Acoustic Impedance in Terms of Energy Functions and the Nonlinear Reactance of Orifices

Abstract

Acoustical systems which are driven at a single terminal pair are considered. The acoustical driving-point impedance Z of such a system is the ratio of the pressure to the volume velocity at the driving terminal. If the system comprises a limited region R bounded by a surface S, Z can be expressed in terms of the volume integrals over R of the dissipation function, and the average lagrange density in addition to the surface integral of the complex power flowing across S. The acoustic impedance of a circular orifice, small compared with the wavelength, is evaluated by the methods discussed above. It is found that about 60 percent of the contribution to the mass reactance of a thin orifice arises from the average kinetic energy contained within the region bounded by two hemispherical caps which have radii equal to the radius of the orifice and which are placed concentrically on opposite sides of the orifice. At sound levels so great that the particle displacement amplitude through the orifice exceeds the radius of the orifice, the flow through the orifice may become jetlike in such a manner that vortex rings which form within the hemispherical caps detach themselves periodically. The coherence of the motion within the caps is destroyed by this jet-flow, so that the average kinetic energy in the caps no longer contributes to the reactance of the orifice. Measurements of the sound field in the neighborhood of the orifice indicate that the configuration of the field outside the jet region is essentially unaffected by the local jet action, thus the kinetic energy in this region will contribute to the mass reactance. The 60 percent reduction in mass reactance predicted by this theory will be compared with recent measurements performed by Ingard.