The method of wave derivatives

Abstract

Workers on the water‐level problem have been hampered by the lack of a method of computing two‐dimensional time‐dependent flow of water. The equations of two‐dimensional time‐dependent flow of an incompressible fluid with a free surface are reformulated in this paper. This formulation, made up of linear combinations of the equations of motion and continuity, results in relations similar to the followingequation imagewhere D1/dt = ∂/∂t + (u + c)∂/∂x + (v + c)∂/∂y and A1 = u + v+ c. Similar relationships for non‐homogeneous equations are described.It is demonstrated that D1A1/dt + D2A2/dt = 0 means that the value of A1 from one point plus the value of A2 at another point at time t is the same as A1 + A2 a t still another point at time t + dt. A computing method, based on motion, addition, and subtraction of isopleths of quantities similar to A1, is described. A name, ‘the method of wave derivatives,’ is suggested for this method, and it is applied to the motion of a mound of water in a flat‐bottom sea.