Computer extension and analytic continuation of Stokes’ expansion for gravity waves

1974/02/11 by Leonard W. Schwartz · 15 citations
Earth and Planetary Sciences · Mathematics · #Ocean Waves and Remote Sensing #Coastal and Marine Dynamics #Oceanographic and Atmospheric Processes #Continuation #Infinitesimal #Mathematical analysis #Analytic continuation #Wavelength #Computation #Convergence (economics) #Extension (predicate logic) #Square root #Series (stratigraphy) #Mathematics #Physics #Geometry #Optics #Computer science #Algorithm #Geology

paper · doi:10.1017/s0022112074000802

openalex publication_date 1974/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Stokes’ infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic. Stokes’ expansion has been found to be incapable of yielding the highest wave for any value of the water depth since convergence is limited by a square-root branch-point some distance short of the maximum. By reformulating the problem using a different independent parameter, the highest waves are obtained correctly. Series summation and analytic continuation are facilitated by the use of Padé approximants. The method is valid in principle for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth. An alternative expansion procedure proposed by Havelock for the computation of waves short of the highest has been reconsidered and found to be defective.

Citations

• The singularity at the crest of a finite amplitude progressive Stokes wave
• XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves
• The disintegration of wave trains on deep water Part 1. Theory

Cited by

• Instabilities of finite-amplitude water waves
• On the Lagrangian description of steady surface gravity waves
• New asymptotic description of nonlinear water waves in Lagrangian coordinates
• A close one-term approximation to the highest Stokes wave on deep water
• Why is a water wave like a grandfather clock?
• Steep gravity waves in water of arbitrary uniform depth
• The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics
• Integral properties of periodic gravity waves of finite amplitude
• Limiting gravity waves in water of finite depth
• Three-Dimensional Instability of Finite-Amplitude Water Waves
• On the existence of a wave of greatest height and Stokes’s conjecture
• Superharmonic instability of stokes waves
• Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Padé approximation
• Strongly Nonlinear Waves
• Stokes wave [wikipedia]