Further Notes on the Gaussian Beam Expansion

We provide alternatively a simple way of computing the Fresnel field integral, a further extension to the Gaussian-beam expansion. With a known result that the circ function is approximately decomposed into a sum of Gaussian functions, the zero-order Bessel function of the first kind is similarly expanded by the Bessel–Fourior transform. Two expansions are together inserted in this integral, which is then expressible in terms of the simple algebraic functions. The approach is useful in treatment of the field radiation problem for a large and important group of piston sources in acoustics. As examples, the calculation results for the uniform and the simply supported piston sources are presented, in a good agreement with those evaluated by numerical integration.