The solitary and cnoidal waves in shallow water

The problems associated with shallow-water waves has received considerable attention in the recent years. In the case of solitary and cnoidal waves, it is often assumed that the nonlinear and dispersion effects balance whilst dissipation is slight or totally neglected in the areas of the shallow water concerned. As a follow-up, this model attempts to provide a description of the classical solitary waves and the related cnoidal oscillations using the traditional shallow-water equations (WHITHAN G. B., Linear and Non-Linear Waves (Wiley and Sons) 1973, pp. 460-470). The derivations in this case do not involve any series expansion; and thus, they differ significantly from the approach using the Korteweg-de Vries (KDV) equations (WHITHAN G. B., Linear and Non-Linear Waves (Wiley and Sons) 1973, pp. 460-470; ZABUSKY N. J, and GALVIN C. J., Shallow waves. The KDV equation and solitons, J. Fluid Mech., 47 (1971) 811-824). Consequently, this model attempts to describe the wave pattern even when the wave height grows as high as very close in magnitude to the depth of the corresponding undisturbed water layer or before breaking begins. In this consideration, realistic shallow-water parameters are used to examine the structure of the wave processes concerned.