Abstract
In this paper, the mathematical toeory on progressive translation waves of the permanent type is developed. The treatment of the second approximation based upon the momentum and energy approaches considering the effects of the vertical acceleration shows that the surface profile of waves is expressed in terms of an elliptic function which is designated as the cn-function, the name cnoidal being derived therefrom as analogous to sinusoidal, and the solitary wave is included as a special case of the cnoidal wave. It should be noticed that the behaviour of cnoidal waves may be reduced to that of classical sinusoidal waves at a special limiting case.
