On the solution of the dispersion relation for water waves

A range of analytic approximations is derived for the roots of the water wa ve dispersion relation, by formulating it as an eigenvalue problem and usin g standard variational methods in conjunction with the Rayleigh-Ritz proced ure. The analytic approximations may be enhanced by a numerical iterative p rocedure derived by means of the same process. Thus, for example, a relativ ely simple explicit expression is derived for the real, positive root of th e 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 mac hine accuracy. (C) 1999 Elsevier Science Ltd. All rights reserved.