A numerical model for wave propagation over three-dimensional (3D) bathymetry was developed based on High Order Spectral (HOS) method, considering the wave-maker boundary and variable bathymetry. The numerical model was validated by the comparison of numerical results and the 3D elliptical mound's experiment results. Based on this model, wave spatial transformation over a submerged shoal was investigated. Strong wave diffractions were observed after the shoal when incident waves passed across this terrain. Moreover, the effects of incident wave length, topographic height and topographic slope on the spatial evolution of waves over the shoal were analyzed respectively. The results show that the wave focusing effect, that is, the concentration of wave energy and the maximum wave height, decreases with the increase of incident wave length. And it increases with the increase of topographic height and the decrease of topographic slope.


Offshore wave propagation, a key issue in port design and construction, is generated by shallow water deformation, refraction, diffraction and reflection because of seabed topography in the process of wave propagating from deep sea to offshore area. As seabed topography determines wave-propagation direction, wavelength and wave height, it deserves a full consideration for the safety of engineering structures in offshore area. Therefore, the interaction between waves and topography has attracted intensive interest. It is significant to investigate the interaction between waves and topography.

For the study of the correlation between waves and topography, multiple approaches have been proposed in the last a few decades. The physical experiment is one of the main means to study the effect of wave and topography. For the researchers of numerical simulation, the comparison with the experimental results is an effective verification method. In the physical experiments, the submerged bar (Beji, 1993) and waves Bragg reflection experiment (Davis and Heathershaw, 1984) are typical cases for the study of two-dimensional wave and topography. In the numerical models, Teng (2010) and Liu et al. (2005) simulated the wave propagation on submerged bar by VOF method. Zheng (2004) and Liu (2005) simulated the wave propagation over submerged bar through the improved Boussinesq equation, which can fairly analyze nonlinear variation of waves. However, the real sea area is three-dimensional. In three-dimensional sea area, wave refraction and diffraction over topography had attracted many scholars' attention (Vincent and Briggs, 1989; Chawla and Kirby, 1998). Mile-slope equation was one of the main methods to study wave refraction and diffraction over topography. It was derived from Berkhoff's (1972) perturbation expansion under the assumption that the seabed topography changed slowly. Based on this, a new analytical solution of the mile-slope long wave equation was presented (Yu and Zhang, 2003). Then Liu and Li (2007) analyzed simple harmonic waves scattered over a submerged circular truncated shoal by an analytical solution of longwave equation in closed-form. Zhu and Harun (2009) and Niu and Yu (2011) further explored wave transformation over a submerged circular hump by an analytical solution of the shallow water wave equation. However, the mile-slope equation has certain restrictions for slope, and the computational error is large when the seabed topography changes dramatically. Most numerical models (Chen and Kirby, 2000; Zhang, 2014) can achieve effective simulation results, but the time-consuming of three-dimensional model is relatively large.

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