Some extensions of the Gaussian beam expansion: Radiation fields of the rectangular and the elliptical transducer

A straightforward extension of Gaussian beam expansion is presented for calculation of the Fresnel field integral [J. J. Wen and M. A. Breazeale, J. Acoust. Soc. Am. 83, 1752-1756 (1988)]. The source distribution function is expanded into the superposition of a series of two-dimensional Gaussian functions. The corresponding radiation field is expressed as the superposition of these two-dimensional Gaussian beams and is then reduced to the computation of these simple functions. This treatment overcomes the limit that the shape of source is of circular axial-symmetry. The numerical examples are presented for the field of the (uniform) elliptical and the rectangular piston transducers and agree well with the results given by complicated computation.