A Useful Analytical Model for the Parametric Acoustic Array

Abstract

In 1963, Westervelt presented a simple analytical model for the parametric acoustic array. Making a number of simplifying assumptions, among them that nonlinear attenuation is negligible, Westervelt arrives at the following: (1) quadratic dependence of secondary level on primary level and (2) Rutherford scattering angle dependence, independent of primary level. Now, experiments show that: (1) the quadratic level dependence goes over to a linear dependence at high levels and (2) the difference frequency beamwidth broadens with increasing level. These effects are not predicted by Westervelt's model. In this letter, it is shown that it is easy to substitute the attenuation of a shocked sound wave in place of the simple linear attenuation used by Westervelt. The resulting Green's function integral is no longer elementary, but has a simply expressed hypergeometric series solution. On axis, the series sums to a simple closed form solution, correctly predicting the transfer from the quadratic to linear level dependence. Off axis, the series apparently does not sum to closed form, but is readily evaluated with a digital computer. The results agree rather well with experiment.