New Analytical Solution for Nonlinear Shallow Water-Wave Equations

Kanoğlu, Utku
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.


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...
Analytical modeling of nonlinear evolution of long waves
Aydın, Baran; Kanoğlu, Utku (2015-06-22)
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 wav...
Analytical solution for the propagation of finite crested long waves over a sloping beach
Yağmur, Ahmed Sabri; Kanoğlu, Utku; Department of Aerospace Engineering (2022-2-10)
The analytical solution of shallow water-wave equations, both linear and nonlinear, is widely used to provide an insightful understanding of the coastal effect of long waves. These solutions are generally carried out for two-dimensional (1 space + 1 time) propagation, even though there are a limited number of analytical solutions for the three-dimensional (2 space + 1 time) propagation. Three-dimensional propagation of long waves over a sloping beach is considered here. The analytical solution is obtained u...
Initial value problem solution of nonlinear shallow water-wave equations
Kanoğlu, Utku (2006-10-06)
The initial value problem solution of the nonlinear shallow water-wave equations is developed under initial waveforms with and without velocity. We present a solution method based on a hodograph-type transformation to reduce the nonlinear shallow water-wave equations into a second-order linear partial differential equation and we solve its initial value problem. The proposed solution method overcomes earlier limitation of small waveheights when the initial velocity is nonzero, and the definition of the init...
New Formulations for Tsunami Runup Estimation
Kanoğlu, Utku; Ceylan, Nihal (null; 2017-12-11)
We evaluate shoreline motion and maximum runup in two folds: One, we use linear shallow water-wave equations over a sloping beach and solve as initial-boundary value problem similar to the nonlinear solution of Aydın and Kanoglu (2017, Pure Appl. Geophys., Methodology we present here is simple; it involves eigenfunction expansion and, hence, avoids integral transform techniques. We then use several different types of initial wave profiles with and without initial ...
Citation Formats
B. AYDIN and U. Kanoğlu, “New Analytical Solution for Nonlinear Shallow Water-Wave Equations,” PURE AND APPLIED GEOPHYSICS, pp. 3209–3218, 2017, Accessed: 00, 2020. [Online]. Available: