, b=((zeta)(conjunction)u), apply Helmholtz' equation (zeta)'+(del)(conjunction)((zeta)(conjunction)u)=0, then after some manipulation and reduction integrate over 2-D or 3-D space and use Kelvin's transformations; finally, differentiate w.r.t. t and take the x component. For the quadrupole, form (y(centered dot)x)(cursive beta)[sub y](centered dot)[x(a(centered dot)b)] and proceed similarly, introducing (integral)y(centered dot)((zeta)(conjunction)u)dV(y)'=(kinetic energy)'=0 in 3D but (integral)y(centered dot)((zeta)(conjunction)u)dS(y)'=(constant or 0)'=0 in 2D. Only kinematic relationships have been used apart from the foregoing hydrodynamic integral relationships for the quadrupole. Most transparently, in 2D for the dipole, if no vorticity is generated, (zeta)'+(cursive beta)(conjunction)((zeta)(conjunction)u)=D(zeta)/Dt=0; so for moving vorti-city (zeta)=(zeta)[sub 0](delta)(y-y[sub 0]), y[sub 0]=y[sub 0](t), and (zeta)[sub 0]=constant. Then the vorticity-alone form reduces directly to the vortex force form: (integral)y(conjunction)((zeta)[sub 0](delta))'' dS(y)=(integral)(y[sub 0](conjunction)(zeta)[sub 0])dS(y[sub 0])''=-(integral)((zeta)[sub 0](conjunction)u[sub 0][sup '])dS(y[sub 0]).