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

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.

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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.

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Correspondence to Swaroop Nandan Bora.

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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

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Keywords

  • Two-layer fluid
  • Oblique waves
  • Elastic bottom undulation
  • Cutoff frequency
  • Fourier transform
  • Perturbation technique
  • Reflection coefficient
  • Transmission coefficient

Mathematics Subject Classification

  • 76B15
  • 76B55