Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 36
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Pujara, Nimish Liu, Philip L.-F. and Yeh, Harry 2015. The swash of solitary waves on a plane beach: flow evolution, bed shear stress and run-up. Journal of Fluid Mechanics, Vol. 779, p. 556.

    Apotsos, Alex Gelfenbaum, Guy and Jaffe, Bruce 2012. Time-dependent onshore tsunami response. Coastal Engineering, Vol. 64, p. 73.

    Apotsos, Alex Jaffe, Bruce and Gelfenbaum, Guy 2011. Wave characteristic and morphologic effects on the onshore hydrodynamic response of tsunamis. Coastal Engineering, Vol. 58, Issue. 11, p. 1034.

    Grimshaw, R. and Yuan, C. 2016. Depression and elevation tsunami waves in the framework of the Korteweg–de Vries equation. Natural Hazards,

    Zainali, Amir and Weiss, Robert 2015. Boulder dislodgement and transport by solitary waves: Insights from three-dimensional numerical simulations. Geophysical Research Letters, Vol. 42, Issue. 11, p. 4490.

    Park, Hyoungsu Cox, Daniel T. and Petroff, Catherine M. 2015. An empirical solution for tsunami run-up on compound slopes. Natural Hazards, Vol. 76, Issue. 3, p. 1727.

    Wang, Benlong and Liu, Hua 2013. Kinematic dynamo by large scale tsunami waves in open ocean. Theoretical and Applied Mechanics Letters, Vol. 3, Issue. 3, p. 032003.

    Park, Hyoungsu and Cox, Daniel T. 2016. Empirical wave run-up formula for wave, storm surge and berm width. Coastal Engineering, Vol. 115, p. 67.

    Madsen, P.A. 2010. On the evolution and run-up of tsunamis. Journal of Hydrodynamics, Ser. B, Vol. 22, Issue. 5, p. 1.

    Shimozono, Takenori 2016. Long wave propagation and run-up in converging bays. Journal of Fluid Mechanics, Vol. 798, p. 457.

    DONG, Jie WANG, Ben-long and LIU, Hua 2015. Run-up of non-breaking double solitary waves with equal wave heights on a plane beach. Journal of Hydrodynamics, Ser. B, Vol. 26, Issue. 6, p. 939.

    Park, Hyoungsu and Cox, Daniel T. 2016. Probabilistic assessment of near-field tsunami hazards: Inundation depth, velocity, momentum flux, arrival time, and duration applied to Seaside, Oregon. Coastal Engineering, Vol. 117, p. 79.

    Wu, Wei Liu, Hua and Fang, Yongliu 2015. A Study on Runup of Nonbreaking Double Solitary Waves on Plane Slope. Procedia Engineering, Vol. 116, p. 738.

    XUAN, Rui-tao WU, Wei and LIU, Hua 2013. An experimental study on runup of two solitary waves on plane beaches. Journal of Hydrodynamics, Ser. B, Vol. 25, Issue. 2, p. 317.

    Liang, Dongfang Gotoh, Hitoshi Khayyer, Abbas and Chen, Jack Mao 2013. Boussinesq modelling of  solitary wave and N-wave runup on coast. Applied Ocean Research, Vol. 42, p. 144.

    Lo, Hong-Yueh Park, Yong Sung and Liu, Philip L.-F. 2013. On the run-up and back-wash processes of single and double solitary waves — An experimental study. Coastal Engineering, Vol. 80, p. 1.

    Zhao, Xi Wang, Benlong and Liu, Hua 2012. Characteristics of tsunami motion and energy budget during runup and rundown processes over a plane beach. Physics of Fluids, Vol. 24, Issue. 6, p. 062107.

    Chan, I-Chi and Liu, Philip L.-F. 2012. On the runup of long waves on a plane beach. Journal of Geophysical Research: Oceans, Vol. 117, Issue. C8, p. n/a.

    Charvet, Ingrid Eames, Ian and Rossetto, Tiziana 2013. New tsunami runup relationships based on long wave experiments. Ocean Modelling, Vol. 69, p. 79.

    Sepúlveda, Ignacio and Liu, Philip L.-F. 2016. Estimating tsunami runup with fault plane parameters. Coastal Engineering, Vol. 112, p. 57.

