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Nonlinear wave run-up in bays of arbitrary cross-section: generalization of the Carrier–Greenspan approach

Published online by Cambridge University Press:  30 April 2014

Alexei Rybkin
Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK 99775, USA
Efim Pelinovsky
Department of Nonlinear Geophysical Processes, Institute of Applied Physics, Uljanov Street 46, 603950, Nizhny Novgorod, Russia Institute for Analysis, J. Kepler University, Altenbergerstr. 69, 4040 Linz, Austria National Research University Higher School of Economics, Gr. Pechersky Street 25/12, 603950, Nizhny Novgorod, Russia Nizhny Novgorod State Technical University n.a. R. E. Alekseev, Minin Street 24, 603950, Nizhny Novgorod, Russia
Ira Didenkulova*
Nizhny Novgorod State Technical University n.a. R. E. Alekseev, Minin Street 24, 603950, Nizhny Novgorod, Russia Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618, Tallinn, Estonia MARUM – Center for Marine Environmental Sciences, University of Bremen, Leobener Str. D-28359 Bremen, Germany
Email address for correspondence:


We present an exact analytical solution of the nonlinear shallow water theory for wave run-up in inclined channels of arbitrary cross-section, which generalizes previous studies on wave run-up for a plane beach and channels of parabolic cross-section. The solution is found using a hodograph-type transform, which extends the well-known Carrier–Greenspan transform for wave run-up on a plane beach. As a result, the nonlinear shallow water equations are reduced to a single one-dimensional linear wave equation for an auxiliary function and all physical variables can be expressed in terms of this function by purely algebraic formulas. In the special case of a U-shaped channel this equation coincides with a spherically symmetric wave equation in space, whose dimension is defined by the channel cross-section and can be fractional. As an example, the run-up of a sinusoidal wave on a beach is considered for channels of several different cross-sections and the influence of the cross-section on wave run-up characteristics is studied.

© 2014 Cambridge University Press 

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