A simplified algorithm for the second-order sound fields

An analytical description for the second-order field of the sum-, difference-frequency and second harmonic components is derived. This treatment is based on the fact that an arbitrary axially symmetric beam can be expressed as the linear superposition of a set of Gaussian beams [J. J. Wen and M. A. Breazeale, J. Acoust. Soc. Am. 83, 1752–1756 (1988)], so that the three-dimensional integral for the second-order components is typically expressed as a combination of a set of interaction terms of the Gaussian beams. Correspondingly, the evaluation of the field distribution is reduced to the summation of exponential integral functions. From some examples, the present approach provides the results for the second-order field, which are in good agreement with those obtained directly by numerical integration.