Dispersive and Nondispersive Nonlinear Long Wave Transformations: Numerical and Experimental Results
The description of gravity waves propagating on the water surface is considered from a historical point of view, with specific emphasis on the development of a theoretical framework and equations of motion for long waves in shallow water . This provides the foundation for a subsequent discussion about tsunami wave propagation and runup on a sloping beach, and in particular the role of wave dispersion for this problem. Wave tank experiments show that wave dispersion can play a significant role for the propagation and wave transformation of wave signals that include some higher frequency components. However, the maximum runup height is less sensitive to dispersive effects, suggesting that runup height can be adequately calculated by use of nondispersive model equations.
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This work was supported by Estonian Research Council (ETAg) grant PUT1378. Authors also thank the PHC PARROT project No 37456YM, which funded the authors’ visits to France and Estonia and allowed this collaboration.
- Airy, G.B.: Tides and waves. Encyclopaedia Metropolitana (1845)Google Scholar
- Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. Journal de Mathématiques Pures et Appliquées 17, 55–108 (1872)MathSciNetzbMATHGoogle Scholar
- Craik, D.D.: The origins of water wave theory. Annu. Rev. Fluid Mech. 36, 1–28 (2004). https://doi.org/10.1146/annurev.fluid.36.050802.122118 MathSciNetCrossRefGoogle Scholar
- Craik, D.D .: Georges Gabriel Stokes on water wave theory. Annu. Rev. Fluid Mech. 37, 23–42 (2005). https://doi.org/10.1146/annurev.fluid.37.061903.175836 MathSciNetCrossRefGoogle Scholar
- Fenton, J.D.: The long wave equations. Alternative Hydraulics Paper 1, http://johndfenton.com/Papers/01-The-long-wave-equations.pdf (2010)
- Madsen, P.A., Banijamali, B., Schäffer, H.A., Sørensen, O.R.: Boussinesq type equations with high accuracy in dispersion and nonlinearity. In: Edge, B. (ed.) Coastal Engineering 25th Conference 1996, vol. 1, pp. 95–108. ASCE, Hørsholm, Denmark (1997). https://doi.org/10.1061/9780784402429.008 CrossRefGoogle Scholar
- [Lord] Rayleigh: On waves. Phil. Mag. 1, 257–279 (1876)Google Scholar
- Russell, J.S.: Report on waves. In: Report of the Fourteenth Meeting of the British Association for the Advancement of Science, pp. 311–390 (1844)Google Scholar
- Stokes, G.G.: Report on recent researches in hydrodynamics. Rep. Br. Assoc. Adv. Sci. 1–20 (1846)Google Scholar