Hostname: page-component-6b989bf9dc-476zt Total loading time: 0 Render date: 2024-04-13T11:07:25.058Z Has data issue: false hasContentIssue false

The runup of solitary waves

Published online by Cambridge University Press:  21 April 2006

Costas Emmanuel Synolakis
School of Engineering, University of Southern California, Los Angeles, California 90089–0242, USA


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.

Research Article
© 1987 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Battjes, J. A. & Roos, A. 1971 Characteristics of flow in the run-up of periodic waves. Rep. 75–3. Dept. of Civil Engineering, Delft University of Technology. 160 pp.
Camfield, F. E. & Street, R. L. 1969 Shoaling of solitary waves on small slopes. Proc. ASCE WW95, 122.Google Scholar
Carrier, G. F. 1966 Gravity waves of water of variable depth. J. Fluid Mech. 24, 641659.Google Scholar
Carrier, G. F. 1971 The dynamics of tsunamis. In Mathematical Problems in the Geophysical Sciences. Proc. 6th Summer Seminar on Applied Mathematics, Rennselar Polytechnic Institute, Troy, NY, 1970. American Mathematical Society.
Carrier, G. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 17, 97110.Google Scholar
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw Hill. 438 pp.
Gjevik, B. & Pedersen, G. 1981 Run-up of long waves on an inclined plane. Preprint Series Inst. of Math. Univ. of Oslo, ISBN 82–553–0453–3.
Goring, D. G. 1978 Tsunamis - the propagation of long waves onto a shelf. Rep. KH-R-38. W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, CA. 337 pp.
Guza, R. T. & Thornton, E. B. 1982 Swash oscillations on a natural beach. J. Geophys. Res. 87, 483491.Google Scholar
Hall, J. V. & Watts, J. W. 1953 Laboratory investigation of the vertical rise of solitary waves on impermeable slopes. Tech. Memo. 33, Beach Erosion Board, US Army Corps of Engineers. 14 pp.
Hammack, J. L. 1972 Tsunamis - A model of their generation and propagation. Rep. KH-R-28. W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, CA. 261 pp.
Heitner, K. L. & Housner, G. W. 1970 Numerical model for tsunami run-up. Proc. ASCE WW3, 701719.Google Scholar
Hibbard, S. & Peregrine, D. H. 1979 Surf and run-up on a beach: a uniform bore. J. Fluid Mech. 95, 323345.Google Scholar
Holman, R. A. 1986 Extreme value statistics for wave runup on a natural beach. Coastal Engng 9, 527544.Google Scholar
Keller, J. B. & Keller, H. B. 1964 Water wave run-up on a beach. ONR Research Rep. Contract NONR-3828(00). Dept. of the Navy, Washington, D.C. 40 pp.
Kim, S. K., Liu, P. L-F. & Liggett, J. A. 1983 Boundary integral equation solutions for solitary wave generation propagation and run-up. Coastal Engng 7, 299317.Google Scholar
Kishi, T. & Saeki, H. 1966 The shoaling, breaking and runup of the solitary wave on impermeable rough slopes. In Proc. ASCE, Tenth Conference on Coastal Engineering, Tokyo, Japan, pp. 322348.
Lewy, H. 1946 Water waves on sloping beaches. Bull. Am. Math. Soc. 52, 737755.Google Scholar
Meyer, R. E. 1986 On the shore singularity of water wave theory. II. Small waves do not break on gentle beaches. Phys. Fluids 29, 31643171.Google Scholar
Pedersen, G. & Gjevik, B. 1983 Run-up of solitary waves. J. Fluid Mech. 135, 283290.Google Scholar
Spielvogel, L. Q. 1974 Single-wave runup on sloping beaches. J. Fluid Mech. 74, 685694.Google Scholar
Stoker, J. J. 1947 Surface waves in water of variable depth. Q. Appl. Maths 5, 154.Google Scholar
Synolakis, C. E. 1986 The runup of long waves, Ph.D. thesis, California Institute of Technology, Pasadena, California, 91125. 228 pp.
Synolakis, C. E. 1988 On the zeros of J0(z)-iJ1(z). Q. Appl. Maths (In press).Google Scholar
Tuck, E. O. & Hwang, L. 1972 Long wave generation on a sloping beach, J. Fluid Mech. 51, 449461.Google Scholar
Zelt, Z. A. 1986 Tsunamis: The response of harbors with sloping boundaries to long wave excitation. Rep. KH-R-47. W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, CA. 318 pp.