Pure and Applied Geophysics

, Volume 174, Issue 8, pp 3209–3218 | Cite as

New Analytical Solution for Nonlinear Shallow Water-Wave Equations

Article

Abstract

We solve the nonlinear shallow water-wave equations over a linearly sloping beach as an initial-boundary value problem under general initial conditions, i.e., an initial wave profile with and without initial velocity. The methodology presented here is extremely simple and allows a solution in terms of eigenfunction expansion, avoiding integral transform techniques, which sometimes result in singular integrals. We estimate parameters, such as the temporal variations of the shoreline position and the depth-averaged velocity, compare with existing solutions, and observe perfect agreement with substantially less computational effort.

Keywords

Tsunami long wave runup shallow water-wave 

References

  1. Anderson, D., Harris, M., Hartle, H., Nicolsky, D., Pelinovsky, E. N., Raz, A., et al. (2017). Runup of long waves in piecewise sloping U-shaped bays. Pure and Applied Geophysics. doi:10.1007/s00024-017-1476-3.
  2. Antuono, M., & Brocchini, M. (2010). Solving the nonlinear shallow-water equations in physical space. Journal of Fluid Mechanics, 643, 207–232. doi:10.1017/S0022112009992096.CrossRefGoogle Scholar
  3. Aydın, B. (2011). Analytical solutions of shallow-water wave equations. Ph.D. Thesis, Middle East Technical University, Ankara, Turkey.Google Scholar
  4. Aydın, B., & Kânoğlu, U. (2007). Wind set-down relaxation. Computer Modeling in Engineering and Sciences (CMES), 21(2), 149–155. doi:10.3970/cmes.2007.021.149.Google Scholar
  5. Bernard, E. N., & Titov, V. V. (2015). Evolution of tsunami warning systems and products. Philosophical Transactions of the Royal Society A, 373, 20140371. doi:10.1098/rsta.2014.0371.CrossRefGoogle Scholar
  6. Bowman, F. (1958). Introduction to Bessel functions. New York: Dover Publications Inc.Google Scholar
  7. Brocchini, M. (1997). Eulerian and Lagrangian aspects of the longshore drift in the surf and swash zones. Journal of Geophysical Research: Oceans, 102(C10), 23,155–23,168. doi:10.1029/97JC01882.
  8. Brocchini, M., & Peregrine, D. H. (1996). Integral flow properties in the swash zone and averaging. Journal of Fluid Mechanics, 317, 241–273. doi:10.1017/S0022112096000742.CrossRefGoogle Scholar
  9. Carrier, G. F., & Greenspan, H. P. (1958). Water waves of finite amplitude on a sloping beach. Journal of Fluid Mechanics, 4, 97–109. doi:10.1017/S0022112058000331.CrossRefGoogle Scholar
  10. Carrier, G. F., & Noiseux, C. F. (1983). The reflection of obliquely incident tsunamis. Journal of Fluid Mechanics, 133, 147–160. doi:10.1017/S0022112083001834.CrossRefGoogle Scholar
  11. Carrier, G. F., Wu, T. T., & Yeh, H. (2003). Tsunami run-up and draw-down on a plane beach. Journal of Fluid Mechanics, 475, 79–99. doi:10.1017/S0022112002002653.CrossRefGoogle Scholar
  12. Choi, B. H., Pelinovsky, E., Kim, D. C., Didenkulova, I., & Woo, S.-B. (2008). Two- and three-dimensional computation of solitary wave runup on non-plane beach. Nonlinear Processes in Geophysics, 15, 489–502. doi:10.5194/npg-15-489-2008.CrossRefGoogle Scholar
  13. Didenkulova, I., & Pelinovsky, E. (2011a). Nonlinear wave evolution and runup in an inclined channel of a parabolic cross-section. Physics of Fluids, 23, 086602. doi:10.1063/1.3623467.CrossRefGoogle Scholar
  14. Didenkulova, I., & Pelinovsky, E. (2011b). Runup of tsunami waves in U-shaped bays. Pure and Applied Geophysics, 168, 1239–1249. doi:10.1007/s00024-010-0232-8.CrossRefGoogle Scholar
  15. Fritz, H. M., Phillips, D. A., Okayasu, A., Shimozono, T., Liu, H. J., Mohammed, F., et al. (2012). The 2011 Japan tsunami current velocity measurements from survivor videos at Kesennuma Bay using LiDAR. Geophysical Research Letters, 39(7), L00G23. doi:10.1029/2011GL050686.CrossRefGoogle Scholar
  16. Fuentes, M. A., Ruiz, J. A., & Riquelme, S. (2015). The runup on a multilinear sloping beach model. Geophysical Journal International, 201, 915–928. doi:10.1093/gji/ggv056.CrossRefGoogle Scholar
  17. Harris, M. W., Nicolsky, D. J., Pelinovsky, E. N., Pender, J. M., & Rybkin, A. V. (2016). Run-up of nonlinear long waves in U-shaped bays of finite length: Analytical theory and numerical computations. Journal of Ocean Engineering and Marine Energy, 2(2), 113–127. doi:10.1007/s40722-015-0040-4.CrossRefGoogle Scholar
  18. Hibberd, S., & Peregrine, D. H. (1979). Surf and run-up on a beach: A uniform bore. Journal of Fluid Mechanics, 95(2), 323–345. doi:10.1017/S002211207900149X.CrossRefGoogle Scholar
