Volume 31 Issue 4 - March 3, 2017 PDF
Evolution of breaking waves on sloping beaches
Yang-Yih Chen1,2,3,*, Bin-Da Yang4
1 Department of Marine Environment and Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
2 International Wave Dynamics Research Center, National Cheng Kung University, Tainan 70101, Taiwan
3 Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan 70101, Taiwan
4 Department of Hydraulics & Ocean Engineering, National Cheng Kung University, Tainan 70101, Taiwan
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For the shoaling process of wave propagating on a gently sloping bottom, authors have systematically discussed on a single progressive wave; however, wave–wave interaction usually occurs and is important in ocean. Especially the evolution of shortwave riding on long wave will need to be further gone into.

We provide an analytic solution for two progressive waves propagating on a sloping bottom, which the writers believe was not treated appropriately in previous available reports. In explicit form both the velocity potential and the surface elevation are expanded as a three-parameter function of rendering the two wave ordering parameter ε1, ε2 and the bottom slope α perturbed to the third order in the Eulerian coordinate system. The results possess the configuration of the general solution; therefore, it can degenerate into a single propagating wave on sloping beach, and wave–wave interaction in uniform depth. We present that the ratio of wave number ki/λi0 increases with the decrease in dimensionless water depth λi0h. It also relates to the bottom slope and incident wave steepness. That is, increasing bottom slope or decreasing incident wave steepness increases the ratio. The angular frequency is also affected by another propagating wave. Decreasing water depth or increasing the ratio λ10/λ20 will enhance the effect mentioned above. Our solutions included the wave–wave interaction, wave shoaling, nonlinearity, and the effect of bottom slope. Those were observed by means of the discussion of the free-surface displacement.

The resonance occurring in specified water depth between the ratioλ1020 ≈ 2.3 and λ1020 = 4 can be interpreted as being due to the wave–wave interaction. However, the singularity of free-surface displacement under the resonant conditions will be further studied in the future.
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