Analytical modeling of nonlinear evolution of long waves

Aydın, Baran
Kanoğlu, Utku
We present an initial-boundary value problem formulation for the solution of the nonlinear shallow-water wave (NSW) equations. We transform the nonlinear equations into a linear problem by using the Carrier-Greenspan transformation. Then, we obtain the solution through the separation of variables method rather than integral transform techniques, which is the usual practice (Carrier et al., J Fluid Mech 2003; Kanoglu, J Fluid Mech 2004). This formulation allows the use of any physically realistic initial waveform, with and without initial velocity. We consider propagation of different incident wave profiles over a sloping beach, such as solitary waves and N-waves. Comparison of the results of our model with the existing integral transform solutions (Carrier et al., 2003; Tinti and Tonini, J Fluid Mech 2005; Kanoglu and Synolakis, Phys Rev Lett 2006) demonstrates its versatility. We then consider the forced NSW equations to model underwater landslides. We introduce approximations to the governing equations, as the forced equations cannot be transformed into a linear governing equation exactly. We solve the resultant equation using both integral transform and separation of variables techniques. We compare our solution with the linear analytical and nonlinear numerical results of Liu et al. (J Fluid Mech 2003) for a deforming Gaussian disturbance on the ocean floor and determine the ranges of different parameters for which the approximate nonlinear theory is valid.
Citation Formats
B. Aydın and U. Kanoğlu, “Analytical modeling of nonlinear evolution of long waves,” presented at the International Union of Geodesy and Geophysics (2015), Prag, Çek Cumhuriyeti, 2015, Accessed: 00, 2021. [Online]. Available: