Relationship between the diffraction constants, the reciprocity parameter, and the mutual radiation impedance of a pair of compact radiators: An improved Pritchard approximation

Abstract

By the principle of reciprocity, the mutual radiation impedance matrix of an acoustical array in a stationary fluid must be symmetric. However, use of the classical ‘‘Pritchard approximation’’ to compute array mutual (mechanical) radiation impedance (Zmut=j Re[Zself]e−jkd/kd; ejωt assumed) [R. L. Pritchard, J. Acoust. Soc. Am. 32, 730–737 (1960)] leads to a matrix which is not symmetric for nonidentical radiators. Also, it is restricted to low frequencies (ka≪1). By a simple application of the principle of reciprocity, it is shown that the mutual mechanical radiation impedance of a pair of reasonably compact radiators in an otherwise unbounded stationary fluid is expressible in terms of their (complex) diffraction constants D1 and D2 and the (complex) spherical wave reciprocity parameter Js as Zmut=S1S2D1D2/Js, where S is the radiator surface area, Jsf=2d/jρf, d is the separation between radiator acoustic centers, ρ is the fluid density, and f is the frequency. (For mutual acoustical radiation impedance, simply drop the factors S1 and S2.) For spherical radiators of radius a, D(1+n=ejkan/jkan). Note, this newly derived form for Zmut is guaranteed to be reciprocal. Also, for small ka, it reduces to S1S2/Js, which is equivalent to the classical Pritchard approximation, corrected for nonidentical radiators.