Non-paraxial model for a parametric acoustic array.

This study is concerned with parametric radiation from an arbitrary axisymmetric planar source with a special focus on low-frequency difference-frequency fields. As a model equation accounting for nonlinearity, diffraction, and dissipation, the Westervelt equation is used. The difference-frequency-field patterns are calculated in the quasi-linear approximation by the method of successive approximations. A multi-layer integral for calculation of the acoustic field is reduced to a three-dimensional one by employing an approximate analytical description of the primary field with the use of a multi-Gaussian beam expansion. This integral is subsequently reduced in the paraxial approximation to a one-dimensional form which has previously been published in literature and which represents a means for fast calculations of secondary acoustic fields. The three-dimensional integral is calculated numerically and the numerical results predict nonzero amplitude of the low-frequency field in the vicinity of the source which is an effect that cannot be correctly encompassed in the paraxial approximation.