Modified Transformation and Integration of 1D Wave Equations

This technical note introduces an alternative method of transforming hyperbolic partial-differential equations into characteristic form. The method is based on transforming the governing equations to a reference frame moving with finite speed u . Thus, the method is analogous to the “moving observers” used traditionally in graphical water-hammer theory to solve the equations of motion [e.g., Parmakian (1963) and Bergeron (1961)] or to the method of deriving simplified governing equations by using a translating reference frame [e.g., Henderson (1966)]. The difference in the present case is that although the governing equations are assumed to be known, they are transformed into characteristic form by a shift in reference frame. In essence, the transformation uses the total derivative concept and is both simple and insightful. In fact, for both open-channel flow and water-hammer applications, it is shown that by transforming only the continuity equation along a characteristic curve, the dynamic equation naturally arises during the transformation. A mathematical justification and generalization of the proposed method is provided.