Theoretical analysis of surface waves propagation on sloping bottoms: Part 1

A new mathematical derivation is developed to describe the periodic gravity surface waves propagating on sloping bottoms. In the Eulerian coordinate system, the velocity potential is obtained as a function of the wave steepness ɛ to the first order and the bottom slope α to the third order. The wave profile is then transformed into the Lagrangian system, and the analytical solution of wave profile and asymmetry parameter on a given bottom can thus be obtained. These enable the description of the features of wave shoaling in the direction of wave propagation from deep to shallow water, particularly, the process of successive deformation of a wave profile. Furthermore, by comparing the theoretical values of wave asymmetry with experimental data, it is found that theoretical results of the present solution has similar tendency with the experimental data.