A new way for the calculation of steady periodic capillary-gravity waves on deep water

The method developed by Longuet-Higgins (1978) for the calculation of steady gravity waves is extended to steady capillary-gravity waves in deep water. Introduction of surface tension provides an additional term in the quadratic identities between the Fourier coefficients in Stokes's expansion. This term is reduced to a simple function of the slope of the local tangent to the profile of the free surface. A set of nonlinear algebraic equations is derived and solved using Newton's method. Good agreement is found between the present results and those given by previous methods. Limiting profiles of gravity-like waves are accurately calculated and a new family of limiting gravity waves is found.