Journal ArticleDOI
Calculations of the development of an undular bore
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In this paper, the growth of an undular bore from a long wave is described, which forms a gentle transition between a uniform flow and still water, and a physical account of its development is followed by the results of numerical calculations.Abstract:
If a long wave of elevation travels in shallow water it steepens and forms a bore. The bore is undular if the change in surface elevation of the wave is less than 0·28 of the original depth of water. This paper describes the growth of an undular bore from a long wave which forms a gentle transition between a uniform flow and still water. A physical account of its development is followed by the results of numerical calculations. These use finite-difference approximations to the partial differential equations of motion. The equations of motion are of the same order of approximation as is necessary to derive the solitary wave. The results are in general agreement with the available experimental measurements.read more
Citations
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The Inverse scattering transform fourier analysis for nonlinear problems
TL;DR: In this article, a systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering, where the form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator.
Journal ArticleDOI
Long waves on a beach
TL;DR: In this paper, the Boussinesq equations for long waves in water of varying depth are derived for small amplitude waves, but do include non-linear terms, and solutions have been calculated numerically for a solitary wave on a beach of uniform slope, which is also derived analytically by using the linearized long-wave equations.
Journal ArticleDOI
The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations
Adrian Constantin,David Lannes +1 more
TL;DR: In this article, the authors prove that the nonlinear dispersive partial differential equations (NPDPDE) and Korteweg-de Vries (KDE) arise in the modeling of the propagation of shallow water waves over a flat bed.
Journal ArticleDOI
A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves
TL;DR: In this paper, a high-order numerical model based on the Boussinesq model was developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed.
Book
Handbook of Nonlinear Partial Differential Equations
TL;DR: In this paper, the authors present a general framework for nonlinear Equations of Mathematical Physics using a general form of the form wxy=F(x,y,w, w, wx, wy) wxy.
References
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Journal ArticleDOI
XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves
D. J. de Korteweg,G. de Vries +1 more
TL;DR: In this article, the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves were discussed, and a new model of long wave propagation was proposed.
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On cnoidal waves and bores
TL;DR: In this paper, it is suggested that, in addition to the volume flow per unit span Q and the total head R, one may usefully study a third constant S, the rate of flow of horizontal momentum(corrected for pressure force, and divided by the density).
Journal ArticleDOI
Mathematical theory of irrotational translation waves
G.H. Keulegan,G.W. Patterson +1 more
Journal ArticleDOI
Experiments on the Flow of Water from a Reservoir through an Open Horizontal Channel. II. The Formation of Hydraulic Jumps
TL;DR: In this paper, the authors studied smooth and broken undular jumps with a weir at the channel outlet providing the necessary obstruction, and the boundary between these two types of jump was at F 1 = 1·26, F 1 being the Froude number of the approaching stream.