Journal of Engineering Mathematics

, Volume 59, Issue 1, pp 61–82 | Cite as

Middle-field formulation for the computation of wave-drift loads

  • Xiao-Bo ChenEmail author
Original Paper


New formulations of second-order wave loads contributed by a first-order wave field are developed by applying two variants of Stokes’s theorem and Gauss’s theorem to a formulation consisting of direct pressure integrations on a body’s hull which is called the near-field formulation. In addition to this direct formulation and the formulation derived from the momentum theorem called the far-field formulation, for the computation of drift (surge/sway) forces in horizontal directions and drift (yaw) moment around the vertical axis, one of new formulations is defined on the control surfaces surrounding the body and called the middle-field formulation. After a brief summary of both pressure-integration (near-field) and momentum (far-field) formulations, the development of the middle-field formulation involving control surfaces is described and complemented in detail in the appendices. The application of the new formulation shows that the near-field and far-field formulations are mathematically equivalent for wall-sided, as well as non-wall-sided bodies and under the condition that the mean yaw moments are expressed with respect to a space-fixed reference point. It is shown that the middle-field formulation is as robust as the far-field formulation and as general as the near-field formulation of second-order loads on a single body as well as on multiple bodies. Furthermore, the extension to the computation of a second-order oscillatory load, which is so far accessed only by the near-field formulation, is envisioned.


Far-field formulation Middle-field formulation Near-field formulation Wave-drift load 


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  1. 1.
    Pinkster JA, van Oortmerssen G (1977) Computation of the first- and second-order wave forces on oscillating bodies in regular waves. Proc. 2nd Int. Conf. Num. Ship Hydrodynamics, Berkley, USA, pp 136–156Google Scholar
  2. 2.
    Molin B (1979a) Computations of wave drift forces. Proc. OTC Conf., Houston, USA, paper No. 3627Google Scholar
  3. 3.
    Molin B (1979b) Second-order drift forces upon large bodies in regular waves. Proc. BOSS’79., London, UK, pp 363–370Google Scholar
  4. 4.
    Pinkster JA (1980) Low frequency second order wave exciting forces on floating structures. H. Veenman En Zonen B.V., The Netherlands, WageningenGoogle Scholar
  5. 5.
    Ogilvie TF (1983) Second-order hydrodynamic effects on ocean platforms. Proc. Intl Workshop on Ship & Platform Motions, Berkeley, USA, pp 205–265Google Scholar
  6. 6.
    Molin B, Hairault JP (1983) On second-order motion and vertical drift forces for three-dimensional bodies in regular waves. Proc. Intl Workshop on Ship & Platform Motions, Berkeley, USA, pp 344–362Google Scholar
  7. 7.
    Maruo H (1960) The drift of a body floating on waves. J Ship Res 4:1–10ADSGoogle Scholar
  8. 8.
    Newman JN (1967) The drift force and moment on ships in waves. J Ship Res 11:51–60Google Scholar
  9. 9.
    Newman JN, Lee CH (2001) Boundary-element methods in offshore structure analysis. Proc. 20th Intl Conf. Off. Mech. Arc. Engeng, Rio de Janeiro, BrazilGoogle Scholar
  10. 10.
    Dai YS (1998) Potential flow theory of ship motions in waves in frequency and time domain (in Chinese). Press of the National Defense Industries, Beijing, ChinaGoogle Scholar
  11. 11.
    Ferreira MD, Lee CH (1994) Computation of second-order mean wave forces and moments in multibody interaction. Proc. 7th Intl Conf. Behaviour Off. Structures, BOSS’94, Boston, USA, vol 2, pp 303–313Google Scholar
  12. 12.
    Noblesse F (1982) The Green function in the theory of radiation and diffraction of regular waves by a body, J Engg Math 16:137–169zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Newman JN (1992) The approximation of free-surface Green functions, Wave asymptotics—Proc. F. Ursell Retirement Meeting, Cambridge University Press, pp 107–135Google Scholar
  14. 14.
    Chen XB (1993) Evaluation de la fonction de Green du problème de diffraction/radiation en profondeur d’eau finie—Une nouvelle méthode rapide and précise, Proc. 4e Journées de l’Hydrodynamique, Nantes, France, pp 371–384Google Scholar
  15. 15.
    Guével P (1982) Le problème de diffraction-radiation—Premième partie: Théorèmes fondamentaux. Ecole Supérieure de Mécanique de Nantes, Nantes, FranceGoogle Scholar
  16. 16.
    Kashiwagi M, Endo K, Yamaguchi H (2005) Wave drift forces and moments on two ships arranged side by side in waves. J Ocean Eng 32:529–555CrossRefGoogle Scholar
  17. 17.
    Chen XB (2004) Hydrodynamics in Offshore and Naval Applications—Part I. Keynote lecture at the 6th Intl Conference on Hydrodynamics, Perth, AustraliaGoogle Scholar
  18. 18.
    Chen XB (1988) Etude des réponses du second ordre d’une structure soumise à une houle aléatoire. Ph.D. Thesis, ENSM, Univ. Nantes, Nantes, FranceGoogle Scholar
  19. 19.
    Bronshtein IN, Semendyayev KA (1998) Handbook of Mathematics. Springer-Verlag, Berlin, HeidelbergGoogle Scholar
  20. 20.
    Newman JN (1977) Marine Hydrodynamics. The MIT Press, Cambridge, USAGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Research DepartmentBUREAU VERITASCourbevoieFrance

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