Role of dimensionless numbers in wave analysis

Abstract Dimensionless numbers, including the Reynolds number R e , the wave number $\hat{\sigma}$ , the Froude number F o , the quasi‐shallow wave number $\hat{\sigma}_{\rm v}$ , the kinematic wave number K , Vedernikov number V , and others, are frequently encountered in analytical and numerical analyses of waves occurring in frictional and frictionless channels. Depending on their derivation, these numbers can be classified into physical, geometrical, derived, input‐based numbers, and their combinations. It can be shown that (a) the waves under friction are primarily governed by R e , pointing to friction laws and, consequently, the rating exponent; (b) the wave number characterizes the wave type for a given R e ; (c) the numbers describing the reference state of flow do not have the efficacy of characterizing a wave type on its own, e.g. F o that describes the significance of inertial forces in wave motion; (d) coupling of F o with $\hat{\sigma}$ produces a more efficacious number K ; (e) Vedernikov number V , a combination of the rating exponent or, in turn, R e and F o , is more powerful than either R e or F o ; and (f) the numbers emanating from analytical solutions describe the wave types, whereas those emanating from numerical analyses usually prescribe the limits for stable solutions, such as Courant number C r and cell Reynolds number D . The significance of these numbers is described using the case of evolution and propagation of a dam‐break flood wave. Furthermore, it is shown that as $\hat{\sigma}$ forms the basic unit of shallow wave description, then quasi‐shallow waves are described by the quasi‐shallow wave number $\hat{\sigma}_{\rm v}$ , the wave length parameter normalized by the normal depth of flow. Copyright © 2002 John Wiley & Sons, Ltd.