Gaussian apodization weighting functions matched to the time delay distribution for focusing of phased arrays are introduced. The matching concept is based upon the theory of eigenmodes of freely propagating acoustic beams with a finite cross-section. The effect of this matched gaussian apodization on the beam cross-section is analysed for several acoustic phased array configurations generating acoustic wavefronts in a weakly attenuating and homogeneous medium. A strength of focusing parameter is introduced. The relation between apodization parameters and the strength of focusing is analyzed and illustrated. For strong focusing the parameters of the gaussian matched apodization become independent of the focal range. A simulator was developed which allows the study of acoustic pressure waves in the near-field and far-field generated by a linear phased array excited with short pulses. The simulator provides 2-D and 3-D representations of acoustic wave fronts generated by a pulsed phased array, illustrating the effects of apodization for pulsed acoustic beams in the time-space domain. © 1980, All rights reserved.