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The parabolic equation (PE) acoustic model uses a Fourier spectral method of solving a second-order differential equation. The solution is ``evolved'' by solving a series of fast Fourier transforms whereby the source is convolved with the environment of the propagating energy. Each convolution marks a step of the march until the solution is marched out to the final range. This evolution has been described mathematically many times in previous published papers [Hardin and Tappert, SIAM Rev. 15, 423 (1973), Thomson and Chapman, J. Acoust. Soc. Am. 74, 1848--1854 (1983)]. A graphical evolution of the convolution of the source with the environment is important in providing a fresh perpective and further insight into the refinement and the understanding of the source function and the sensitivity to the reference sound speed. In addition, the observation of the geometry as it marches the propagation forward in wave space is an important graphic tool for understanding the analytical description of the Tappert split step PE approach.