Periodic Water Waves: Cnoidal and Solitary Profiles

This research formulates the water surface elevation equation for water waves, yielding periodic cnoidal and solitary wave profiles. The equation is derived by integrating the Kinematic Free Surface Boundary Condition with respect to time. The relationships among wave period, wave amplitude, and proportional wavelength facilitate the generation of both cnoidal and solitary wave profiles. In deep water, where wave dynamics are unaffected by the sea bottom, only cnoidal wave profiles are produced. In contrast, solitary profiles emerge through the shoaling-breaking process as waves approach shallower depths.