This paper addresses characteristics of compressible flow dynamics inside a pipe with an accumulator and an inlet orifice. It also presents a simple but stable numerical method associated with the accumulator-orifice calculation. In particular, a focus is given to developing a method of finding an optimum design of the accumulator-orifice system (i.e., the accumulator size and the throttle resistance) that gives the most effective dissipation of the water-hammering problem. It is found that there exists indeed an optimum set of parameter values for the most effective dissipation of the wave energy.