New Analytical Solution for Nonlinear Shallow Water-Wave Equations
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.
KeywordsTsunami long wave runup shallow water-wave
This work is partially supported through the grant provided by Middle East Technical University with the project no: BAP-08-11-DPT2002K120510. We acknowledge the partial support from the Scientific and Technological Research Council of Turkey project no. 109Y387 of the joint research program between Turkey and Greece. This contribution was also partially supported by the project ASTARTE (Assessment, STrategy And Risk Reduction for Tsunamis in Europe), Grant 603839, 7th FP (ENV.2013.6.4-3).
- 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.
- Aydın, B. (2011). Analytical solutions of shallow-water wave equations. Ph.D. Thesis, Middle East Technical University, Ankara, Turkey.Google Scholar
- Bowman, F. (1958). Introduction to Bessel functions. New York: Dover Publications Inc.Google Scholar
- 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.
- 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
- 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
- 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
- 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).