Accurate series solutions for gravity-driven Stokes waves

Michael C. Dallaston, Scott W. McCue

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)
35 Downloads (Pure)


In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that a higher order behavior of the series generated by the solution casts doubt over arguments that rely on estimating the series' radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Padé approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Padé approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.

Original languageEnglish
Article number082104
JournalPhysics of Fluids
Issue number8
Publication statusPublished - Aug 2010
Externally publishedYes


  • Dispersion relations
  • Numerical solutions
  • Bernoulli's principle
  • Exact solutions
  • Solution processes

ASJC Scopus subject areas

  • Condensed Matter Physics


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