  • Journal of Fluid Mechanics, Volume 645
  • February 2010, pp. 27-57

Analytical solutions for tsunami runup on a plane beach: single waves, N-waves and transient waves

  • DOI:
  • Published online: 22 February 2010

In the literature it has so far been common practice to consider solitary waves and N-waves (composed of solitary waves) as the appropriate model of tsunamis approaching the shoreline. Unfortunately, this approach is based on a tie between the nonlinearity and the horizontal length scale (or duration) of the wave, which is not realistic for geophysical tsunamis. To resolve this problem, we first derive analytical solutions to the nonlinear shallow-water (NSW) equations for the runup/rundown of single waves, where the duration and the wave height can be specified separately. The formulation is then extended to cover leading depression N-waves composed of a superposition of positive and negative single waves. As a result the temporal variations of the runup elevation, the associated velocity and breaking criteria are specified in terms of polylogarithmic functions. Finally, we consider incoming transient wavetrains generated by monopole and dipole disturbances in the deep ocean. The evolution of these wavetrains, while travelling a considerable distance over a constant depth, is influenced by weak dispersion and is governed by the linear Korteweg–De Vries (KdV) equation. This process is described by a convolution integral involving the Airy function. The runup on the plane sloping beach is then determined by another convolution integral involving the incoming time series at the foot of the slope. A good agreement with numerical model results is demonstrated.

Corresponding author
Email address for correspondence:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

M. J. Briggs , C. E. Synolakis , G. S. Harkins & D. R. Green 1995 Laboratory experiments of tsunami runup on a circular island. Pure Appl. Geophys. 144 (3/4), 569593.

I. I. Didenkulova , A. A. Kurkin & E. N. Pelinovsky 2007 Run-up of solitary waves on slopes with different profiles. Atmos. Ocean. Phys. 43 (3), 384390.

I. I. Didenkulova , N. Zahibo , A. A. Kurkin , B. V. Levin , E. N. Pelinovsky & T. Soomere 2006 Runup of nonlinearly deformed waves on a coast. Dokl. Earth Sci. 411 (8), 12411243.

D. R. Fuhrman & H. B. Bingham 2004 Numerical solutions of fully nonlinear and highly dispersive Boussinesq equations in two horizontal dimensions. Intl J. Numer. Methods Fluids 44, 231255.

D. R. Fuhrman & P. A. Madsen 2008 Simulation of nonlinear wave runup with a high-order Boussinesq model. Coast. Engng 55 (2), 139154.

E. L. Geist 1999 Local tsunamis and earthquake source parameters. Adv. Geophys. 39, 117209.

E. L. Geist & S. Yoshioka 1996 Source parameters controlling the generation and propagation of potential local tsunamis along the Cascadia Margin. Nat. Hazards 13, 151177.

L. Lewin 1991 Structural Properties of Polylogarithms. American Mathematical Society.

H. Lewy 1946 Water waves on sloping beaches. Bull. Am. Math. Soc. 52, 737755.

Y. Li & F. Raichlen 2001 Solitary wave runup on plane slopes. J. Waterway Port Coast. Ocean Engng 127 (1), 3344.

P. A. Madsen & D. R. Fuhrman 2008 Runup of tsunamis and long waves in terms of surf-similarity. Coast. Engng 55 (3), 209224.

P. A. Madsen , D. R. Fuhrman & H. A. Schäffer 2008 On the solitary wave paradigm for tsunamis. J. Geophys. Res. 113, C12012, 122.

P. A. Madsen , D. R. Fuhrman & B. Wang 2006 A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Engng 53, 487504.

E. N. Pelinovsky & R. K. Mazova 1992 Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles. Nat. Hazards 6, 227249.

C. E. Synolakis & E. N. Bernard 2006 Tsunami science before and beyond Boxing Day 2004. Phil. Trans. R. Soc. A 364, 22312265.

C. E. Synolakis , M. K. Deb & J. E. Skjelbreia 1988 The anomaleous behaviour of the runup of cnoidal waves. Phys. Fluids 31 (1), 35.

S. Tadepalli & C. E. Synolakis 1994 The run-up of N-waves on sloping beaches. Proc. R. Soc. Lond. A 445, 99112.

S. Tadepalli & C. E. Synolakis 1996 Model for the leading waves of tsunamis, Phys. Rev. Lett. 77, 21412145.

H. Yeh , P. L.-F. Liu , M. Briggs & C. E. Synolakis 1994 Propagation and amplification of tsunamis at coastal boundaries. Nature 372, 353355.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *