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J Acoust Soc Am. 2013 Aug;134(2):933-8. doi: 10.1121/1.4813223.

Non-paraxial model for a parametric acoustic array.

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Czech Technical University in Prague, Faculty of Electrical Engineering, Technicka 2, 166 27 Prague 6, Czech Republic.


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.

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