  19. Kânoğlu, U. (2004). Nonlinear evolution and runup-rundown of long waves over a sloping beach. Journal of Fluid Mechanics, 513, 363–372. doi:10.1017/S002211200400970X.CrossRefGoogle Scholar
  20. Kânoğlu, U., & Synolakis, C. E. (2006). Initial value problem solution of nonlinear shallow water-wave equations. Physical Review Letters, 97, 148501. doi:10.1103/PhysRevLett.97.148501.CrossRefGoogle Scholar
  21. Kânoğlu, U., & Synolakis, C. E. (1998). Long wave runup on piecewise linear topographies. Journal of Fluid Mechanics, 374, 1–28. doi:10.1017/S0022112098002468.CrossRefGoogle Scholar
  22. Kânoğlu, U., Titov, V. V., Aydın, B., Moore, C., Stefanakis, T. S., Zhou, H., et al. (2013). Focusing of long waves with finite crest over constant depth. Proceedings of the Royal Society A, 469, 20130015. doi:10.1098/rspa.2013.0015.CrossRefGoogle Scholar
  23. Kânoğlu, U., Titov, V. V., Bernard, E. N., & Synolakis, C. E. (2015). Tsunamis: Bridging science, engineering and society. Philosophical Transactions of the Royal Society A, 373, 20140369. doi:10.1098/rsta.2014.0369.CrossRefGoogle Scholar
  24. Madsen, P. A., & Fuhrman, D. R. (2008). Run-up of tsunamis and long waves in terms of surf-similarity. Coastal Engineering, 55(3), 209–223. doi:10.1016/j.coastaleng.2007.09.007.CrossRefGoogle Scholar
  25. Madsen, P. A., & Schäffer, H. G. (2010). Analytical solutions for tsunami runup on a plane beach: Single waves, N-waves and transient waves. Journal of Fluid Mechanics, 645, 27–57. doi:10.1017/S0022112009992485.CrossRefGoogle Scholar
  26. O’Brien, L., Christodoulides, P., Renzi, E., Stefanakis, T., & Dias, F. (2015). Will oscillating wave surge converters survive tsunamis? Theoretical and Applied Mechanics Letters, 5(4), 160–166. doi:10.1016/j.taml.2015.05.008.CrossRefGoogle Scholar
  27. Okada, Y. (1985). Surface deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 75, 1135–1154.Google Scholar
  28. Okada, Y. (1992). Internal deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 82, 1018–1040.Google Scholar
  29. Okal, E. A. (2015). The quest for wisdom: Lessons from 17 tsunamis, 2004–2014. Philosophical Transactions of the Royal Society A, 373, 20140370. doi:10.1098/rsta.2014.0370.CrossRefGoogle Scholar
  30. Postacioglu, N., Özeren, M. S., & Canlı, U. (2016). On the resonance hypothesis of tsunami and storm surge runup. Natural Hazards and Earth System Sciences. doi:10.5194/nhess-2016-334 (in review).
  31. Pritchard, D., & Dickinson, L. (2007). The near-shore behaviour of shallow-water waves with localized initial conditions. Journal of Fluid Mechanics, 591, 413–436. doi:10.1017/S002211200700835X.CrossRefGoogle Scholar
  32. Rybkin, A., Pelinovsky, E. N., & Didenkulova, I. (2014). Nonlinear wave run-up in bays of arbitrary cross-section: Generalization of the Carrier-Greenspan approach. Journal of Fluid Mechanics., 748, 416–432. doi:10.1017/jfm.2014.197.CrossRefGoogle Scholar
  33. Sepulveda, I., & Liu, P. L. F. (2016). Estimating tsunami runup with fault plane parameters. Coastal Engineering, 112, 57–68. doi:10.1016/j.coastaleng.2016.03.001.CrossRefGoogle Scholar
  34. Synolakis, C. E. (1987). The runup of solitary waves. Journal of Fluid Mechanics, 185, 523–545. doi:10.1017/S002211208700329X.CrossRefGoogle Scholar
  35. Synolakis, C. E., & Bernard, E. N. (2006). Tsunami science before and beyond Boxing Day 2004. Philosophical Transactions of the Royal Society A, 364, 2231–2265. doi:10.1098/rsta.2006.1824.CrossRefGoogle Scholar
  36. Synolakis, C. E., Bernard, E. N., Titov, V. V., Kânoğlu, U., & González, F. I. (2008). Validation and verification of tsunami numerical models. Pure and Applied Geophysics, 165(11–12), 2197–2228. doi:10.1007/s00024-004-0427-y.CrossRefGoogle Scholar
  37. Tadepalli, S., & Synolakis, C. E. (1994). The run-up of N-waves on sloping beaches. Proceedings of the Royal Society A, 445, 99–112. doi:10.1098/rspa.1994.0050.CrossRefGoogle Scholar
  38. Tinti, S., & Tonini, R. (2005). Analytical evolution of tsunamis induced by near-shore earthquakes on a constant-slope ocean. Journal of Fluid Mechanics, 535, 33–64. doi:10.1017/S0022112005004532.CrossRefGoogle Scholar
  39. Titov, V. V., Kânoğlu, U., & Synolakis, C. E. (2016). Development of MOST for real-time tsunami forecasting. Journal of Waterway, Port, Coastal, and Ocean Engineering, 142, 03116004. doi:10.1061/(ASCE)WW.1943-5460.0000357.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Civil EngineeringAdana Science and Technology UniversityAdanaTurkey
  2. 2.Department of Engineering SciencesMiddle East Technical UniversityAnkaraTurkey

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