Journal of Engineering Mathematics

, Volume 57, Issue 1, pp 23–40 | Cite as

The linear-wave response of a periodic array of floating elastic plates

Original Paper


The problem of an infinite periodic array of identical floating elastic plates subject to forcing from plane incident waves is considered. This study is motivated by the problem of trying to model wave propagation in the marginal ice zone, a region of ocean consisting of an arbitrary packing of floating ice sheets. It is shown that the problem considered can be formulated exactly in terms of the solution to an integral equation in a manner similar to that used for the problem of wave scattering by a single elastic floating plate, the key difference here being the use of a modified periodic Green function. The convergence of this Green function in its original form is poor, but can be accelerated by a transformation. It is shown that the results from the method satisfy energy conservation and that in the particular case of a fixed rigid rectangular plate which spans the periodicity uniformly the solution reduces to that for a two-dimensional rigid dock. Solutions for a range of elastic-plate geometries are also presented.


Elastic plates Ice sheet Periodic Green function Water waves 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kashiwagi M (2000) Research on hydroelastic response of vlfs: Recent progress and future work. Int J Offsh Polar Engng 10(2):81–90Google Scholar
  2. 2.
    Watanabe E, Utsunomiya T, Wang CM (2004) Hydroelastic analysis of pontoon-type VLFS: A literature survey. Engng Struct 26:245–256CrossRefGoogle Scholar
  3. 3.
    Squire VA, Dugan JP, Wadhams P, Rottier PJ, Liu AJ (1995) Of ocean waves and sea ice. Annu Rev Fluid Mech 27:115–168CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Meylan MH, Squire VA (1996) Response of a circular ice floe to ocean waves. J Geophys Res 101(C4): 8869–8884CrossRefADSGoogle Scholar
  5. 5.
    Meylan MH (2002) The wave response of ice floes of arbitrary geometry. J Geophys Res - Oceans 107(C6): article no. 3005Google Scholar
  6. 6.
    Peter MA, Meylan MH (2004) Infinite depth interaction theory for arbitrary bodies with application to wave forcing of ice floes. J Fluid Mech 500:145–167MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Evans DV, Porter R (1995). Wave scattering by periodic arrays of waters. Wave Motion 23:95–120MathSciNetGoogle Scholar
  8. 8.
    Chou T (1998) Band structure of surface flexural-gravity waves along periodic interfaces. J Fluid Mech 369:333–350MATHADSMathSciNetGoogle Scholar
  9. 9.
    Twersky V (1952) Multiple scattering or radiation by an arbitrary configuration of parallel cylinders. J Acoust Soc Am 24:42–46CrossRefGoogle Scholar
  10. 10.
    Linton CM, Evans DV (1993) The interaction of waves with a row of circular cylinders. J Fluid Mech 251:687–708MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Fernyhough M, Evans DV (1995) Scattering by a periodic array of rectangular blocks. J Fluid Mech 305:263–279MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Linton CM (1998) The green’s function for the two-dimensional Helmholtz equation in periodic domains. J Engng Maths 33:377–402MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Porter R, Evans DV (1999) Rayleigh-bloch surface waves along periodic grating and their connection with trapped modes in waveguides. J Fluid Mech 386:233–258MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Maniar H, Newman JN (1997) Wave diffraction by long arrays of cylinders. J Fluid Mech 339:309–330MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Kashiwagi M (1991) Radiation and diffraction forces acting on an offshore-structure model in a towing tank. Int J Offs Polar Engng 1(2):101–106Google Scholar
  16. 16.
    Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill, New YorkGoogle Scholar
  17. 17.
    Porter D, Porter R (2004) Approximations to wave scattering by an ice sheet of variable thickness over undulating bed topography. J Fluid Mech 509:145–179MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    de Veubeke BF (1979) Cours d’elasticite. Springer-Verlag, New YorkGoogle Scholar
  19. 19.
    Tayler AB (1986) Mathematical models in applied mathematics. Clarendon Press, OxfordGoogle Scholar
  20. 20.
    Wehausen J, Laitone E (1960) Surface waves. In: Flügge S, Truesdell C (eds) Fluid dynamics III vol. 9 of Handbuch der Physik. Berlin, Heidelberg, Springer Verlag, pp 446–778Google Scholar
  21. 21.
    Scott C (1998) Introduction to optics and optical imaging. IEEE Press, New YorkGoogle Scholar
  22. 22.
    Kim WD (1965) On the harmonic oscillations of a rigid body on a free surface. J Fluid Mech 21:427–451MATHCrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Abramowitz M, Stegun IA (1970) Handbook of mathematical functions. Dover, New YorkGoogle Scholar
  24. 24.
    Wang CD, Meylan MH (2004) Higher order method for the wave forcing of a floating thin plate of arbitrary geometry. J Fluids and Struct 19(4):557–572CrossRefADSGoogle Scholar
  25. 25.
    Petyt M (1990) Introduction to finite element vibration analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  26. 26.
    Meylan MH (2001) A variation equation for the wave forcing of floating thin plates. J Appl Ocean Res 23(4):195–206CrossRefGoogle Scholar
  27. 27.
    Hildebrand FB (1965) Methods of applied mathematics (2nd edn.). Prentice-Hall, Englewood CliffsGoogle Scholar
  28. 28.
    Jorgenson RE, Mittra R (1990) Efficient calculation of the free-space periodic green’s function. IEEE Trans Anten Propag 38:633–642MATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Singh S, Richards W, Zinecker JR, Wilton DR (1990) Accelerating the convergence of series representing the free-surface periodic green’s function. IEEE Trans Anten Propag 38:1958–1962CrossRefGoogle Scholar
  30. 30.
    Wang C (2004) The linear wave response of a single and a line-array of floating elastic plates. PhD thesis, Massey University, New ZealandGoogle Scholar
  31. 31.
    Linton CM, McIver P (2001) Handbook of mathematical techniques, for wave/structure interactions. Chapman & Hall/CRC, Boca RatonMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Civil EngineeringNational University of SingaporeKent Ridge, SingaporeSingapore
  2. 2.Department of MathematicsUniversity of AucklandAucklandNew Zealand
  3. 3.School of MathematicsUniversity of BristolBristolUK

Personalised recommendations