# Elastic Bottom Effect on Trapped Waves in a Two-Layer Fluid

24 Apr 2015-International Journal of Applied Mechanics (Imperial College Press)-Vol. 7, Iss: 2, pp 1550028

TL;DR: In this paper, a hydroelastic model is considered to examine the trapped modes supported by a horizontal submerged cylinder placed in either of the layers of a two-layer fluid flowing over an elastic bottom at a finite depth.

Abstract: In this paper, a hydroelastic model is considered to examine the trapped modes supported by a horizontal submerged cylinder placed in either of the layers of a two-layer fluid flowing over an elastic bottom at a finite depth. The elastic bottom is modeled as a thin elastic plate and is based on the Euler–Bernoulli beam equation. Using multipole expansion method, an infinite system of homogenous linear equations is obtained. For a fixed geometrical configuration and a specific arrangement of a set of other parameters, the frequencies for which the value of the truncated determinant is zero are numerically computed and the trapped wavenumbers corresponding to those frequencies are obtained by using the dispersion relation. These trapped modes are compared with those for which the lower layer is of infinite depth. We also look into the effect of the variation of the elastic plate parameters on the existence of trapped modes. Significant difference is observed with respect to the existence and also in the pattern of the trapped modes between the present case and the one when the cylinder is placed in an infinite depth lower layer of a two-layer fluid.

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TL;DR: In this paper, a hydroelastic model is considered to examine the proliferation of water waves over little deformation on a versatile seabed, where the Euler-Bernoulli beam equation is modelled as a thin large plate.

11 citations

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TL;DR: In this article, a hydrodynamic model with 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.

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.

11 citations

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TL;DR: In this paper, a general three-dimensional hydroelastic model is developed to study the effect of elastic bottom on surface gravity wave motion in three-dimensions under the action of uniform compressive force based on linearized theory of water wave in finite water depth.

11 citations

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TL;DR: In this article, a hydrodynamic model, with the incorporation of elasticity, is considered to study oblique incident waves propagating over a small undulation on an elastic bed in a two-layer fluid.

Abstract: A hydrodynamic model, with the incorporation of elasticity, is considered to study oblique incident waves propagating over a small undulation on an elastic bed in a two-layer fluid, where the upper...

9 citations

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TL;DR: In this paper, a flexible base surface is modelled as a thin elastic plate under the acceptance of Euler-Bernoulli beam equation, and four Fredholm-type integral equations are obtained from the boundary value problem.

Abstract: Within the framework of linearised theory of water waves, a model of oblique wave scattering by obstacles in the form of thin multiple surface-piercing porous barriers having non-uniform porosity is analysed. Herein, we consider a flexible base in an ocean of uniform finite depth. The flexible base surface is modelled as a thin elastic plate under the acceptance of Euler–Bernoulli beam equation. With the aid of eigenfunction expansion method along with mode-coupling relations, four Fredholm-type integral equations are obtained from the boundary value problem. The multi-term Galerkin approximations in terms of Chebychev polynomials multiplied by suitable weight functions are used for solving those integral equations. Analytic solutions for different hydrodynamic quantities (viz. reflection coefficients, transmission coefficients, dissipated wave energy and non-dimensional wave force) are determined, and those quantities are displayed graphically for various values of the dimensionless parameters. It is observed from the graphical representations that the permeability of the barriers and thickness of the bottom surface play a crucial role in modelling of efficient breakwaters.

8 citations

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TL;DR: In this article, the authors consider trapped modes in waveguides governed by the Helmholtz equation, and examine situations from water-wave theory in which the potential satisfies Laplace's equation with the frequency parameter now appearing in the free-surface boundary condition.

256 citations

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TL;DR: In this paper, a linear free hydroelastic vibration analysis of a frictionless liquid with a free surface contained in a cylindrical tank with a flexible bottom has been performed, where the side-wall has been treated as rigid and the effect of surface tension taken into consideration.

51 citations

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TL;DR: In this article, expansion formulae for flexural gravity wave problems in two-layer fluids are developed in both the cases of water of finite and infinite depths, and relations based on Green's identity are derived for the reflection and transmission coefficients in surface and interface modes.

44 citations

01 Jan 2007

TL;DR: In this article, the existence of trapped modes for a freely-floating structure without moorings is considered and the conditions necessary for such a trapped mode are discussed for the case of a structure able to move in heave.

Abstract: It has been known for about ten years that, within the framework of the linearised water-wave problem, certain fixed structures can support fluid oscillations of finite energy known as “trapped modes”. In this work the open question of the existence of trapped modes for a freely-floating structure without moorings is considered. For the case of a structure able to move in heave, the conditions necessary for the existence of such a trapped mode are discussed.

42 citations

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TL;DR: In this article, it was shown that trapped modes also exist in the linearized water-wave problem for a freely floating structure that is able to move in response to the hydrodynamic forces acting upon it.

Abstract: Trapped modes in the linearized water-wave problem are free oscillations of an unbounded fluid with a free surface that have finite energy; it has been known for some time that such modes are supported by certain structures when held fixed. This paper investigates the problem of a freely floating structure that is able to move in response to the hydrodynamic forces acting upon it and it is shown that trapped modes also exist in this problem. For a freely floating structure, a trapped mode is a coupled oscillation of the fluid and the structure.

38 citations