When a column of fluid moving with uniform velocity is instantaneously stopped at the downstream end a pressure wave is propagated upstream. In an inviscid fluid the wave is a step discontinuity, and the pressure so calculated serves as an easily obtained upper bound for all practical “water-hammer” problems, the exact solution of which may be either difficult or impossible to obtain. This paper describes an analysis of viscous dispersion in relation to the upper bound. The conclusion is reached that in problems of practical interest the bound is not significantly changed by the dispersive effects of viscosity.