arXiv:1204.3354v2 [physics.flu-dyn] 23 Apr 2012 1 On a new type of solitary surface waves in finite water depth Shijun LIAO State Key Laboratory of Ocean Engineering, Dept. of Mathematics School of Naval Architecture, Ocean and Civil Engineering Shanghai Jiao Tong University, Shanghai 200240, China (Email address: sjliao@sjtu.edu.cn) Abstract In this paper, a new type of solitary surface waves in a finite water depth is found by analytically solving the fully nonlinear wave equations. Using a new type of base functions which decays exponentially in the horizontal direction, this new type of solitary surface waves is gained first by means of linear wave equations, and then confirmed by the fully nonlinear wave equations. The new type of solitary surface waves have many unusual characteristics. First, it has a peaked crest. Secondly, it may be in the form of depression, which has been often reported for internal solitary waves but never for free-surface solitary ones, to the best of author’s knowledge. Third, its phase speed has nothing to do with wave height, say, the peaked solitary waves are non-dispersive. Finally, its horizontal velocity at bottom is always larger than that on surface. All of these are so different from the traditional periodic and solitary waves that they clearly indicate the novelty of the peaked solitary waves. Based on the new peaked solitary surface waves, a new explanation to the so-called rogue waves and some theoretical predictions are given. All of these are helpful to deepen our understandings and enrich our knowledge about solitary waves. Key words Solitary wave, peaked crest, progressive wave, fully nonlinear 1 Introduction Since the solitary surface wave was discovered by John Scott Russell in 1834, various types of solitary waves have been found. The mainstream models of shallow water waves, such as the Boussinesq equation [1], the KdV equation [2], the BBM equation [3] and so on, admits dispersive smooth periodic and solitary waves of permanent form: the wave elevation is infinitely differentiable in the whole domain. Especially, the phase speed of these smooth water waves is closely related to the wave height: in general, a wave with higher amplitude travels faster than a lower one. Such kind of smooth periodic and solitary waves have been the mainstream of the teaching and investigating of water waves for quite a long time. However, in theory, the discontinuity of water wave elevation appears accidentally. It is well-known that the limiting gravity wave has a corner crest with 120 degree, as