The Fourier Coefficients of Stokes’ Waves

  • Pavel I. Plotnikov
  • John F. Toland
Part of the International Mathematical Series book series (IMAT, volume 1)


It is common to formulate the Stokes wave problem as Nekrasov’s nonlinear integral equation to be satisfied by a periodic function θ which gives the angle between the tangent to the wave and the horizontal. The function θ is odd for symmetric waves. In that case, numerical calculations using spectral methods reveal the coefficients in the sine series of θ to form a sequence of positive terms that converges monotonically to zero. In this paper, we prove that the Fourier sine coefficients of θ form a log-convex sequence that converges monotonically to zero. In harmonic analysis there are many very beautiful theorems about the behavior of functions whose Fourier sine series form a convex monotone sequence tending to zero.


Harmonic Function Fourier Coefficient Stokes Wave Symmetric Wave Sine Series 
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© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Pavel I. Plotnikov
  • John F. Toland

There are no affiliations available

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