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.

This work presents a versatile computational approach for the numerical modelling of a parametrically generated low-frequency sound. The proposed method is based on the quasi-linear approximation, and it does not employ the paraxial approximation. The primary acoustic field is calculated by the Rayleigh integral or the boundary element method; the secondary difference-frequency field is calculated by the finite element method. As governing wave equations, a general second-order wave equation for acoustic pressure, the Westervelt equation, and Kuznetsov equation are tested, and the corresponding numerical results are compared. The proposed approach allows studying the near-field, far-field, as well as the off-axis field of the difference-frequency wave parametrically radiated from complex emitters. As numerical examples, parametric radiation from a baffled piston and a piston combined with a horn are examined.