Cnoidal-type surface waves in deep water

Two-dimensional potential flows due to progressive surface waves in deep water are considered. For periodic waves, only gravity is included in the dynamic boundary condition, but both gravity and surface tension are taken into account for solitary waves. The validity of the steady first-order cnoidal wave approximation, i.e. the periodic solution of KdV, is extended to infinite depth by renormalizations. This renormalized cnoidal wave (RCW) solution is expressed as a Fourier–Padé approximation. It is analytically simpler and more accurate than fifth-order Stokes approximations. It is also capable of describing the recently discovered sharp-crested wave. A sharp-crested wave is obtained when the fluid velocity at the crest is larger than the phase speed. When the wavelength is infinite, RCW yields an algebraic solitary wave. Depending on the surface tension, the solitary wave involves one or two interfaces: a wave of depression; a wave of depression with a pocket of air; a wave of elevation with a pocket of air. Solitary waves are found for all values of the surface tension. However, this does not necessarily mean that these waves are solutions of the exact equations. Moreover, RCW approximate solitary waves always present a dipole singularity. It is also shown that a cnoidal wave in deep water can be rewritten as a periodic distribution of dipoles, each dipole representing an algebraic solitary wave. This provides a new paradigm for descriptions of water wave phenomena.