Abstract
This paper describes the application of a recently developed analytic approach known as the homotopy analysis method to derive a solution for the classical problem of nonlinear progressive waves in deep water. The method is based on a continuous variation from an initial trial to the exact solution. A Maclaurin series expansion provides a successive approximation of the solution through repeated application of a differential operator with the initial trial as the first term. This approach does not require the use of perturbation parameters and the solution series converges rapidly with the number of terms. In the framework of this approach, a new technique to apply the Padé expansion is implemented to further improve the convergence. As a result, the calculated phase speed at the 20th-order approximation of the solution agrees well with previous perturbation solutions of much higher orders and reproduces the well-known characteristics of being a non-monotonic function of wave steepness near the limiting condition.
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Liao, SJ., Cheung, K.F. Homotopy analysis of nonlinear progressive waves in deep water. Journal of Engineering Mathematics 45, 105–116 (2003). https://doi.org/10.1023/A:1022189509293
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DOI: https://doi.org/10.1023/A:1022189509293