A numerical study of the second-order wave excitation of ship springing in infinite water depth

Yanlin Shao, Odd M. Faltinsen

Research output: Contribution to journalJournal articleResearchpeer-review


Assuming the wave steepness of the incident waves and the ship motions are small, the second-order weakly-nonlinear hydrodynamic problem of a ship moving with constant forward speed is studied numerically in a consistent way. The boundary value problem is formulated in a body-fixed coordinate system and the perturbation scheme is used. This formulation does not include any derivatives of the velocity potential on the right-hand side of the body-boundary conditions, and thus avoid the difficulties associated with the terms similar to the so-called mj-terms and their derivatives. The second-order sum-frequency wave excitation of ship springing is studied in both monochromatic and bichromatic head-sea waves. Different Froude numbers are considered. A time-domain Higher-Order Boundary Element Method based on cubic shape function is used as a numerical tool. An upstream finite difference scheme is used for longitudinal derivative terms in the free-surface conditions. For a modified Wigley hull in head-sea waves, it is found that the second-order velocity potential gives dominant contribution to second-order wave excitation of ship springing in the wave frequency region where sum-frequency springing occurs. Quadratic velocity terms in the Bernoulli equation have a relatively small contribution. The numerical results also demonstrate strong dependency of the second-order wave excitation of ship springing on the Froude numbers for small wave lengths. The effect of beam and draft is investigated. © The Author(s) 2011.
Original languageEnglish
JournalProceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment
Issue number2
Pages (from-to)103-119
Publication statusPublished - 2011
Externally publishedYes


  • Higher-Order Boundary Element Method
  • mj-terms
  • Nonlinear ship springing
  • Second-order wave theory
  • Body-fixed coordinate system


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