Abstract: This paper investigates the principles and practices of the foundamental analytical method. for computing water hammer pressures, namely, method by superposition of elastic waves. This method is lisle known or adopted in practical work due to the fact that though its principles are simple and direct, the required computing work is tedious and difficult, because the number of waves generated at the operating gate and propagated 11I the pipe line system increases indefinitely as they reflect and divide at each joints, except for the most simple case. The purposes of this paper are: (1) Describe the foundamental principles of this method and derive useful formulas in more complete form; (2) Present general working procedure. and convenient computing tables to simplify the numerical work and bring the method more adoptable for practical use. Typical numerical examples are given to show the procedure of computation in detail and finally some conclusions about this method are discussed. Analysis of water hammer pressures by using another important method "the conjugate-equation method" either graphycally or arithmetically has also been studied by the writer and the results will be discussed in an another paper.