Shoaling of Solitary Waves on Plane Beaches
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 120, Issue 6
Abstract
Shoaling of solitary waves on both gentle (1:35) and steeper slopes (≤1:6.50) is analyzed up to breaking using both a fully nonlinear wave model and high‐accuracy laboratory experiments. For the mildest slope, close agreement is obtained between both approaches up to breaking, where waves become very asymmetric and breaking indices reach almost twice the value for the largest stable symmetric wave. Bottom friction does not seem to affect the results at all. Wave celerity decreases during shoaling and slightly increases before breaking. At breaking, the crest particle velocity is almost horizontal and reaches 90% of the crest celerity, which is two to three times larger than the bottom velocity. The nonlinear shallow water (NSW) equations and the Boussinesq approximation both fail to predict these results. Finally, shoaling rates for various wave heights and bottom slopes differ from the predictions of Green's or Boussinesq shoaling laws. On the mildest slope, shoaling rates roughly follow a “two‐zone” model proposed earlier but on steeper slopes reflection becomes significant and wave heights change little during shoaling.
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References
1.
Brebbia, C. A., and Walker, S. (1978). Boundary element techniques in engineering. Newnes‐Butterworths, London, England.
2.
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), 1–22.
3.
Carrier, G. F. (1966). “Gravity waves on water of variable depth.” J. Fluid Mech., 24(4), 641–659.
4.
Carrier, G. F., and Greenspan, H. P. (1958). “Water waves of finite amplitude on a sloping beach.” J. Fluid Mech., 4(1), 97–110.
5.
Chan, R. K., and Street, R. L. (1970). “A computer study of finite‐amplitude water waves.” J. Comp. Phys., 6, 68–94.
6.
Chan, E. S., and Melville, W. K. (1988). “Deep‐water plunging breakers: a comparison between potential theory and experiments.” J. Fluid Mech., 189, 423–442.
7.
Dold, J. W., and Peregrine, D. H. (1986). “An efficient boundary integral method for steep unsteady water waves.” Numerical methods for fluid dynamics II, K. W. Morton and M. J. Baines, eds., Clarendon Press, Oxford, England, 671–679.
8.
Fenton, J. D., and Rienecker, M. M. (1982). “A Fourier method for solving nonlinear water‐wave problems: application to solitary‐wave interactions.” J. Fluid Mech., 118, 411–443.
9.
Freilich, M. H., and Guza, R. T. (1984). “Nonlinear effects on shoaling surface gravity waves.” Phil. Trans., Royal Society, London, England, A311, 1–41.
10.
Goring, D. G. (1978). “Tsunamis—the propagation of long waves onto a shelf.” Rep. No. KH‐R‐38, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, Calif.
11.
Grilli, S., Losada, M. A., and Martin, F. (1994a). “Characteristics of solitary wave breaking induced by breakwaters.” J. Wtrwy., Port, Coast., and Oc. Engrg., ASCE, 120(1), 74–92.
12.
Grilli, S., Skourup, J., and Svendsen, I. A. (1989). “An efficient boundary element method for nonlinear water waves.” Engrg. Analysis with Boundary Elements, 6(2), 97–107.
13.
Grilli, S. T., and Subramanya, R. (1993). “Nonlinear wave modeling in very shallow water.” Proc., 15th Int. Conf. on Boundary Elements in Engrg., J. Rencis and C. A. Brebbia, eds., Elsevier Applied Science, Southampton, England, 194–206.
14.
Grilli, S. T., and Subramanya, R. (1994). “Quasi‐singular integrals in the modeling of nonlinear water waves in shallow water.” Engrg. Analysis with Boundary Elements, 13(2), 181–191.
15.
Grilli, S. T., Subramanya, R., Kirby, J. T., and Wei, G. (1994b). “Comparison of modified Boussinesq and fully nonlinear potential models for shoaling solitary waves.” Proc., Int. Symp. on Waves—Physical and Numerical Modelling, R. Isaacson and R. Quick, eds., Vancouver, Canada, 524–533.
16.
Grilli, S., and Svendsen, I. A. (1990). “Corner problems and global accuracy in the boundary element solution of nonlinear wave flows.” Engrg. Analysis with Boundary Elements, 7(4), 178–195.
17.
Grilli, S., and Svendsen, I. A. (1991a). “The propagation and runup of solitary waves on steep slopes.” Rep. No. 91‐4, Ctr. for Appl. Coast. Res., Univ. of Delaware, Newark, Del.
18.
Grilli, S., and Svendsen, I. A. (1991b). “Long wave interaction with steeply sloping structures.” Proc., 22nd Int. Conf. on Coast. Engrg.; ICCE22, ASCE, New York, N.Y., Vol. 2, 1200–1213.
