Propagating Stationary Surface Potential Waves in a Deep Ideal Fluid

A new exact solution of the problem for propagating stationary potential wave of an arbitrary amplitude in a deep ideal homogeneous fluid was constructed. Calculated wavy surface is represented by transcendental Lambert’s complex functions. For a physical interpretation of the results real linear combinations of the solutions were formed. The range of the wave steepness values, in which the real sum of constructed comprehensive solutions describes waves with smooth crests, is defined. In the limiting case of waves with small but finite amplitude as well as infinitesimal amplitude, the real combinations of the solutions are transferred in classical nonlinear and linear asymptotic Stokes expressions. Another real combination of constructed complex solutions describing waves with cusped crests do not fall within the range of conditions for the existence of stationary waves.