On the Formulation of Mass, Momentum and Energy Conservation in the KdV Equation
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On the Formulation of Mass, Momentum and Energy Conservation in the KdV Equation Alfatih Ali · Henrik Kalisch
Received: 4 December 2012 / Accepted: 25 November 2013 © The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present article focuses on the formulation of mass, momentum and energy balance laws in the context of the KdV approximation. The densities and the associated fluxes appearing in these balance laws are given in terms of the principal unknown variable η representing the deflection of the free surface from rest position. The formulae are validated by comparison with previous work on the steady KdV equation. In particular, the mass flux, total head and momentum flux in the current context are compared to the quantities Q, R and S used in the work of Benjamin and Lighthill (Proc. R. Soc. Lond. A 224:448–460, 1954) on cnoidal waves and undular bores. Keywords KdV equation · Surface waves · Mechanical balance laws · Energy conservation · Hydraulic head
1 Introduction The Korteweg-de Vries (KdV) equation is a model equation describing the evolution of long waves at the surface of a body of fluid. The KdV equation was derived in 1895 by Korteweg and de Vries [18], but was already featured in earlier work by Boussinesq [7]. The main assumptions on the waves to be represented by solutions of the KdV equation are that they be of small amplitude and long wavelength when compared to the undisturbed depth of the fluid, that the wave motion be predominantly one-directional, and that transverse effects be
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A. Ali · H. Kalisch ( ) Department of Mathematics, University of Bergen, P.O. box 7800, 5020 Bergen, Norway e-mail: [email protected] A. Ali e-mail: [email protected]
A. Ali, H. Kalisch
weak. In dimensional variables, the KdV equation is given by ηt + c0 ηx +
3 c0 c0 h20 ηxxx = 0, ηηx + 2 h0 6
(1.1)
where η(x, t) represents the excursion of the free surface, h0 is the undisturbed water depth, √ g denotes the gravitational acceleration, and c0 = gh0 is the limiting long-wave speed. The equation arises in the so-called Boussinesq scaling regime where wavelength and wave amplitude are balanced in such a way as to allow the formation of traveling-wave solutions. Denoting by a typical wavelength and by a a typical amplitude of the wavefield to be described, the number α = a/ h0 represents the relative amplitude, and β = h20 /2 measures the relative wavenumber. The waves fall into the Boussinesq regime if both α and β are small, and of similar size. In this case, the KdV-equation arises as a simplified asymptotic model describing the wavemotion. In other words, solutions of the full waterwave problem may be ap
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