Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-01T01:34:29.639Z Has data issue: false hasContentIssue false
  • English
  • Français

On the analytical solutions for water waves generated by a prescribed landslide

Published online by Cambridge University Press:  16 May 2017

Hong-Yueh Lo Open the ORCID record for Hong-Yueh Lo [Opens in a new window]  and
Philip L.-F. Liu
Hong-Yueh Lo*
Affiliation:
Department of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14850, USA
Philip L.-F. Liu
Affiliation:
Department of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14850, USA Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore Institute of Hydrological and Oceanic Sciences, National Central University, Jhongli, Taoyuan, 320, Taiwan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents a suite of analytical solutions, for both the free-surface elevation and the flow velocity, for landslide-generated water waves. The one-dimensional (horizontal, 1DH) analytical solutions for water waves generated by a solid landslide moving at a constant speed in constant water depth were obtained for the linear and weakly dispersive wave model as well as the linear and fully dispersive wave model. The area enclosed by the landslide was shown to have stronger lasting effects on the generated water waves than the exact landslide shape. In addition, the resonance solution based on the fully dispersive wave model was examined, and the growth rate was derived. For the 1DH linear shallow water equations (LSWEs) on a constant slope, a closed-form analytical solution, which could serve as a useful benchmark for numerical models, was found for a special landslide forcing function. For the two-dimensional (horizontal, 2DH) LSWEs on a plane beach, we rederived the solutions using the quiescent water initial conditions. The difference between the initial conditions used in the new solutions and those used in previous studies was found to have a permanent effect on the generated waves. We further noted that convergence rate of the 2DH LSWE analytical solutions varies greatly, and advised that case-by-case convergence tests be conducted whenever the modal analytical solutions are numerically evaluated using a finite number of modes.

JFM classification

Geophysical and Geological Flows: Coastal engineering Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 821 , 25 June 2017 , pp. 85 - 116
Check for updates
Copyright
© 2017 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.)

References

Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers. Springer.Google Scholar
Didenkulova, I., Nikolkina, I., Pelinovsky, E. & Zahibo, N. 2010 Tsunami waves generated by submarine landslides of variable volume: analytical solutions for a basin of variable depth. Nat. Hazards Earth Syst. Sci. 10, 2407–2419.Google Scholar
Didenkulova, I. I., Nikolkina, I. F. & Pelinovsky, E. N. 2011 Resonant amplification of tsunami waves from an underwater landslide. Dokl. Earth Sci. 436, 66–69.Google Scholar
Gottlieb, S., Shu, C.-W. & Tadmor, E. 2001 Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (1), 89–112.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 2007 Table of Integrals, Series, and Products, 7th edn. Academic.Google Scholar
Liu, P. L.-F., Lynett, P. & Synolakis, C. E. 2003 Analytical solutions for forced long waves on a sloping beach. J. Fluid Mech. 478, 101–109.Google Scholar
Lynett, P. & Liu, P. L.-F. 2002 A numerical study of submarine-landslide-generated waves and run-up. Proc. R. Soc. Lond. A 458, 2885–2910.Google Scholar
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. World Scientific.Google Scholar
Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. ASCE J. Waterway Port Coastal Ocean Engng 119 (6), 618–638.Google Scholar
Poularikas, A. D. 2000 The Transforms and Applications Handbook, 2nd edn. CRC Press.Google Scholar
Renzi, E.2010 Landslide tsunami. PhD thesis, Università degli Studi di Roma Tor Vergata.Google Scholar
Renzi, E. & Sammarco, P. 2010 Landslide tsunamis propagating around a conical island. J. Fluid Mech. 650, 251–285.CrossRefGoogle Scholar
Renzi, E. & Sammarco, P. 2012 The influence of landslide shape and continental shelf on landslide generated tsunamis along a plane beach. Nat. Hazards Earth Syst. Sci. 12, 1503–1520.CrossRefGoogle Scholar
Rosenheinrich, W.2015 Table of some indefinite integrals of Bessel functions. http://www.eah-jena.de/∼rsh/Forschung/Stoer/besint.pdf.Google Scholar
Sammarco, P. & Renzi, E. 2008 Landslide tsunami propagating along a plane beach. J. Fluid Mech. 598, 107–119.Google Scholar
Seo, S. N. & Liu, P. L.-F. 2013 Edge waves generated by the landslide on a sloping beach. Coast. Engng 73, 133–150.Google Scholar
Tinti, S. & Bortolucci, E. 2000 Energy of water waves induced by submarine landslides. Pure Appl. Geophys. 157, 281–318.Google Scholar
Tinti, S., Bortolucci, E. & Chiavettieri, C. 2001 Tsunami excitation by submarine slides in shallow-water approximation. Pure Appl. Geophys. 158, 759–797.Google Scholar
Tuck, E. O. & Hwang, L.-S. 1972 Long wave generation on a sloping beach. J. Fluid Mech. 51, 449–461.Google Scholar
Wei, G., Kirby, J. T., Grilli, S. T. & Subramanya, R. 1995 A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71–92.Google Scholar