Some New Relations Between Stokes's Coefficients in the Theory of Gravity Waves

Abstract

Stokes's method of calculating the form of steady finite-amplitude, gravity waves in deep water involves a series of coefficients C_n related to the Fourier coefficients of the free surface elevation. The condition of constant pressure at the free surface yields a series of cubic relations between the C_n, which are normally used for calculations. In this paper it is shown that the C_n also satisfy some simpler, quadratic relations, which render the calculation of the profile faster and more accurate. The new relations are equivalent to certain integral properties involving the square of the particle speed, integrated along a streamline. This enables a generalization to be readily made to waves in water of finite depth.