Abstract: In this paper, an analytical form is obtained for describing ultrasonic field of piston source by using entirely new "Gaussian function's polynomial". By this method, the pressure distribution is calculated along the axis and on cross section in near and far field of the circular piston. The result is in good agreement with the one predicted by Rayleigh's integral, We also demonstrate that the "function's polynomial" or the set of Gaussian function is complete to the expansion of some functions, and it indicates that all axis -symmetric sources are taken as the superposition of sets of Gaussian source. This method not only simplifies mathematical representation of the sound field but also deals with the problems such as the beam -limited sound field scattering and focusing source etc.