Expressions in closed form are derived for a number of local properties of a random, irrotational wave field. They are: (i) the mean potential and kinetic energies per unit projected area; (ii) the energy balance among the processes of energy input from the surface pressure fluctuations, rate of growth of potential and kinetic energy and horizontal energy flux; and (iii) the partition between potential and kinetic energy. These expressions are mainly in terms of quantities measured at the free surface, which are therefore functions of only two spatial variables (x, y) and of time t.

Approximations for these expressions can be found simply by subsequent expansion methods; the fourth order being the highest for which the assumption of irrotational motion is appropriate in a real fluid. It is shown that the mean product of any three first-order quantities is of fourth or higher order in the root-mean-square wave slope, and this result is applied in estimating the magnitude of some higher order effects. In particular, the skewness of the surface displacement is of the order of the root-mean-square surface slope, which has been confirmed observationally by Kinsman (1960).