Skip to main content
SpringerLink
Log in

Scattering of linear oblique water waves by an elastic bottom undulation in a two-layer fluid

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

A hydrodynamic model, with incorporation of elasticity, is considered to study oblique incident waves propagating over a small undulation on an elastic bed in a two-layer fluid, with the upper layer exposed to a free surface. Following the Euler–Bernoulli beam equation, the elastic bed is approximated as a thin elastic plate. The surface tension at the interface of the layers is completely ignored since its contribution will be minimal. While considering water waves passing over a deformable bottom, a significant change in the wave characteristics is observed due to the elasticity of the bottom which has an immense impact on the water wave kinematics and dynamics in addition to demonstrating the elastic behavior of the soil beneath. Time-harmonic waves propagate over an elastic bed with two different modes: the one corresponding to the smaller wavenumber propagates along the interface and the other one corresponding to the higher wavenumber along the free surface for any given frequency. Considering an irrotational motion in an incompressible and inviscid fluid, and applying perturbation technique, the first-order corrections to the velocity potentials are evaluated by an appropriate application of Fourier transform and, subsequently, the corresponding reflection and transmission coefficients are computed through integrals containing a shape function which depicts the bottom undulation. To validate the theory developed, one particular undulating bottom topography is taken up as an example in order to evaluate the hydrodynamic coefficients which are represented through graphs to establish the water wave energy conversion between those modes. The observation is that when the oblique wave is incident on the interface, energy transfer takes place to the free surface, but for free-surface oblique incident waves, no such energy transfer to the interface takes place because of the parameter ranges. It is noticed that reasonable changes in the elasticity of the bed have a significant impact when the propagating wave encounters a small elastic bottom undulation. Further, the values of reflection and transmission coefficients obtained for both the interfacial wave mode as well as the free-surface wave mode in the fluid are found to satisfy the important energy balance relations almost accurately. Such problems with a deformable bed, to be precise elastic here, will enable researchers to take up problems which take into account the characteristics of the infinite depth of soil beneath the bed, and the present study is expected to provide the necessary background.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  1. Bauer, H.F.: Hydroelastic vibrations in a rectangular container. Int J. Solids Struct. 17, 639–652 (1981)

    Article  Google Scholar 

  2. Chamberlain, P.G., Porter, D.: Wave scattering in a two-layer fluid of varying depth. J. Fluid Mech. 524, 207–228 (2005)

    Article  MathSciNet  Google Scholar 

  3. Chanda, A., Bora, S.N.: Propagation of oblique waves over a small undulating elastic bottom topography in a two-layer fluid flowing through a channel. Int. J. Appl. Mech. (2020). https://doi.org/10.1142/S1758825120500234

    Article  Google Scholar 

  4. Chen, X., Wu, Y., Cui, W., Jensen, J.J.: Review of hydroelasticity theories for global response of marine structures. Ocean Eng. 33, 439–457 (2006)

    Article  Google Scholar 

  5. Chiba, M., Watanabe, H., Bauer, H.F.: Hydroelastic coupled vibrations in a cylindrical container with a membrane bottom containing liquid with surface tension. J. Sound Vib. 251(4), 717–740 (2002)

    Article  Google Scholar 

  6. Cho, I.H., Kim, M.H.: Interaction of a horizontal flexible membrane with oblique incident waves. J. Fluid Mech. 367, 139–161 (1998)

    Article  Google Scholar 

  7. Das, D., Mandal, B.N., Chakrabarti, A.: Energy identities in water wave theory for free-surface boundary condition with higher-order derivatives. Fluid Dyn. Res. 40, 253–272 (2008)

    Article  MathSciNet  Google Scholar 

  8. Davies, A.G.: The reflection of wave energy by undulations on the sea bed. Dyn. Atmos. Oceans 6, 207–232 (1982)

    Article  Google Scholar 

  9. Davies, A.G., Heathershaw, A.D.: Surface-wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419–443 (1984)

    Article  Google Scholar 

  10. Dawson, T.H.: Wave propagation over a deformable sea floor. Ocean Eng. 5(4), 227–234 (1978)

    Article  Google Scholar 

  11. Devillard, P., Dun-lop, F., Souillard, B.: Localization of gravity waves on a channel with a random bottom. J. Fluid Mech. 186, 521–538 (1988)

    Article  Google Scholar 

  12. Eyov, E., Klar, A., Kadri, U., Stiassnie, M.: Progressive waves in a compressible-ocean with an elastic bottom. Wave Motion 50, 929–39 (2013)

    Article  MathSciNet  Google Scholar 

  13. Fox, C., Squire, V.A.: Reflection and transmission characteristics at the edge of shore fast sea ice. J. Geophys. Res. 95(C7), 1629–1639 (1990)

    Article  Google Scholar 

  14. Fox, C., Squire, V.A.: Coupling between an ocean and an ice shelf. Ann. Glaciol. 15, 101–108 (1991)

    Article  Google Scholar 

  15. Fox, C., Squire, V.A.: On the oblique reflection and transmission of ocean waves at shore fast sea ice. Philos. Trans. R. Soc. Lond. A 347, 185–218 (1994)

    Article  Google Scholar 

  16. Guazzelli, E., Guyon, E., Souillard, B.: On the localization of shallow water waves by a random bottom. J. Phys. Lett. 44, 837–841 (1983)

    Article  Google Scholar 

  17. Guazzelli, E., Rey, V., Belzons, M.: Higher-order bragg reflection of gravity surface waves by periodic beds. J. Fluid Mech. 245, 301–317 (1992)

    Article  Google Scholar 

  18. Hassan, U.L.M., Meylan, M.H., Peter, M.A.: Water-wave scattering by submerged elastic plates. Q. J. Mech. Appl. Math. 62(3), 321–344 (2009)

    Article  MathSciNet  Google Scholar 

  19. Heathershaw, A.D.: Seabed-wave resonance and sand bar growth. Nature 296, 343–345 (1982)

    Article  Google Scholar 

  20. Ibrahim, R.A.: Liquid Sloshing Dynamics Theory and Applications. Cambridge University Press, New York (2005)

    Book  Google Scholar 

  21. Lamb, H.: Hydrodynamics, 6th edn. Cambridge University Press, Cambridge (1932)

    MATH  Google Scholar 

  22. Linton, C.M., Cadby, J.R.: Scattering of oblique waves in a two-layer fluid. J. Fluid Mech. 461, 343–364 (2002)

    Article  MathSciNet  Google Scholar 

  23. Maiti, P., Mandal, B.N.: Scattering of oblique waves by bottom undulations in a two-layer fluid. J. Appl. Math. Comput. 22, 21–39 (2006)

    Article  MathSciNet  Google Scholar 

  24. Mallard, W., Dalrymple, R.: Water waves propagationg over a deformeable bottom. In: Offshore Technology Conference, Houston, Texas, pp. 141–146 (1977)

  25. Martha, S.C., Bora, S.N., Chakrabarti, A.: Oblique water wave scattering by small undulation on a porous sea-bed. Appl. Ocean Res. 29, 86–90 (2007)

    Article  Google Scholar 

  26. Mei, C.C.: Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335 (1985)

    Article  Google Scholar 

  27. Miles, J.W.: Oblique surface wave diffraction by a cylindrical obstacle. Dyn. Atmos. Oceans 6, 121–123 (1981)

    Article  Google Scholar 

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

    Article  Google Scholar 

  29. Mohapatra, S., Bora, S.N.: Oblique wave scattering by an impermeable ocean-bed of variable depth in a two-layer fluid with ice cover. Z. Angew. Math. Phys. 63, 879–903 (2012)

    Article  MathSciNet  Google Scholar 

  30. Mohapatra, S.: Effects of elastic bed on hydrodynamic forces for a submerged sphere in an ocean of finite depth. Z. Angew. Math. Phys. 68, 91 (2017). https://doi.org/10.1007/s00033-017-0837-1

    Article  MathSciNet  MATH  Google Scholar 

  31. Panda, S., Martha, S.C.: Oblique wave scattering by undulating porous bottom in a two-layer fluid: Fourier transform approach. Geophys. Astrophys. Fluid Dyn. 108, 587–614 (2014)

    Article  MathSciNet  Google Scholar 

  32. Porter, R., Porter, D.: Scattered and free waves over periodic beds. J. Fluid Mech. 483, 129–163 (2003)

    Article  MathSciNet  Google Scholar 

  33. Saha, S., Bora, S.N.: Elastic bottom effect on trapped waves in a two-layer fluid. Int. J. Appl. Mech. 7(2), 1550028 (2015)

    Article  Google Scholar 

  34. Sarangi, M.R., Mohapatra, S.: Investigation on the effects of versatile deformating bed on a water wave diffraction problem. Ocean Eng. 164, 377–387 (2018)

    Article  Google Scholar 

  35. Sarangi, M.R., Mohapatra, S.: Hydro-elastic wave proliferation over an impermeable seabed with bottom deformation. Geophys. Astrophys. Fluid Dyn. 113, 303–325 (2019)

    Article  MathSciNet  Google Scholar 

  36. Silva, R., Salles, P., Palacio, A.: Linear wave propagating over a rapidly varying finite porous bed. Coast Eng. 44, 239–260 (2002)

    Article  Google Scholar 

  37. Stokes, G. G.: On the theory of oscillatory waves. Trans. Camb. Philos. Soc. 8, 441–455. Reprinted in Math. Phys. Pap., Cambridge University Press, London, vol. 1, pp. 314–326 (1847)

  38. Zhu, S.: Water waves within a porous medium on an undulating bed. Coast Eng. 42, 87–101 (2001)

    Article  Google Scholar 

  39. An, Z., Ye, Z.: Band gaps and localization of water waves over one-dimensional topographical bottoms. Appl. Phys. Lett. 84, 2952–2954 (2004)

    Article  Google Scholar 

Download references

Acknowledgements

Both authors immensely appreciate the efforts of the anonymous learned reviewer for thoroughly going through the manuscript and providing his insightful comments two times which allowed the authors to produce a much more improved manuscript. Authors are also very much thankful to the Editor Dr. Alfred Kluwick for allowing a revision. The first author expresses his gratefulness to Indian Institute of Technology Guwahati, India, for providing him a graduate research fellowship to pursue PhD during which this work has been completed.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    Ayan Chanda & Swaroop Nandan Bora

Authors
  1. Ayan Chanda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Swaroop Nandan Bora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Swaroop Nandan Bora.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chanda, A., Bora, S.N. Scattering of linear oblique water waves by an elastic bottom undulation in a two-layer fluid. Z. Angew. Math. Phys. 71, 107 (2020). https://doi.org/10.1007/s00033-020-01331-7

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-020-01331-7

Keywords

Mathematics Subject Classification

Search

Navigation