R. Gayen

The Dock Problem Revisited

The well-known semi-infinite dock problem of the theory of scattering of surface water waves is reexamined and known results are recovered by utilizing a Fourier type of analysis, giving rise to Carleman-type singular integral equations over semi-infinite ranges.

Water-wave scattering by two symmetric circular-arc-shaped thin plates

The problem of surface water-wave scattering by two symmetric circular-arc-shaped thin plates submerged in deep water is investigated in this paper assuming linear theory. The problem is formulated in terms of hypersingular integral equations which are solved approximately using finite series involving Chebyshev polynomials of the second kind. The coefficients of the finite series are obtained numerically by a collocation method. Very accurate numerical estimates for the reflection and the transmission coefficients are then obtained. The numerical results are depicted graphically against the wave number for different arc lengths of the plates, the depth of their submergence and the separation length. Known results for a circular cylinder and horizontal straight plate are recovered.

Water wave diffraction by a surface strip

The two-dimensional problem of wave diffraction by a strip of arbitrary width is investigated here in the context of linearized theory of water waves by reducing it to a pair of Carleman-type singular integral equations. These integral equations have been solved earlier by an iterative process which is valid only for a sufficiently wide strip. A new method is described here by which solutions of these integral equations are determined by solving a set of four Fredholm integral equations of the second kind, and the process is valid for a strip of arbitrary width. Numerical solutions of these Fredholm integral equations are utilized to obtain fairly accurate numerical estimates for the reflection and transmission coefficients. Previous numerical results for a wide strip are recovered from the present analysis. Additional results for the reflection coefficient are presented graphically for moderate values of the strip width which exhibit a less oscillatory nature of the curve than the case of a wide strip.

Scattering of Surface Water Waves by a Floating Elastic Plate in Two Dimensions

A new method is developed to study the problem of water wave scattering by a thin elastic plate of arbitrary width floating in deep water assuming linear theory. Using Havelocks expansion of water wave potentials, the boundary value problem describing the potentials is reduced to solving singular integral equations of Carleman type. With the introduction of some integral operators the problem is further reduced to twelve Fredholm integral equations of second kind with regular kernels, and the numerical solutions of these integral equations are used to compute the reflection and transmission coefficients. The numerical estimates for the reflection coefficient are presented in a number of figures given varying different physical parameters. It is shown that the present analysis produces known results for the reflection coefficient.

Scattering of water waves by a pair of vertical porous plates

The two-dimensional problem of scattering of small-amplitude surface water-waves by two symmetric vertical thin porous plates is investigated here assuming the linear theory. The problem is formulated in terms of two hypersingular integral equations of the second kind involving the discontinuities in the unknown symmetric and antisymmetric potential functions describing the motion in the fluid across one of the plates. Exploiting the conditions at the two tips of the plate, the discontinuity is approximated by expanding it in terms of a finite series involving Chebyshev polynomials of the second kind multiplied by an appropriate weight function, and then the integral equations are solved numerically by a collocation method. Using the solutions, the reflection and the transmission coefficient are computed numerically. The numerical results for the reflection and the transmission coefficients, amount of energy dissipation and the hydrodynamic forces are depicted graphically against the wave number. Known results for the reflection coefficient for two impermeable plates are recovered when the porous-effect parameter is set to be equal to zero. An energy identity for the permeable plates is determined.

Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier

The problem of scattering of two dimensional surface water waves by a partially immersed rigid plane vertical barrier in deep water is re-examined. The associated mixed boundary value problem is shown to give rise to an integral equation of the first kind. Two direct approximate methods of solution are developed and utilized to determine approximate solutions of the integral equation involved. The all important physical quantity, called the Reflection Coefficient, is evaluated numerically, by the use of the approximate solution of the integral equation. The numerical results, obtained in the present work, are found to be in an excellent agreement with the known results, obtained earlier by Ursell (1947), by the use of the closed form analytical solution of the integral equation, giving rise to rather complicated expressions involving Bessel functions