Weakly nonlinear edge waves are investigated using an asymptotic procedure in the framework of shallow-water theory. The structure and nonlinear dispersion relation is found explicitly for periodic edge waves of different modes over cylindrical shelf with arbitrary profile. Coefficients of nonlinear correction in dispersion relation are calculated and the nonlinear Shrodinger equation is derived.
Edge waves belong to the class of coastally trapped waves, which play dominant role in the dynamics of oceanic coastal zone. Such waves have been detected in the wave field in the oceanic coastal zone many times; see for instance (Huntley and Bowen 1973; Bryan Hows and Bowen 1998). Edge waves are often considered as the major factor of the long-term evolution of irregular coastal line, forming the rhythmic crescentic bars (Bowen and Inman 1971; Komar 1998). (Komar 1998) gives several excellent pictures of coastal line structures induced by edge waves. These waves in the recent years are the object of numerous studies in hydrodynamics, hydraulic and coastal engineering. The linear long-wave theory of edge waves can be found in books (Le Blond and Mysak 1981; Rabinovich 1993). Weakly nonlinear Stokes edge waves were studied in (Whitham 1976), and the paper (Minzoni 1976) shows that the results of shallow-water theory are in a good agreement with the full theory. Properties of nonlinear edge waves were studied in (Yeh 1985) experimentally and theoretically in the framework of the Schrodinger equation. In the present work weakly nonlinear edge waves are investigated using an asymptotic procedure similar to that (Whitham 1976) in the framework of shallow-water theory. The structure and nonlinear dispersion relation are found explicitly for periodic edge waves of different modes over cylindrical shelf with arbitrary profile. Several forms of bottom geometry, such as linearly sloped shelf and concave exponential shelf, are considered as examples.