Hydroelastic analysis of surface gravity wave interacting with elastic plate resting on a linear viscoelastic foundation


This article investigates the interaction between a surface gravity wave that propagates over an elastic plate based on linear viscoelastic foundation. The plate is considered to be thin and infinite and is modeled based on the Euler–Bernoulli beam theory. Static and dynamic boundary conditions are applied to the Laplace equation of the fluid domain. The dispersion relation of the wave–plate system is derived and ratio of surface wave amplitude and plate deflection is proposed. Considering dimensionless dispersion relation, two modes of propagating wave are attained. Problem is analyzed for two cases of presence and absence of viscous damping coefficient in the foundation of the elastic plate. It is shown that flexural rigidity of the submerged plate has considerable effect on wave decay and plate vibration. It is illustrated that shallowness has noticeable effect on the wave propagation frequency and a critical shallowness demarcates damped or overdamped excitation of the elastic plate based on the viscoelastic foundation. Moreover, effects of flexural rigidity of the plate, foundation stiffness coefficient, and foundation viscous coefficient on phase and group velocities of wave are discussed in the present study.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12


x :

Direction of wave propagation

h :

Mean water depth


Velocity potential

t :



Laplacian operator

η :

Surface wave amplitude


Surface wave amplitude

g :

Gravity acceleration

ω :

Frequency of propagating wave


Real part of complex number

i :

Unit imaginary number

η o :

Surface wave amplitude at the conventional origin

A, B :

Arbitrary constants

γ :

Dimensionless spring-restoring force

μ :


τ :

Dimensionless thickness

Ωs :

Surface mode dimensionless frequency

v PS :

Surface mode phase velocity of wave

v gS :

Surface mode group velocity of wave

y :

Direction of water depth

P b :

Pressure applied on the viscoelastic bottom

ρ :

Fluid density

El :

Flexural rigidity of plate

ρ b :

Density of plate

d :

Thickness of plate

k * :

Stiffness of plate's foundation

c * :

Viscous damping coefficient of plate's foundation

ϕ :

Spatial velocity potential

e :

Euler's number

k :

Wave number

ζ 0 :

Wave number


Dimensionless frequency

ξ :

Dimensionless damping ratio

γ b :

Dimensionless elasticity-restoring force

ε :

Dimensionless flexural rigidity

Ωb :

Bottom mode dimensionless frequency

v Pb :

Bottom mode phase velocity of wave

v gb :

Bottom mode group velocity of wave


  1. 1.

    H. Behera, S. Das, T. Sahoo, Wave propagation through mangrove forests in the presence of a viscoelastic bed. Wave Motion 78, 162–175 (2018)

    MathSciNet  Article  Google Scholar 

  2. 2.

    S. Elgar, B. Raubenheimer, Wave dissipation by muddy seafloors. Geophys. Res. Lett. (2008). https://doi.org/10.1029/2008GL033245

    Article  Google Scholar 

  3. 3.

    K. Parvathy, P.K. Bhaskaran, Wave attenuation in presence of mangroves: a sensitivity study for varying bottom slopes. Int. J. Ocean Climate Syst. 8, 126–134 (2017)

    Article  Google Scholar 

  4. 4.

    D.V. Evans, R. Porter, Wave scattering by narrow cracks in ice sheets floating on water of finite depth. J. Fluid Mech. 484, 143–165 (2003)

    MathSciNet  Article  Google Scholar 

  5. 5.

    A. Kohout et al., Linear water wave propagation through multiple floating elastic plates of variable properties. J. Fluids Struct. 23, 649–663 (2007)

    Article  Google Scholar 

  6. 6.

    X. Feng, L. DongQiang, Hydroelastic interaction between water waves and a thin elastic plate of arbitrary geometry. Sci. China Phys. Mech. Astron. 54, 59–66 (2011)

    Article  Google Scholar 

  7. 7.

    R. Chakraborty, B. Mandal, Scattering of water waves by a submerged thin vertical elastic plate. Arch. Appl. Mech. 84, 207–217 (2014)

    Article  Google Scholar 

  8. 8.

    J.E. Mosig, F. Montiel, V.A. Squire, Comparison of viscoelastic-type models for ocean wave attenuation in ice-covered seas. J. Geophys. Res. Oceans 120, 6072–6090 (2015)

    Article  Google Scholar 

  9. 9.

    W.W. Mallard, R.A. Dalrymple, Water waves propagating over a deformable bottom. Houston, Offshore Technology Conference (1977), pp. 141–146.

  10. 10.

    T. Dawson, Wave propagation over a deformable seafloor. Ocean Eng. 5, 227–234 (1978)

    Article  Google Scholar 

  11. 11.

    S. Mohapatra, T. Sahoo, Surface gravity wave interaction with elastic bottom. Appl. Ocean Res. 33, 31–40 (2011)

    Article  Google Scholar 

  12. 12.