19.
Grilli, S. T., Svendsen, I. A., and Subramanya, R. (1994c). “Breaking criterion and characteristics for solitary waves on plane beaches.” J. Wtrwy. Port, Coast, and Oc. Engrg., ASCE (submitted).
20.
Hibberd, S., and Peregrine, D. H. (1979). “Surf and runup on a beach: a uniform bore.” J. Fluid Mech., 95(2), 323–345.
21.
Ippen, A. T., and Kulin, G. (1954). “The shoaling and breaking of the solitary waves.” Proc., 5th Int. Conf. on Coast. Engrg.; ICCE5, ASCE, New York, N.Y., 27–47.
22.
Kim, S. K., Liu, P. L.‐F., and Liggett, J. A. (1983). “Boundary integral equation solutions for solitary wave generation propagation and run‐up.” Coast. Engrg., 7, 299–317.
23.
Kirby, J. T. (1991). “Intercomparison of truncated series solutions for shallow water waves.” J. Wtrwy., Port, Coast. and Oc. Engrg., ASCE, 117(2), 143–155.
24.
Kobayashi, N., and DeSilva, G. S., and Watson, K. D. (1989). “Wave transformation and swash oscillation on gentle and steep slope.” J. Geoph. Res., 94(C1), 951–966.
25.
Liu, P. L. F., Yoon, S. B., and Kirby, J. T. (1985). “Nonlinear refraction‐diffraction of waves in shallow water.” J. Fluid Mech., 209, 567–589.
26.
Longuet‐Higgins, M. S., and Cokelet, E. D. (1976). “The deformation of steep surface waves on water. I: a numerical method of computation.” Proc., Royal Society, London, England, A350, 1–26.
27.
Otta, A. K., Svendsen, I. A., and Grilli, S. T. (1993). “The breaking and runup of solitary waves on beaches.” Proc., 23rd Int. Conf. on Coast. Engrg.; ICCE23, ASCE, New York, N.Y., Vol. 2, 1461–1474.
28.
Papanicolaou, P., and Raichlen, F. (1987). “Wave characteristics in the surf zone.” Proc., Coast. Hydrodynamics, R. A. Dalrymple, ed., ASCE, New York, N.Y., 765–780.
29.
Pedersen, G., and Gjevik, B. (1983). “Runup of solitary waves.” J. Fluid Mech., 135, 283–299.
30.
Skjelbreia, J. E. (1987). “Observations of breaking waves on sloping bottoms by use of laser Doppler velocimetry.” Rep. No. KH‐R‐48, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, Calif.
31.
Subramanya, R., and Grilli, S. T. (1994). “Kinematics and integral properties of fully nonlinear periodic waves shoaling over a gentle slope.” Proc., Int. Symp. on Waves—Physical and Numerical Modelling, R. Isaacson and R. Quick, eds., Vancouver, Canada, 1106–1115.
32.
Svendsen, I. A., and Grilli, S. (1990). “Nonlinear waves on steep slopes.” J. Coast. Res., SI 7, 185–202.
33.
Svendsen, I. A., Otta, A. K., and Grilli, S. (1992). “Unsteady free surface waves.” Proc., I.U.T.A.M. Symp. on Breaking Waves, M. L. Banner and R. H. J. Grimshaw, eds., Springer‐Verlag, Berlin, Germany, 229–236.
34.
Synolakis, C. E. (1987). “The runup of solitary waves.” J. Fluid Mech., Vol. 185, 523–545.
35.
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(3), 323–342.
36.
Tanaka, M. (1986). “The stability of solitary waves.” Phys. Fluids, 29(3), 650–655.
37.
Tanaka, M., Dold, J. W., and Peregrine, D. H. (1987). “Instability and breaking of a solitary wave.” J. Fluid Mech., Vol. 185, 235–248.
38.
Vinje, T., and Brevig, P. (1981). “Numerical simulation of breaking waves.” Adv. Water Resour., 4, 77–82.
39.
Yasuda, T., Sakakibara, Y., and Hara, M. (1992). “BIM simulation on deformation up to breaking of solitary waves over uneven bottoms.” Proc., 4th Int. Conf. on Hydr. Engrg. Software; HYDROSOFT92, W. R. Blain and E. Cabrera, eds., Elsevier Applied Science, Southampton, England, 523–535.
40.
Zelt, J. A. (1991). “The runup of non‐breaking and breaking solitary waves.” Coast. Engrg., 15, 205–246.
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Copyright © 1994 American Society of Civil Engineers.
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Received: Oct 12, 1993
Published online: Nov 1, 1994
Published in print: Nov 1994
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