Skip to main content
SpringerLink
Log in

The analytic form of Green’s function in elliptic coordinates

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

The purpose of this study is the derivation of a closed-form formula for Green’s function in elliptic coordinates that could be used for achieving an analytic solution for the second-order diffraction problem by elliptical cylinders subjected to monochromatic incident waves. In fact, Green’s function represents the solution of the so-called locked wave component of the second-order velocity potential. The mathematical analysis starts with a proper analytic formulation of the second-order diffraction potential that results in the inhomogeneous Helmholtz equation. The associated boundary-value problem is treated by applying Green’s theorem to obtain a closed-form solution for Green’s function. Green’s function is initially expressed in polar coordinates while its final elliptic form is produced through the proper employment of addition theorems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. John F (1950) On the motion of floating bodies II. Simple harmonic motions. Commun Pure Appl Math 3: 45–101

    Article  Google Scholar 

  2. Wehausen JV, Laitone EV (1960) Surface waves. In: Flügge S, Truesdell C (eds) Handbuch der Physik, vol IX. Springer, Berlin, pp 446–778

    Google Scholar 

  3. Garrison CJ, Chow PY (1972) Wave forces on submerged bodies. J Waterw Harb Div ASCE 98: 375–392

    Google Scholar 

  4. Black JL (1975) Wave forces on vertical axisymmetric bodies. J Fluid Mech 67: 369–376

    Article  ADS  MATH  Google Scholar 

  5. Fenton JD (1978) Wave forces on vertical bodies of revolution. J Fluid Mech 85: 241–255

    Article  ADS  MATH  Google Scholar 

  6. Kim M-H, Yue DKP (1989) The complete second-order diffraction for an axisymmetric body. Part 1. Monochromatic incident waves. J Fluid Mech 200: 235–264

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Kim M-H, Yue DKP (1990) The complete second-order diffraction solution for an axisymmetric body. Part 2. Bichromatic incident waves and body motions. J Fluid Mech 211: 557–593

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Chau FP, Eatock Taylor R (1992) Second-order wave diffraction by a vertical cylinder. J Fluid Mech 240: 571–599

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Eatock Taylor R, Huang JB (1997) Semi-analytical formulation for second-order diffraction by a vertical cylinder in bichromatic waves. J Fluids Struct 11: 465–484

    Article  Google Scholar 

  10. Huang JB, Eatock Taylor R (1996) Semi-analytical solution for second-order wave diffraction by a truncated circular cylinder in monochromatic waves. J Fluid Mech 319: 171–196

    Article  ADS  MATH  Google Scholar 

  11. Huang JB, Eatock Taylor R (1997) Second-order interaction of irregular waves with a truncated column. Acta Mech Sin 13: 130–142

    Article  ADS  MATH  Google Scholar 

  12. Mavrakos SA, Chatjigeorgiou IK (2006) Second-order diffraction by a bottom-seated compound cylinder. J Fluids Struct 22: 463–492

    Article  Google Scholar 

  13. Mavrakos SA, Chatjigeorgiou IK (2009) Second-order hydrodynamic effects on an arrangement of two concentric truncated vertical cylinders. Mar Struct 22: 545–575

    Article  Google Scholar 

  14. Chen HS, Mei CC (1973) Wave forces on a stationary platform of elliptical shape. J Ship Res 17: 61–71

    Google Scholar 

  15. Williams AN (1985) Wave forces on an elliptic cylinder. J Waterw Port Coast Ocean Eng Div ASCE 111: 433–449

    Article  Google Scholar 

  16. Williams AN (1985) Wave diffraction by elliptical breakwaters in shallow water. Ocean Eng 12: 25–43

    Article  Google Scholar 

  17. Williams AN, Darwiche MKD (1988) Three dimensional wave scattering by elliptical breakwaters. Ocean Eng 15: 103–118

    Article  Google Scholar 

  18. Zhang S, Williams AN (1990) Wave scattering by submerged elliptical disk. J Waterw Port Coast Ocean Eng ASCE 122: 38–45

    Article  Google Scholar 

  19. Tao L, Song H (2008) Solving water wave diffraction by an elliptical cylinder using scaled boundary finite element method. ANZIAM J 50: C474–C489

    MathSciNet  Google Scholar 

  20. Chatjigeorgiou IK, Mavrakos SA (2009) Hydrodynamic diffraction by multiple elliptical cylinders. In: Proceedings of the 14th international workshop on water waves and floating bodies, Zelonogorsk, Russia, pp 38–41

  21. Chatjigeorgiou IK, Mavrakos SA (2010) An analytical approach for the solution of the hydrodynamic diffraction by arrays of elliptical cylinders. Appl Ocean Res 32: 242–251

    Article  Google Scholar 

  22. Dettman JW (1988) Mathematical methods in physics and engineering. Dover Publications, New York

    MATH  Google Scholar 

  23. Abramowitz M, Stegun IA (1970) Handbook of mathematical functions. Dover Publications, New York

    Google Scholar 

  24. Watson GN (1966) A treatise theory on the theory of Bessel functions (2nd edn). Cambridge University Press, London

    MATH  Google Scholar 

  25. Særmark KA (1959) A note on addition theorems for Mathieu functions. ZAMP 10: 426–428

    Article  ADS  MATH  Google Scholar 

  26. Meixner J, Schäfke FW (1954) Mathieusche funktionen und sphäroidfunktionen. Springer, Berlin

    Google Scholar 

  27. Zhang S, Jin J (1996) Computation of special functions. Wiley, New York

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Floating Structures and Mooring Systems, School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Heroön Polytechniou Ave, Zografos Campus, 157-73, Athens, Greece

    Ioannis K. Chatjigeorgiou & Spyros A. Mavrakos

Authors
  1. Ioannis K. Chatjigeorgiou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Spyros A. Mavrakos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ioannis K. Chatjigeorgiou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chatjigeorgiou, I.K., Mavrakos, S.A. The analytic form of Green’s function in elliptic coordinates. J Eng Math 72, 87–105 (2012). https://doi.org/10.1007/s10665-011-9464-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-011-9464-6

Keywords

Search

Navigation