On Some Nonlinear Wave Diffraction and Refraction Solutions in Shallow Waters

Abstract

Abstract Diffraction and refraction of nonlinear shallow water waves due to uneven bathymetry is studied numerically in two and three dimensions. The numerical tank consists of a wavemaker at the upwave side of the domain, the submerged obstacles in the middle of the domain, and a numerical wave absorber on the downwave of the domain. The numerical wavemaker is capable of generating solitary and cnoidal waves as solutions of the Green-Naghdi (GN) equations. The nonlinear wave refraction and diffraction is studied by use of the Level I GN equations. The system of equations are solved numerically in time domain by use of a second-order finite difference approach, and in a boundary-fitted coordinate system. Various forms of three-dimensional bathymetry with large slopes, including flat and curved ramps from deep to shallow regions are considered. Results include solitary and cnoidal wave surface elevation and particle velocities and are compared with the existing solutions where possible. Overall very good agreement is observed. Discussion is provided on the nonlinearity and dispersion effects on the wave diffraction and refraction, as well as on the performance of the GN equations in solving these problems.