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1

Edge waves over a shelf

Dolai P., Dolai D. P.

International Journal of Applied Mechanics and Engineering

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2019

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Vol. 24, no. 2

453--460

EN

The problem considered in this paper is the derivation of properties of edge waves travelling along a submerged horizontal shelf. The problem is formulated within the framework of the linearized theory of water waves and Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the dispersion relation for edge waves in terms of an integral. Appropriate multi-term Galerkin approximations involving ultra spherical Gegenbauer polynomials are utilized to obtain a very accurate numerical estimate for the integral and hence to derive the properties of edge waves over a shelf. The numerical results are illustrated in a table and curves are presented showing the variation of frequency of the edge waves with the width of the shelf.

2

Scattering of oblique water waves by an infinite step

Dolai P., Dolai D. P.

International Journal of Applied Mechanics and Engineering

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2018

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Vol. 23, no. 2

327--338

EN

The present paper is concerned with the problem of scattering of obliquely incident surface water wave train passing over a step bottom between the regions of finite and infinite depth. Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the physical parameters reflection and transmission coefficients in terms of integrals. Appropriate multi-term Galerkin approximations involving ultra spherical Gegenbauer polynomials are utilized to obtain very accurate numerical estimates for reflection and transmission coefficients. The numerical results are illustrated in tables.

3

Internal wave diffraction by a strip of an elastic plate on the surface of a stratified fluid

Dolai P., Dolai D. P.

International Journal of Applied Mechanics and Engineering

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2013

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Vol. 18, no. 1

5--26

EN

The problem of internal wave diffraction by a strip of an elastic plate of finite width present on the surface of an exponentially stratified liquid is investigated in this paper. Assuming linear theory, the problem is formulated in terms of a function related to the stream function describing the motion in the liquid. The related boundary value problem involves a hyperbolic type partial differential equation (PDE), known as the Klein Gordon equation. The method of Wiener-Hopf is utilized in the mathematical analysis to a slightly generalized boundary value problem (BVP) by introducing a small parameter, and the problem is solved approximately for large width of the plate. In the final results, this small parameter is made to tend to zero. The diffracted field is obtained in terms of integrals, which are then evaluated asymptotically in different regions for a large distance from the edges of the plate and the results are interpreted physically.

4

Interface wave diffraction by bottom undulations in the presence of a thin plate submerged in lower fluid

Dolai D. P., Dolai P.

International Journal of Applied Mechanics and Engineering

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2010

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Vol. 15, no 4

1017-1036

EN

The problem of interface wave diffraction by bottom undulations in the presence of a thin vertical plate is investigated in this paper. The plate is submerged in the lower fluid of finite depth with bottom undulations and the upper fluid is of infinite height separated by a common interface. In the method of solution, we use a simplified perturbation analysis and suitable applications of Green's integral theorem in the two fluid regions produce first-order reflection and transmission coefficients in terms of integrals involving the shape function describing the bottom undulations and solution of the scattering problem involving a submerged vertical plate present in the lower fluid of uniform finite depth. For sinusoidal bottom undulations, the first-order transmission coefficient vanishes identically. The corresponding first-order reflection coefficient is computed numerically by solving the zero-order reflection coefficient and suitable application of multi-term Galerkin approximations. The numerical results of zero-order and first-order reflection coefficients are depicted graphically against the wave number in a number of figures. An oscillatory nature of first-order reflection coefficient due to multiple interaction of the incident wave with bottom undulations is observed. The first-order reflection coefficient has a peak value for some particular value of the ratio of the incident wavelength and the bottom wavelength. The presence of the upper fluid has some significant effect on the reflection coefficients.