The behavior of the longitudinal wake fields excited by a very short bunch in the SLAC S-band constant gradient accelerating structures has been studied. Wake potential calculations were performed for a bunch length of 10 microns using the author's code to obtain a numerical solution of Maxwell's equations in the time domain. We have calculated six accelerating sections in the series (60-ft) to find the stationary solution. While analyzing the computational results we have found a new formula for the Green's function. Wake potentials, which are calculated using this Green's function are in amazingly good agreement with numerical results over a wide range of bunch lengths. The Green's function simplifies the wake potential calculations and can be easily incorporated into the tracking codes. This is very useful for beam dynamics studies of the linear accelerators of LCLS and FACET. Show Partial Abstract