Nonlinear evolution and runup–rundown of long waves over a sloping beach

The initial value problem of the nonlinear evolution, shoreline motion and flow velocities of long waves climbing sloping beaches is solved analytically for different initial waveforms. A major difficulty in earlier work utilizing hodograph-type transformation when solving either boundary value or initial value problems has been the specification of equivalent boundary or initial condition in the transformed space. Here, in solving the initial value problem, the transformation is linearized in space at t = 0, then the full nonlinear transformation is used to solve the initial value problem of the nonlinear shallow-water wave equations. A solution method is presented to describe the most physically realistic initial waveforms and simplified equations for the runup-rundown motions and shoreline velocities. This linearization of the initial condition does not appear to affect the subsequent nonlinear evolution, as shown through comparisons with earlier studies. Comparisons with runup results from solutions of the boundary value problem suggest the same variation with the runup laws. The methodology presented here appears simpler than earlier work as it does not involve the numerical calculation of singular elliptic integrals.
Journal of Fluid Mechanics


Analytical solutions for evolution and runup of longwaves over a sloping beach
Ceylan, Nihal; Kanoğlu, Utku; Department of Engineering Sciences (2019)
The initial value problem of the linear evolution and runup of long waves on a plane beach is analyzed analytically. The shallow water-wave equations are solved by integral transform and eigenvalue expansion methodologies. The results from linear solutions are compared with the solution of the nonlinear shallow water-wave equations confirming the runup invariance, i.e. nonlinear and linear theories produce same maximum runup. Then, existing analytical nonlinear solution for shoreline motion is implemented f...
Yalçıner, Ahmet Cevdet; Synolakis, Costas E. (2004-06-12)
The runup of long waves on the sloping planes is described by the analytical solutions of the long wave equations with special initial conditions, proper approximations and boundary conditions. These studies are also compared with experimental data (Yeh et al. (1996); Lin et al. (1999); Yalciner et al. (2003)). Similarly the numerical methods solving governing equations with proper boundary conditions are also developed and compared with either analytical or experimental or field data for long wave propagat...
Analytical solutions of shallow-water wave equations
Aydın, Baran; Kanoğlu, Utku; Department of Engineering Sciences (2011)
Analytical solutions for the linear and nonlinear shallow-water wave equations are developed for evolution and runup of tsunamis –long waves– over one- and two-dimensional bathymetries. In one-dimensional case, the nonlinear equations are solved for a plane beach using the hodograph transformation with eigenfunction expansion or integral transform methods under different initial conditions, i.e., earthquake-generated waves, wind set-down relaxation, and landslide-generated waves. In two-dimensional case, th...
Kanoğlu, Utku (2004-06-12)
The initial value problem solution of the nonlinear shallow water–wave equations developed by Kânoğlu (2004) is applied to the benchmark problem — tsunami runup onto a plane beach — and results are compared. Comparisons with the benchmark solution produce identical results.
Numerical analysis of a projection-based stabilization method for the natural convection problems
Çıbık, Aytekin Bayram; Kaya Merdan, Songül; Department of Mathematics (2011)
In this thesis, we consider a projection-based stabilization method for solving buoyancy driven flows (natural convection problems). The method consists of adding global stabilization for all scales and then anti-diffusing these effects on the large scales defined by projections into appropriate function spaces. In this way, stabilization acts only on the small scales. We consider two different variations of buoyancy driven flows based on the projection-based stabilization. First, we focus on the steady-sta...
Citation Formats
U. Kanoğlu, “Nonlinear evolution and runup–rundown of long waves over a sloping beach,” Journal of Fluid Mechanics, pp. 363–372, 2004, Accessed: 00, 2020. [Online]. Available: