Ranita Roy

Abstract: Scattering of an obliquely incident wave train by two non-identical thin vertical barriers either partially immersed or fully submerged in infinitely deep water was studied by employing Havelock’s expansion of water wave potential and reducing the problem ultimately to the solution of a pair of vector integral equations of the first kind. A one-term Galerkin approximation in terms of a known exact solution of the integral equation corresponding to a single vertical barrier is used to obtain very accurate numerical estimates for the reflection and transmission coefficients. The reflection coefficient is depicted graphically for two different arrangements of the vertical barriers. It is observed that total reflection is possible for some discrete values of the wavenumber only when the barriers are identical, either partially immersed or completely submerged. As the separation length between the two vertical barriers increases, the reflection coefficient becomes oscillatory as a function of the wavenumber, which is due to multiple reflections by the barriers. Also, as the separation length becomes very small, the known results for a single barrier are obtained for normal incidence of the wave train.

Abstract: The problem of water-wave scattering by two thin vertical plates of unequal lengths submerged beneath the free surface of an infinitely deep water is studied here assuming linear theory. The problem is reduced to a pair of vector integral equations of first kind which are solved approximately by using single-term Galerkin approximation. Very accurate numerical estimates for the reflection and transmission coefficients for different values of the wave number and other parameters are obtained. The numerical results for the reflection coefficient are plotted against the wave number in a number of figures for different configurations of the two plates. It is observed from these figures that the reflection coefficient vanishes for a sequence of values of the wave number only for two identical submerged plates. However, for two non-identical plates, the reflection coefficient never becomes zero, although there exists a few wave numbers at which this becomes small for some particular configurations of the plates. When the two plates become very close to each other, known numerical results for a single plate are deduced.

Abstract: A study of obliquely incident water wave scattering by two, three and four unequal partially immersed vertical barriers in water of uniform finite depth has been carried out in this paper employing Havelock’s expansion of water wave potential. A formulation involving integral equations in terms of either horizontal component of velocities across the gap below each barrier or difference of potentials across each barrier are obtained using the Havelock’s inversion formulae. A multi-term Galerkin approximation technique with Chebychev’s polynomials (multiplied by appropriate weights) as basis functions is adapted to solve these integral equations and to compute the reflection and transmission coefficients numerically. The numerical results are depicted graphically against the wavenumber in several figures for various arrangements of the vertical barriers. From these figures zeros of reflection coefficient are observed when the vertical barriers are immersed upto equal depths below the mean free surface. However, this observation is not true always for non-identical vertical barriers. For two non-identical partially immersed barriers reflection coefficient never vanishes whereas for three non-identical partially immersed barriers reflection coefficient vanishes at discrete frequencies if the two outer barriers have equal lengths of submergence. For four partially immersed barriers arranged symmetrically about a vertical line, zeros of reflection coefficient are always observed. The known results of a single barrier are recovered as special cases so as to establish the correctness of the present method.

Abstract: The scattering of obliquely incident water waves by two thin vertical barriers with gaps at different depths has been studied assuming linear theory. Using Havelock’s expansion of water wave potential, the problem is reduced to two pairs of integral equations of the first kind, one pair involving a horizontal component of velocity across the gaps and the other pair involving the difference of potentials across each wall. These two pairs of integral equations can be solved approximately by employing a Galerkin single-term approximation technique to obtain numerical estimates for the reflection and transmission coefficients. These estimates for the reflection and transmission coefficients thus obtained are seen to satisfy the energy identity. The reflection coefficient is plotted against wave number in a number of figures for different values of various parameters involved in the problem. It is observed that the reflection coefficient vanishes at discrete frequencies when the vertical barriers are identical. For nonidentical vertical barriers the reflection coefficient never vanishes, though at some wave number it becomes close to zero. The results for a single barrier and fully submerged two barriers, and for a single barrier with a narrow gap, are also recovered as special cases.

Abstract: Oceans crossing the Solar System attract people’s attention: the Earth, Enceladus, and Titan. On a variable-coefficient nonlinear dispersive-wave system for the shallow oceanic environment, our symbolic computation yields two non-auto-Backlund transformations and auto-Backlund transformations with some solitons, with regard to the wave elevation and surface velocity of the water wave, which depend on the variable coefficients. This paper could be of some use for the future oceanic studies in the Solar System.

Abstract: People pay attention to the oceans crossing the Solar System: the Earth, Enceladus and Titan. Hereby, on a variable-coefficient nonlinear dispersive-wave system for the shallow oceanic environment, (A) nonlinear-water-wave symbolic computation and Bell polynomials lead to two hetero-Backlund transformations and (B) nonlinear-water-wave symbolic computation gives rise to a similarity reduction, both depending on the variable coefficients representing the wave elevation and surface velocity of the water wave. This paper might be of some use for the future oceanic studies on the Solar System.

Abstract: In this study, we analyse the effect of two submerged unequal permeable plates in the propagation of water waves under the assumptions of linear water wave theory. The permeability of the plates varies along the depth of submergence of the plates. The plates are submerged in water of uniform depth. The velocity potential is expanded by using Havelock’s expansion of water wave potential. The associated boundary value problem is transformed into two coupled Fredholm type integral equations with the help of the Havelock’s inversion theorem and the porous plate condition. A multi-term Galerkin approximation in terms of Chebyshev polynomials is used to solve the vector integral equations and to obtain the numerical estimates for the reflection and the transmission coefficients. The computed numerical results for the reflection coefficient are depicted graphically for various values of several parameters. The present results are validated against the known results for the case of two identical impermeable plates and a single permeable plate submerged in water of finite depth.

Abstract: The scattering of obliquely incident water waves by two unequal permeable barriers with variable permeability is investigated for two types of barriers, namely partially immersed barriers and bottom-standing barriers, under the consideration of the theory of linear water waves. The barriers are present in water of uniform finite depth. The velocity potential is expanded by using Havelock’s expansion of water wave potential and employing Havelock’s inversion formula together with the conditions on the permeable barriers; the boundary value problem is reduced to a coupled Fredholm-type vector integral equations. The integral equations are solved using the multi-term Galerkin approximation where the unknown functions are approximated in terms of Chebyshev polynomials. The numerical results for the reflection coefficient, the transmission coefficient and the energy dissipation are depicted graphically. Known results for two identical as well as two non-identical impermeable barriers and for a single and twin permeable barriers are recovered in the limiting cases.