Shoaling on Steep Continental Slopes: Relating Transmission and Reflection Coefficients to Green’s Law

  • Jithin George
  • David I. Ketcheson
  • Randall J. LeVequeEmail author


The propagation of long waves onto a continental shelf is of great interest in tsunami modeling and other applications where understanding the amplification of waves during shoaling is important. When the linearized shallow water equations are solved with the continental shelf modeled as a sharp discontinuity, the ratio of the amplitudes is given by the transmission coefficient. On the other hand, when the slope is very broad relative to the wavelength of the incoming wave, then amplification is governed by Green’s Law, which predicts a larger amplification than the transmission coefficient, and a much smaller amplitude reflection than given by the reflection coefficient of a sharp interface. We explore the relation between these results and elucidate the behavior in the intermediate case of a very steep continental shelf.


Shoaling tsunamis Green’s Law reflection transmission continental shelf 



The authors are grateful to Avi Schwarzschild for stimulating discussions in the early phase of this project.


The article was funded by National Aeronautics and Space Administration (NNA10DF26C) and King Abdullah University of Science and Technology.


  1. Adams, L. M., & LeVeque, R. J. (2017). GeoClaw Model Tsunamis Compared to Tide Gauge Results—Final Report.
  2. Berger, M. J., George, D. L., & LeVeque, R. J. (2011). Adaptive mesh refinement techniques for tsunamis and other geophysical flows over topography. Acta Numerica, 20, 211–289.CrossRefGoogle Scholar
  3. Clawpack Development Team. (2017). Clawpack software.,, version 5.5.0.
  4. Davies, G., Griffin, J., Løvholt, F., Glimsdal, S., Harbitz, C., Thio, H. K., et al. (2018). A global probabilistic tsunami hazard assessment from earthquake sources. Geological Society, London, Special Publications, 456, 219–244. Scholar
  5. Dzvonkovskaya, A., Heron, M., Figueroa, D., & Gurgel, K. (2014). Observations and theory of a shoaling tsunami wave. In: 2014 Oceans—St. John’s, 1–5.
  6. Evans, D. V., & Linton, C. M. (1994). On step approximations for water-wave problems. Journal of Fluid Mechanics, 278, 229–249. Scholar
  7. George, J.D. (2018) Green’s law and the Riemann problem in layered media. Master’s thesis, University of Washington.Google Scholar
  8. George, J. D., Ketcheson, D. I., & LeVeque, R. J. (2019a). A characteristics-based approximation for wave scattering from an arbitrary obstacle in one dimension., arXiv:1901.04158.
  9. George, J. D., Ketcheson, D. I., & LeVeque, R. J. (2019b). Code to accompany this paper.,
  10. Glimsdal, S., Løvholt, F., Harbitz, C. B., Romano, F., Lorito, S., Orefice, S., et al. (2019). A new approximate method for quantifying tsunami maximum inundation height probability. Pure and Applied Geophysics. Scholar
  11. Green, G. (1838). On the motion of waves in a variable canal of small depth and width. Transactions of the Cambridge Philosophical Society, 6, 457.Google Scholar
  12. Grezio, A., Babeyko, A., Baptista, M. A., Behrens, J., Costa, A., Davies, G., et al. (2017). Probabilistic tsunami hazard analysis: Multiple sources and lobal applications. Reviews of Geophysics, 55, 1158–1198. Scholar
  13. Grilli, S. T., Subramanya, R., Svendsen, I. A., & Veeramony, J. (1994). Shoaling of solitary waves on plane beaches. Journal of Waterway, Port, Coastal, and Ocean Engineering, 120(6), 609–628. Scholar
  14. Heron, M., & Dzvonkovskaya, A. (2015). Conceptual view of reflection and transmission of a tsunami wave at a step in bathymetry. In OCEANS 2015—MTS/IEEE Washington. (pp 1–4),
  15. Kânoǧlu, U., & Synolakis, C. E. (1998). Long wave runup on piecewise linear topographies. Journal of Fluid Mechanics, 374, 1–28.CrossRefGoogle Scholar
  16. LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems. Cambridge University Press,
  17. Levin, B. W., & Nosov, M. A. (2016). Physics of tsunamis (2nd ed.). New York: Springer.CrossRefGoogle Scholar
  18. Lorito, S., Selva, J., Basili, R., Romano, F., Tiberti, M. M., & Piatanesi, A. (2015). Probabilistic hazard for seismically induced tsunamis: Accuracy and feasibility of inundation maps. Geophysical Journal International, 200(1), 574–588.CrossRefGoogle Scholar
  19. Løvholt, F., Glimsdal, S., Harbitz, C. B., Zamora, N., Nadim, F., Peduzzi, P., et al. (2012). Tsunami hazard and exposure on the global scale. Earth-Science Reviews, 110(1–4), 58–73. Scholar
  20. Løvholt, F., Griffin, J., & Salgado-Gálvez, M. (2016). Tsunami hazard and risk assessment on the global scale. In R. A. Meyers (Ed.), Encyclopedia of complexity and systems science (pp. 1–34). Berlin: Springer. Scholar
  21. Madsen, P. A., Fuhrman, D. R., & Schäffer, H. A. (2008). On the solitary wave paradigm for tsunamis. Journal of Geophysical Research: Oceans, 113(C12), C12012. Scholar
  22. Mei, C. C. (1992). The applied dynamics of ocean surface waves. Singapore: World Scientific.Google Scholar
  23. Miles, J. W. (1967). Surface-wave scattering matrix for a shelf. Journal of Fluid Mechanics, 28, 755–767. Scholar
  24. Synolakis, C. E. (1991). Green’s law and the evolution of solitary waves. Physics of Fluids A: Fluid Dynamics, 3(3), 490–491. Scholar
  25. Ward, S. N., & Asphaug, E. (2000). Asteroid impact tsunami: A probabilistic hazard assessment. Icarus, 145(1), 64–78.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Engineering Sciences and Applied MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.Computer, Electrical, and Mathematical Sciences and Engineering DivisionKing Abdullah University of Science and TechnologyThuwalSaudi Arabia
  3. 3.Department of Applied MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations