Author(s): J. D. Fenton
Linked Author(s): John D. Fenton
Keywords: Wave propagation; Low-inertia approximation; Kinematic wave; Numerical methods; Stability
Abstract: The long wave equations for waves in rivers and canals are considered and some results obtained that contradict current understanding and practice. Some generalisations of traditional coefficients are described, including a simple approximation for non-prismatic channels. Criticism is made of traditional reliance on the Gauckler-Manning equation. The method of characteristics is also criticised for providingmisleading insight into the nature of the equations and the waves they describe. It is shown by linearising the equations that long waves have propagation characteristicsthatdepend on wave period, so that the behaviour is more complicated than often believed. Traditional methods of nondimensionalising the equations also give a misleading picture of them. Terms that have been previously believed to be inertial terms, of the magnitude of the Froude number squared, are in fact of the magnitude of the time rate of change of boundary conditions such as the inflow hydrograph. Accordingly, the nomenclature and application of some well-known approximations are criticised. Considering computational methods, the simplest forward-timecentral-space finite difference scheme is shown to be more stablethan widely believed, and can be used to develop simple simulations. Finally, the problem of an open downstream boundary is considered and a good treatment shown to be to simply treat the end point as if it were an ordinary point in the stream and numerically solve the equations there also. This is in opposition to current theoretical understanding, but it seems to work well.