Keywords
wave shoaling
long wave theory
KdV equation.
How to Cite
Abstract
The waveheight change in surface waves with a sufficiently slow variation in depth is examined. Using a new formulation of the energy flux associated to waves modeled by the Korteweg-de Vries equation, a system of three coupled equations is derived for the determination of the local wave properties as waves propagate over gently changing depth. The system of equations is solved numerically, and the resulting shoaling curves are compared to previous results on long wave shoaling.References
Ali, A. and Kalisch, H., 2010, Energy balance for undular bores, C R Mecanique, 338, 67-70
Ali, A. and Kalisch, H., 2012, Mechanical balance laws for Boussinesq models of surface water waves, J.Nonlinear Sci., 22, 371-398
Ali, A. and Kalisch, H., 2014, On the formulation of mass, momentum and energy conservation in the KdV equation, Acta Appl. Math., 133, 113-131
Bjørkavåg, M. and Kalisch, H., 2011, Wave breaking in Boussinesq models for undular bores, Phys. Lett. A, 375, 1570-1578
Bona, J.L., Chen, M. and Saut, J.-C., 2002, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory, J. Nonlinear Sci., 12, 283-318
Camfield, F.E. and Street, R.L., 1969, Shoaling of solitary waves on small slopes, J. Wtrwy., Port, Coast., and Oc. Engrg, ASCE, 95, 1-22
Dean, R.G. and Dalrymple, R.A., Water wave mechanics for engineers and scientists, (World Scientific, Singapore, 1991)
Grilli, S.T., Subramanya, R., Svendsen, I.A. and Veeramony, J., 1994, Shoaling of solitary waves on plane beaches, J. Wtrwy., Port, Coast., and Oc. Engrg., 120, 609-628
Grilli, S.T., Svendsen, I.A. and Subramanya, R., 1997, Breaking criterion and characteristics for solitary waves on slopes, J. Wtrwy., Port, Coast., and Oc. Engrg., 123, 102-112
Grimshaw, R., 1971, The solitary wave in water of variable depth, part 2, J. Fluid Mech., 46, 611-622
Hibberd, S. and Peregrine, D.H., 1979, Surf and runup on a beach: a uniform bore, J. Fluid Mech., 95, 323-345
Ippen, A.T. and Kulin, G., 1954, The shoaling and breaking of the solitary wave, Proc. 5th Conf. On Coast. Eng., Grenoble, 27-47
Kalisch, H. and Senthilkumar A., 2013, Derivation of Boussinesq's shoaling law using a coupled BBM system, Nonlin. Processes Geophys., 20, 213-219
Lawden, D.F., Elliptic functions and applications, (Springer, New York, 1989)
Ostrovskiy, L.A. and Pelinovskiy, E.N., 1970, Wave transformation on the surface of a fluid on variable depth, Atmos. Ocean. Phys., 6, 552-555
Papanicolaou, P. and Raichlen, F., 1987, Wave characteristics in the surf zone, Proc. Coast. Hydrodynamics, ( R. A. Dalrymple, ed.), ASCE, New York, 765-780
Sakai, T. and Battjes, J., 1980, Wave shoaling calculated from Cokelet's theory, Coastal Engineering, 4, 65-84
Sorensen, R.M., Basic wave mechanics: for coastal and ocean engineers, (Wiley, New York, 1993)
Svendsen, I.A. and Brink-Kjær, O., 1972, Shoaling of cnoidal waves, Proc. 13th Conf. Coastal Engng, Vancouver, 365-383
Svendsen, I.A. and Buhr Hansen, J., 1977, The wave height variation for regular waves in shoaling water, Coastal Engineering, 1, 261-284
Svendsen, I.A. and Buhr Hansen, J., 1978, On the deformation of periodic long waves over a gently sloping bottom, J. Fluid Mech., 87, 433-448
Swift, R.H. and Dixon, J.C., 1987, Transformation of regular waves, Proc. Instn Civ. Engrs, Part2, 83, 359-380
Synolakis, C.E., 1987, The runup of solitary wave, J. Fluid Mech., 185, 523-545
Synolakis, C.E. and Skjelbreia, J.E., 1993, Evolution of maximum amplitude of solitary waves on plane beaches, J. Wtrwy., Port, Coast., and Oc. Engrg., ASCE, 119, 323-342
Wei, G., Kirby, J.T., Grilli, S.T. and Subramanya, R., 1995, A fully nonlinear Boussinesq model for surface waves. Part1. Highly nonlinear unsteady waves, J. Fluid Mech., 294, 71-92
Whitham, G.B., Linear and nonlinear waves, (Wiley, New York, 1974)