On the solution of the dispersion relation for water waves

Abstract

A range of analytic approximations is derived for the roots of the water wave dispersion relation, by formulating it as an eigenvalue problem and using standard variational methods in conjunction with the Rayleigh–Ritz procedure. The analytic approximations may be enhanced by a numerical iterative procedure derived by means of the same process. Thus, for example, a relatively simple explicit expression is derived for the real, positive root of the dispersion relation, which is in error by less than 0.1% across the whole frequency range; two iterations of this approximation give the root to machine accuracy.