Nonlinear Attenuation and the Parametric Array

Abstract

The average intensity of a finite-amplitude wave satisfies a simply expressed energy conservation law [P. J. Westervelt, in L. Cremer, Ed., Proc. Int. Cong. Acoust., 3rd (Elsevier, Amsterdam, 1960), p. 316] in terms of the coefficient of small-amplitude attenuation, the first-order pressure, and the virtual source density. This equation when applied to a plane wave yields an equation for the attenuation of the pressure amplitude, in terms not only of the small amplitude attenuation coefficient, but also the shock formation distance. This latter equation exhibits a smooth transition through the region of shock formation. An approximation applicable to the end-fire array can be developed, and from this a compatible end-fire virtual source distribution. This source distribution results in a directional pattern modified from the low-amplitude Rutherford law [P. J. Westervelt, J. Acoust. Soc. Amer. 35, 535 (1963)] by a multiplicative factor, which is expressible as a hypergeometric series. This series can be summed exactly on axis, and a simple closed-form expression giving the reduction in sound pressure level due to saturation is found. These results, obtained independently by each of us, are to be published, flint in a Letter to the Editor by J. F. B. accepted by J. Acoust. Soc. Amer. and later in greater detail in a joint paper by both of us.