Transformation of progressive waves propagating obliquely on gentle slope

This paper reports the results of a new mathematical derivation for the transformation of a progressive wave propagating obliquely on a gentle slope. On the basis of the conservation principle of wave motion and in a wave-ray coordinate system, an explicit expression for the velocity potential of the wave field is derived as a function of the bottom slope a perturbed to a second order in an Eulerian coordinate system. Wave profile is then obtained in the Lagrangian system. Together, these enable the description of the features of wave shoaling and refraction in the direction of wave propagation from deep to shallow water, particularly, the process of successive deformation of a wave profile.