On the performance of a dual frequency parametric source via matched asymptotic solutions of Burgers' equation

Abstract

In this paper, following a brief review of previous contributions to the field of Parametric Acoustics, plane and spherical wave solutions of Burgers' equation are derived by the method of successive approximations for the difference-frequency signal which results from nonlinear interaction in the medium of two different-frequency primary waves radiated simultaneously by a finite-amplitude source. These solutions are then matched asymptotically obtain a farfield solution which includes the influence of nonlinear interaction within the Fresnel and Fraunhofer zones of the primary waves, from which an expression for the “equivalent difference-frequency source level” is derived. The analysis assumes that the primary-wave amplitudes, which induce the nonlinear interaction are below their respective shock thresholds as defined by a criterion which relates the steepest slope of the waveform to a predetermined upper bound approximation of an ideal discontinuity. In order to determine the predictive reliability of the solutions, a parameter which characterizes the “equivalent difference-frequency source level” is compared with empirical results obtained over a wide range of experimental conditions. Finally, a quasiplane three-dimensional equation (which in one dimension assumes the form of Burgers' equation) is also analyzed in order to obtain a second-order farfield directivity function for the difference-frequency signal.