**8. References**


Bailard, J. A. & Inman, D. L. (1981). An energetics bedload model for a plane sloping beach: Local transport, *J. Geophys. Res*., Vol. 86, C3, pp. 2035-2043

372 Numerical Simulation – From Theory to Industry

those until 2×104 steps continued.

was demonstrated in the present study.

*Public Works Research Center, Taito, Tokyo*  Masumi Serizawa and Shiho Miyahara

2001), pp. 296-300, ISSN 0028-0836

*Coastal Engineering Laboratory Co., Ltd., Shinjuku, Tokyo* 

Vol. 3, pp. 507-528, Wiley, ISBN 978-0674017306, New York

**7. Conclusions** 

**Author details** 

**8. References** 

Takaaki Uda

wave incidence caused shoreward sand transport flux along the contours of the sand bar. On the other hand, on the lee of the submerged sand bar, the wave height was reduced, similarly to in the case of a detached breakwater, inducing longshore sand transport toward the lee of the sand bar from outside the wave-shelter zone. After 2×104 steps, as a result of the landward extension of the slender sand bar and the decrease in water depth, wave breaking occurred there, causing a marked reduction in wave height. At the same time, the oblique wave incidence continued in this area and landward sand transport from the island to the land continuously occurred. After 4×104 and 5×104 steps, the same beach changes as

The BG model was applied to predict the development of a sand spit on a flat shallow seabed and the formation of a cuspate foreland on coasts with an abrupt change in the shoreline orientation. The results of the model were validated by comparison with the experimental results obtained by Uda & Yamamoto (1992). The predicted and measured results were in good agreement. As another type of beach change due to waves on a coast with a shallow flat seabed, the shoreward transport of sand originally supplied from the offshore zone of a tidal flat, forming a slender sand bar, and the landward deposition of such sand were observed on the Kutsuo coast, which has a very wide tidal flat and faces the Suo-nada Sea, part of the Seto Inland Sea. We investigated these phenomena by field observations and then performed a numerical simulation using the BG model. The observed results were successfully explained by the results of the numerical simulation. Although the BG model has been used to predict the development of river mouth bars, a single sand spit and a bay barrier (Serizawa et al., 2009b, 2009a; Uda & Serizawa, 2011), another application

Ashton, A.; Murray, A. B. & Arnault, O. (2001). Formation of coastline features by largescale instabilities induced by high angle waves, *Nature*, Vol. 14, No. 6861, (November

Bagnold, R. A. (1963). Mechanics of Marine Sedimentation, In: *The Sea*, Hill, M. N. (editor),


predicting-opographic-changes-on-coast-composed-of-sand-of-mixed-grain-size-andits-appli

**Chapter 0**

**Chapter 17**

**Numerical Simulations of Water Waves'**

**of Wind and Dissipation**

Additional information is available at the end of the chapter

Julien Touboul and Christian Kharif

http://dx.doi.org/10.5772/48595

**1. Introduction**

**Modulational Instability Under the Action**

The seek of uniform, propagative wave train solutions of the fully nonlinear potential equations has been a major topic for centuries. [5] was the first to propose an expression of such waves, the so called Stokes' waves. However, pioneer works of [6] emphasized that such waves might be unstable, providing a geometric condition for this stability problem. Later on, [1] showed analytically that Stokes' waves of moderate amplitude are unstable to long wave perturbations of small amplitude travelling in the same direction. This instability is named the Benjamin-Feir instability (or modulational instability). This result was derived independently by [7] in an averaged Lagrangian approach, and by [8] who used an Hamiltonian formulation of the water wave problem. Using this approach, the latter author derived the nonlinear

Within the last fifty years, the study of this instability became central for fundamental and applied research. The modulation instability is one of the most important mechanisms for the formation of rogue waves [9]. A complete review on the various phenomena yielding to rogue waves can be found in the book of [10]. In the absence of forcing and damping, Stokes' waves of specific initial steepness are submitted to this instability, when they encounter perturbations of specific wave numbers [11, 12]. In this case, they encounter a nonlinear quasi-recursive evolution, the so called Fermi-Pasta-Ulam recurrence phenomenon ([13]). This phenomenon corresponds to a series of modulation - demodulation cycles, during which initially uniform wave trains become modulated, leading possibly to the formation of a huge wave. Modulation is due to an energy transfer from the wave carrier to the unstable sidebands. In the wave number space, these unstable sidebands are located in a finite narrow band centered around the carrier wave number. During the demodulation, the energy returns to the fundamental component of the original wave train. Using the Zakharov equation, [14] questions the

and reproduction in any medium, provided the original work is properly cited.

©2012 Touboul and Kharif, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

Schrödinger equation (NLS), and confirmed the previous stability results.

cited.

