, Volume 54, Issue 3, pp 429–450 | Cite as

New analytical solutions to water wave radiation by vertical truncated cylinders through multi-term Galerkin method

  • Ai-jun Li
  • Yong LiuEmail author
  • Hua-jun Li


New analytical solutions to water wave radiation by vertical truncated circular cylinders are developed based on linear potential flow theory. Two typical cylinder configurations of a surface-piercing cylinder and a submerged floating cylinder are considered. The multi-term Galerkin method is employed in the solution procedure, in which the fluid velocity on the interface between different regions is expanded into a set of basis function involving the Gegenbauer polynomials, and the cube-root singularity of fluid velocity at the side edges of the truncated cylinders is correctly modeled. The present solutions have the merits of very rapid convergence. The results with six-figure accuracy for added mass and radiation damping can be obtained using a few truncated numbers in the basis function for three motions (surge, heave and roll). The calculated results of the present solutions agree well with that by a higher-order boundary element method solution. Calculation examples are presented to investigate the influence of the motion frequency on the added mass and the radiation damping of the truncated cylinders with different geometric parameters. The present solutions can be used as a reliable benchmark for numerical solutions to water wave radiation by complicated structures.


Wave radiation Truncated cylinders Multi-term Galerkin method Cube-root singularity Added mass Radiation damping 



This study was supported by the Natural Science Foundation of China under Grant Numbers 51725903 and 51490675.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Bhattacharyya SK, Sathyapal S, Vendhan CP (2001) Absorbing boundary condition on elliptic boundary for finite element analysis of water wave diffraction by large elongated bodies. Int J Numer Methods Fluids 37:249–277CrossRefzbMATHGoogle Scholar
  2. 2.
    Teng B, Eatock Taylor R (1995) New higher-order boundary element methods for wave diffraction/radiation. Appl Ocean Res 17:71–77CrossRefGoogle Scholar
  3. 3.
    Utsunomiya T, Watanabe E (2002) Accelerated higher order boundary element method for wave diffraction/radiation problems and its applications. In: Proceedings of the international offshore and polar engineering conference, p 12Google Scholar
  4. 4.
    Meng QC, Zhang CW (2018) Analytical study on a submerged tubular wave energy converter. Renew Energy 118:955–964CrossRefGoogle Scholar
  5. 5.
    Ning DZ, Zhou Y, Zhang CW (2018) Hydrodynamic modeling of a novel dual-chamber OWC wave energy converter. Appl Ocean Res 78:180–191CrossRefGoogle Scholar
  6. 6.
    Yeung RW (1981) Added mass and damping of a vertical cylinder in finite-depth waters. Appl Ocean Res 3:119–133CrossRefGoogle Scholar
  7. 7.
    Bhatta DD, Rahman M (1995) Wave loadings on a vertical cylinder due to heave motion. Int J Math Math Sci 18:151–170MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bhatta DD, Rahman M (2003) On scattering and radiation problem for a cylinder in water of finite depth. Int J Eng Sci 41:931–967MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jiang SC, Gou Y, Teng B (2014) Water wave radiation problem by a submerged cylinder. J Eng Mech 140:06014003CrossRefGoogle Scholar
  10. 10.
    Koh HJ, Cho IH (2016) Heave motion response of a circular cylinder with the dual damping plates. Ocean Eng 125:95–102CrossRefGoogle Scholar
  11. 11.
    Yilmaz O (1998) Hydrodynamic interactions of waves with group of truncated vertical cylinders. J Waterw Port Coast Ocean Eng 124:272–279CrossRefGoogle Scholar
  12. 12.
    Mavrakos SA (2004) Hydrodynamic coefficients in heave of two concentric surface-piercing truncated circular cylinders. Appl Ocean Res 26:84–97CrossRefGoogle Scholar
  13. 13.
    Zheng YH, Shen YM, You YG, Wu BJ, Rong L (2005) Hydrodynamic properties of two vertical truncated cylinders in waves. Ocean Eng 32:241–271CrossRefGoogle Scholar
  14. 14.
    Wu BJ, Zheng YH, You YG, Jie DS, Chen Y (2006) On diffraction and radiation problem for two cylinders in water of finite depth. Ocean Eng 33:679–704CrossRefGoogle Scholar
  15. 15.
    Siddorn P, Eatock Taylor R (2008) Diffraction and independent radiation by an array of floating cylinders. Ocean Eng 35:1289–1303CrossRefGoogle Scholar
  16. 16.
    Porter R (1995) Complementary methods and bounds in linear water waves. Doctoral thesis, University of BristolGoogle Scholar
  17. 17.
    Roy R, Chakraborty R, Mandal BN (2017) Propagation of water waves over an asymmetrical rectangular trench. Q J Mech Appl Math 70:49–64MathSciNetGoogle Scholar
  18. 18.
    Porter R, Evans DV (1995) Complementary approximations to wave scattering by vertical barriers. J Fluid Mech 294:155–180ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Banerjea S, Kanoria M, Dolai DP, Mandal BN (1996) Oblique wave scattering by submerged thin wall with gap in finite-depth water. Appl Ocean Res 18:319–327CrossRefGoogle Scholar
  20. 20.
    Evans DV, Porter R (1997) Efficient calculation of hydrodynamic properties of OWC-Type devices. J Offshore Mech Arct Eng 119:210–218CrossRefGoogle Scholar
  21. 21.
    Martins-Rivas H, Mei CC (2009) Wave power extraction from an oscillating water column at the tip of a breakwater. J Fluid Mech 626:395–414ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chang KH, Tsaur DH, Huang LH (2012) Accurate solution to diffraction around a modified V-shaped breakwater. Coast Eng 68:56–66CrossRefGoogle Scholar
  23. 23.
    Fernyhough M, Evans DV (1995) Scattering by a periodic array of rectangular blocks. J Fluid Mech 305:263–279ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mandal BN, Kanoria M (2000) Oblique wave-scattering by thick horizontal barriers. J Offshore Mech Arct Eng 122:100–108CrossRefGoogle Scholar
  25. 25.
    Kanoria M, Dolai DP, Mandal BN (1999) Water-wave scattering by thick vertical barriers. J Eng Math 35:361–384MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kanoria M (2001) Water wave scattering by thick rectangular slotted barriers. Appl Ocean Res 23:285–298CrossRefGoogle Scholar
  27. 27.
    Linton CM (2009) Accurate solution to scattering by a semi-circular groove. Wave Motion 46:200–209MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Teng B, Gou Y (2017) BEM for wave interaction with structures and low storage accelerated methods for large scale computation. J Hydrodyn 29:748–762CrossRefGoogle Scholar
  29. 29.
    Teng B, Gou Y (2017) Instruction for WAFDUT1.6 program. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian (in Chinese)Google Scholar
  30. 30.
    Teng B, Ning DZ (2004) A fast multipole boundary element method for three-dimensional potential flow problems. Acta Oceanol Sin 23:747–756Google Scholar
  31. 31.
    Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products, 7th edn. Academic Press, New YorkzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of EngineeringOcean University of ChinaQingdaoChina
  2. 2.Shandong Provincial Key Laboratory of Ocean EngineeringOcean University of ChinaQingdaoChina

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