Sudeshna Banerjea

Oblique wave scattering by submerged thin wall with gap in finite-depth water

Abstract The problem of oblique wave scattering by a submerged thin vertical wall with a gap in finite-depth water and its modification when another identical wall is introduced, are investigated in this paper. The techniques of both one-term and multiterm Galerkin approximations have been utilized in the mathematical analysis. The multi-term approximations in terms of appropriate Chebyshev polynomials provide extremely accurate numerical estimates for the reflection coefficient. The reflection coefficient is depicted graphically for a number of geometries. It is found that by the introduction of another identical wall, there occurs zero reflection for certain wave numbers. This may have some bearings on the modelling of a breakwater.

Solution of a singular integral equation in a double interval arising in the theory of water waves

Abstract This paper is concerned with a straightforward method of solving a singular integral equation in a double interval arising in the linear theory of water waves. The kernel of the integral equation involves a combination of logarithmic and Cauchy type singularity. The integral equation is solved by utilizing the solution of a singular integral equation of first kind with a Cauchy type kernel in (0,∞) and in a finite interval.

A Note on some Dual Integral Equations

In the present paper, appropriate solutions of dual integral equations with trigonometric function as kernel are obtained by considering the behaviour of one of the integrals of the pair at the ‘turning point’.

SCATTERING OF WATER WAVES BY A VERTICAL WALL WITH GAPS

This paper is concerned with a reinvestigation of the problem of water wave scattering by a wall with multiple gaps by using the solution of a singular integral equation with a combination of logarithmic and power (Cauchy-type) kernels in disjoint multiple intervals. Use of Havelocks expansion of water wave potential reduces the problem to such an integral equation in the horizontal velocity across the gaps. The solution of the integral equation is obtained by utilizing the solutions of Cauchy-type integral equations in (0, oo) and also in multiple disjoint intervals. An explicit expression for the reflection coefficient is obtained for a wall with n gaps and supplemented by numerical results for up to three gaps.

A note on waves due to rolling of a partially immersed nearly vertical plate

A rigid, nearly vertical, partially immersed wide plate is constrained to rotate about a horizontal axis through it. The waves from small rolling oscillations of the plate are studied. Expressions for the first-order corrections to the amplitudes of the wave motion so set at large distances on the right and left sides of the plate are obtained by the use of Green’s integral theorem. Assuming a Fourier expansion of a function related to the shape of the plate, these corrections are calculated explicitly. Considering some particular explicit forms for the shape function, numerical calculations are performed.

Solution of a hypersingular integral equation in two disjoint intervals

A hypersingular integral equation in two disjoint intervals is solved by using the solution of Cauchy type singular integral equation in disjoint intervals. Also a direct function theoretic method is used to determine the solution of the same hypersingular integral equation in two disjoint intervals. Solutions by both the methods are in good agreement with each other.

Interface wave diffraction by a thin vertical barrier submerged in the lower fluid

This paper is concerned with interface wave diffraction by a thin vertical barrier which is completely submerged in the lower fluid of two superposed infinite fluids and which extends infinitely downwards into the lower fluid. By a suitable application of Greens integral theorem in the two fluid regions, the problem is formulated in terms of a hypersingular integral equation for the difference of potential across the barrier. A numerical procedure is utilized to evaluate the reflection and transmission coefficients directly from this hypersingular integral equation. Also, an integro-differential equation formulation of the problem is considered, wherein the equation is solved approximately up to O (s), s being the ratio of the densities of the upper and lower fluids. Utilizing this approximate solution, the reflection and transmission coefficients are also obtained up to O (s). Numerical results illustrate that the reflection coefficient up to O (s) thus obtained is in good agreement with the same evaluated directly from the hypersingular integral equation for 0 < s 0.5. The advantage of the hypersingular integral equation formulation is that the reflection and transmission coefficients can be evaluated for any value of s such that 0 s 1. It is observed that the presence of the upper fluid reduces the reflection coefficients from their exact values for a single fluid significantly.

A unified approach to problems of scattering of surface water waves by vertical barriers

A unified analysis involving the solution of multiple integral equations via a simple singular integral equation with a Cauchy type kernel is presented to handle problems of surface water wave scattering by vertical barriers. Some well known results are produced in a simple and systematic manner.

On waves due to rolling of a ship in water of finite depth

SummaryThe problem of the generation of waves due to small rolling oscillations of a thin vertical plate partially immersed in uniform finite-depth water is investigated here by utilizing two mathematical methods assuming the linearised theory of water waves. In the first method, the use of eigenfunction expansion of the velocity potentials on the two sides of the plate produces the amplitude of wave motion at infinity in terms of an integral involving the unknown horizontal velocity across the gap, and also in terms of another integral involving the unknown difference of the potential across the plate. These unknown functions satisfy two integral equations. Any one of these, when solved numerically, can be used to compute the amplitude of the wave motion set up at either infinity on the two sides of the plate for various values of the wave number.In the second method, the problem is formulated in terms of a hypersingular integral equation involving the difference of the potential function across the plate. The hypersingular integral equation is solved numerically, and its numerical solution is used to compute the wave amplitude at infinity. The two methods produce almost the same numerical results. The results are illustrated graphically, and a comparison is made with the deep-water result. It is observed that the deep-water result effectively holds good if the plate is partially immersed to the order of one-tenth of the bottom depth.

On gravity‐capillary waves due to interface disturbance

The generation of gravity‐capillary waves at the interface between two superposed fluids due to interface disturbance is considered, assuming linear theory. Fourier and Laplace transform techniques are employed in the mathematical analysis and the form of the interface depression is obtained as an infinite integral involving oscillatory functions when the disturbance is concentrated at the origin. The method of stationary phase is then employed to evaluate this infinite integral asymptotically. The asymptotic form of the interface depression is presented graphically and compared with the non‐capillary case. It is observed that the interface capillarity has some significant effect on the wave motion.