Introduction. In the theory of long waves, and also for infinitesimal "surface" waves, the total wave energy is half kinetic and half potential to a first order of approximation. 1 It was pointed out by Rayleigh (1911), in proceeding by successive approximations to the solution of the problem of oscillatory permanent waves on the surface of deep water, that the kinetic energy exceeds the potential by an amount proportional to a quantity of the fourth order, when quantities of order higher than the fourth are neglected.
Platzman, George W.. 1947. "The partition of energy in periodic irrotational waves of the surface of deep water." Journal of Marine Research 6, (3). https://elischolar.library.yale.edu/journal_of_marine_research/636