    H. Behera, T. Sahoo, Hydroelastic analysis of gravity wave interaction with submerged horizontal flexible porous plate. J. Fluids Struct. 54, 643–660 (2015)

    Article  Google Scholar 

  13. 13.

    C. Nové-Josserand et al., Surface wave energy absorption by a partially submerged bio-inspired canopy. Bioinspir. Biomimetics 13, 036006 (2018)

    Article  Google Scholar 

  14. 14.

    S.C. Mohapatra, T. Sahoo, C.G. Soares, Interaction between surface gravity wave and submerged horizontal flexible structures. J. Hydrodynam. 30, 481–498 (2018)

    Article  Google Scholar 

  15. 15.

    S. Mohapatra, C. Guedes Soares, Interaction of ocean waves with floating and submerged horizontal flexible structures in three-dimensions. Appl. Ocean Res. 83, 136–154 (2019)

    Article  Google Scholar 

  16. 16.

    S. Zheng, M.H. Meylan, D. Greaves, G. Iglesias, Water-wave interaction with submerged porous elastic disks. Phys. Fluids 32, 047106 (2020)

    Article  Google Scholar 

  17. 17.

    M.-R. Alam, A flexible seafloor carpet for high-performance wave energy extraction (2012a). Rio de Janeiro,Brazil

  18. 18.

    M.-R. Alam, Nonlinear analysis of an actuated seafloor-mounted carpet for a high-performance wave energy extraction. Proc. R. Soc. A 468, 3153–3171 (2012)

    MathSciNet  Article  Google Scholar 

  19. 19.

    M. Lehmann et al. An artificial seabed carpet for multidirectional and broadband wave energy extraction: theory and experiment (2013). Aalborg, Denmark, s.n

  20. 20.

    S. Abdelghany, K. Ewis, A. Mahmoud, M.M. Nassar, Dynamic response of non-uniform beam subjected to moving load and resting on non-linear viscoelastic foundation. Beni-Suef Univ. J. Basic Appl. Sci. 4, 192–199 (2015)

    Article  Google Scholar 

  21. 21.

    R. Bogacz, Response of beam on viscoelastic foundation to moving distributed load. J. Theor. Appl. Mech. 46, 763–775 (2008)

    Google Scholar 

  22. 22.

    H. Ding, K.-L. Shi, L.-Q. Chen, Dynamic response of an infinite Timoshenko beam on a nonlinear viscoelastic foundation to a moving load. Nonlinear Dyn 73, 285–298 (2013)

    MathSciNet  Article  Google Scholar 

  23. 23.

    M. Kargarnovin, D. Younesian, D. Thompson, C. Jones, Response of beams on nonlinear viscoelastic foundations to harmonic moving loads. Comput. Struct. 83, 1865–1877 (2005)

    Article  Google Scholar 

  24. 24.

    H.P. Lee, Dynamic response of a Timoshenko beam on a Winkler foundation subjected to a moving mass. Appl. Acoust. 55, 203–215 (1998)

    Article  Google Scholar 

  25. 25.

    B. Li, S. Wang, X. Wu, B. Wang, Dynamic response of continuous beams with discrete viscoelastic supports under sinusoidal loading. Int. J. Mech. Sci. 86, 76–82 (2014)

    Article  Google Scholar 

  26. 26.

    A.D. Senalp, A. Arikoglu, I. Ozkol, V.Z. Dogan, Dynamic response of a finite length euler-bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force. J. Mech. Sci. Technol. 24, 1957–1961 (2010)

    Article  Google Scholar 

  27. 27.

    Y. Wang, Y. Wang, B. Zhang, S. Shepard, Transient responses of beam with elastic foundation supports under moving wave load excitation. Int. J. Eng. Technol. 1, 137–143 (2011)

    Google Scholar 

  28. 28.

    Y. Yang, H. Ding, L.-Q. Chen, Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation. Acta. Mech. Sin. 29, 718–727 (2013)

    MathSciNet  Article  Google Scholar 

  29. 29.

    H. Yu, C. Cai, Y. Yuan, M. Jia, Analytical solutions for Euler-Bernoulli Beam on Pasternak foundation subjected to arbitrary dynamic loads. Int. J. Numer. Anal. Methods Geomech. 41, 1125–1137 (2017)

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Saeed Rashidi-Juybari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rashidi-Juybari, S., Fathi, A. & Afrasiab, H. Hydroelastic analysis of surface gravity wave interacting with elastic plate resting on a linear viscoelastic foundation. Mar Syst Ocean Technol 15, 286–298 (2020). https://doi.org/10.1007/s40868-020-00085-1

Download citation


  • Wave–plate interaction
  • Viscoelastic seabed
  • Linear water waves
  • Euler–bernoulli beam theory
  • Flexural rigidity
  • Dispersion relation