Water waves of finite amplitude on a sloping beach
TL;DR: In this paper, the authors investigated the behavior of a wave as it climbs a sloping beach and obtained explicit solutions of the equations of the non-linear inviscid shallow-water theory for several physically interesting wave-forms.
Abstract: In this paper, we investigate the behaviour of a wave as it climbs a sloping beach. Explicit solutions of the equations of the non-linear inviscid shallow-water theory are obtained for several physically interesting wave-forms. In particular it is shown that waves can climb a sloping beach without breaking. Formulae for the motions of the instantaneous shoreline as well as the time histories of specific wave-forms are presented.
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TL;DR: In this article, an approximate theory is presented for non-breaking waves and an asymptotic result is derived for the maximum runup of solitary waves on plane beaches, and a series of laboratory experiments is described to support the theory.
Abstract: This is a study of the runup of solitary waves on plane beaches. An approximate theory is presented for non-breaking waves and an asymptotic result is derived for the maximum runup of solitary waves. A series of laboratory experiments is described to support the theory. It is shown that the linear theory predicts the maximum runup satisfactorily, and that the nonlinear theory describes the climb of solitary waves equally well. Different runup regimes are found to exist for the runup of breaking and non-breaking waves. A breaking criterion is derived for determining whether a solitary wave will break as it climbs up a sloping beach, and a different criterion is shown to apply for determining whether a wave will break during rundown. These results are used to explain some of the existing empirical runup relationships.
866Â citations
TL;DR: In this article, an extended Boussinesq model for surf zone hydrodynamics in two horizontal dimensions is implemented and verified using an eddy viscosity term.
Abstract: In this paper, we focus on the implementation and verification of an extended Boussinesq model for surf zone hydrodynamics in two horizontal dimensions The time-domain numerical model is based on the fully nonlinear Boussinesq equations As described in Part I of this two-part paper, the energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term into the momentum equations, with the viscosity strongly localized on the front face of the breaking waves Wave runup on the beach is simulated using a permeable-seabed technique We apply the model to simulate two laboratory experiments in large wave basins They are wave transformation and breaking over a submerged circular shoal and solitary wave runup on a conical island Satisfactory agreement is found between the numerical results and the laboratory measurements
659Â citations
Cites methods or result from "Water waves of finite amplitude on ..."
...It was shown by Madsen et al. (1997) that, even though a very narrow width of slot is used, there is still about a 10% error in the computed maximum runup in comparison with the analytical solution by Carrier and Greenspan (1958) ....
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...In the case of 1D runup, the present model was tested against an analytical solution by Carrier and Greenspan (1958) and a wide range of experimental data, including irregular waves....
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TL;DR: In this article, the authors describe experiments and computer simulations of the run-up and return of a solitary wave traveling over shallowing water and then onto a dry beach backed by a vertical wall.
Abstract: In this paper, we describe experiments and computer simulations of the run-up and return of a solitary wave traveling over shallowing water and then onto a dry beach backed by a vertical wall. This topography is similar to that of beaches on the north coast of Crete, where narrow coastal plains are backed by steeply rising mountains. The experiments show that the solitary wave collapses onto the dry beach without curling over. It then flows to the vertical wall where it rises up, splashes, curls over, and finally returns as a turbulent bore. On reaching the original water's edge, the bore produces a zig-zag surface perturbation. The simulations reproduce the experiments accurately.
562Â citations
TL;DR: In this paper, the spacings of some cusps formed under reflective wave conditions both in the laboratory and in certain selected natural situations are shown to be consistent with models hypothesizing formation by either (1) subharmonic edge waves (period twice that of the incident waves) of zero mode number or (2) synchronous edge waves of low mode.
Abstract: Genetically, beach cusps are of at least two types: those linked with incident waves which are surging and mostly reflected (reflective systems) and those generated on beaches where wave breaking and nearshore circulation cells are important (dissipative systems). The spacings of some cusps formed under reflective wave conditions both in the laboratory and in certain selected natural situations are shown to be consistent with models hypothesizing formation by either (1) subharmonic edge waves (period twice that of the incident waves) of zero mode number or (2) synchronous (period equal to that of incident waves) edge waves of low mode. Experiments show that visible subharmonic edge wave generation occurs on nonerodable plane laboratory beaches only when the incident waves are strongly reflected at the beach, and this observation is quantified. Edge wave resonance theory and experiments suggest that synchronous potential edge wave generation can also occur on reflective beaches and is a higher-order, weaker resonance than the subharmonic type. In dissipative systems, modes of longshore periodic motion other than potential edge waves may be important in controlling the longshore scale of circulation cells and beach morphologies. On reflective plane laboratory beaches, initially large subharmonic edge waves rear-rage sand tracers into shapes which resemble natural beach cusps, but the edge wave amplitudes decrease as the cusps grow. Cusp growth is thus limited by negative feedback from the cusps to the edge wave excitation process. Small edge waves can form longshore periodic morphologies by providing destabilizing perturbations on a berm properly located in the swash zone. In this case the retreating incident wave surge is channelized into breeches in the berm caused by the edge waves, and there is an initially positive feedback from the topography to longshore periodic perturbations.
472Â citations
TL;DR: In this article, the exact solutions correspond to time-dependent motions in parabolic basins, where the shoreline is not fixed and must be determined as part of the solution, and the motion is oscillatory and has the appropriate small-amplitude limit.
Abstract: These exact solutions correspond to time-dependent motions in parabolic basins. A characteristic feature is that the shoreline is not fixed. It is free to move and must be determined as part of the solution. In general, the motion is oscillatory and has the appropriate small-amplitude limit. For the case in which the parabolic basin reduces to a flat plane, there is a solution for a flood wave. These solutions provide a valuable test for numerical models of inundating storm tides.
396Â citations
Cites result from "Water waves of finite amplitude on ..."
...This is in agreement with the conclusion of Carrier & Greenspan (1958) that whether or not a wave breaks as i t runs up a beach depends upon its initial shape and velocity distribution....
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References
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Book•
01 Jan 1944
TL;DR: The tabulation of Bessel functions can be found in this paper, where the authors present a comprehensive survey of the Bessel coefficients before and after 1826, as well as their extensions.
Abstract: 1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions of Bessel functions 8. Bessel functions of large order 9. Polynomials associated with Bessel functions 10. Functions associated with Bessel functions 11. Addition theorems 12. Definite integrals 13. Infinitive integrals 14. Multiple integrals 15. The zeros of Bessel functions 16. Neumann series and Lommel's functions of two variables 17. Kapteyn series 18. Series of Fourier-Bessel and Dini 19. Schlomlich series 20. The tabulation of Bessel functions Tables of Bessel functions Bibliography Indices.
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157Â citations