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

1997 ◽  
Vol 39 (1) ◽  
pp. 93-103 ◽  
Author(s):  
A. Chakrabarti ◽  
Sudeshna Banerjea ◽  
B. N. Mandal ◽  
T. Sahoo
Keyword(s):  
Surface Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Scattering ◽  
Multiple Integral ◽  
Cauchy Type ◽  
Unified Approach ◽  
Water Wave Scattering ◽  
Vertical Barriers ◽  
Surface Water Waves

AbstractA 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.

scholarly journals Scattering of water waves by a submerged thin vertical wall with a gap

1998 ◽  
Vol 39 (3) ◽  
pp. 318-331 ◽  
Author(s):  
Sudeshna Banerjea ◽  
B. N. Mandal
Keyword(s):  
Integral Equation ◽  
Surface Water ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Water Waves ◽  
Water Wave ◽  
Vertical Wall ◽  
Multiple Integral ◽  
Cauchy Kernel ◽  
Surface Water Waves

AbstractA train of surface water waves normally incident on a thin vertical wall completely submerged in deep water and having a gap, experiences reflection by the wall and transmission through the gaps above and in the wall. Using Havelock's expansion of water wave potential, two different integral equation formulations of the problem are presented. While the first formulation involves multiple integral equations which are solved here by reducing them to a singular integral equation with Cauchy kernel in a double interval, the second formulation involves a first-kind singular integral equation in a double interval with a combination of logarithmic and Cauchy kernel, the solution of which is obtained by utilizing the solution of a singular integral equation with Cauchy kernel in (0, ∞) and also in a double interval. The reflection coefficient is evaluated by both the methods.

scholarly journals On the Wiener–Hopf solution of water-wave interaction with a submerged elastic or poroelastic plate

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200360
Author(s):  
M. J. A. Smith ◽  
M. A. Peter ◽  
I. D. Abrahams ◽  
M. H. Meylan
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Wave Scattering ◽  
Wave Interaction ◽  
Thin Elastic Plate ◽  
Cauchy Type ◽  
Water Wave Scattering ◽  
Transmitted Waves ◽  
Submerged Plate ◽  
Cauchy Type Integrals

A solution to the problem of water-wave scattering by a semi-infinite submerged thin elastic plate, which is either porous or non-porous, is presented using the Wiener–Hopf technique. The derivation of the Wiener–Hopf equation is rather different from that which is used traditionally in water-waves problems, and it leads to the required equations directly. It is also shown how the solution can be computed straightforwardly using Cauchy-type integrals, which avoids the need to find the roots of the highly non-trivial dispersion equations. We illustrate the method with some numerical computations, focusing on the evolution of an incident wave pulse which illustrates the existence of two transmitted waves in the submerged plate system. The effect of the porosity is studied, and it is shown to influence the shorter-wavelength pulse much more strongly than the longer-wavelength pulse.

scholarly journals Scattering of water waves by a vertical wall with gaps

1996 ◽  
Vol 37 (4) ◽  
pp. 512-529 ◽  
Author(s):  
Sudeshna Banerjea
Keyword(s):  
Integral Equation ◽  
Reflection Coefficient ◽  
Water Waves ◽  
Water Wave ◽  
Wave Scattering ◽  
Vertical Wall ◽  
Cauchy Type ◽  
Cauchy Type Integral ◽  
Water Wave Scattering ◽  
Type Integral

AbstractThis 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 Havelock's 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,∞) 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.

scholarly journals Water wave scattering by two submerged nearly vertical barriers

2006 ◽  
Vol 48 (1) ◽  
pp. 107-117 ◽  
Author(s):  
B. N. Mandal ◽  
Soumen De
Keyword(s):  
Water Wave ◽  
Wave Scattering ◽  
Shape Function ◽  
Integral Theorem ◽  
Water Wave Scattering ◽  
First Order ◽  
Transmission Coefficients ◽  
Definite Integrals ◽  
Vertical Barriers ◽  
Two Barriers

AbstractThe problem of surface water wave scattering by two thin nearly vertical barriers submerged in deep water from the same depth below the mean free surface and extending infinitely downwards is investigated here assuming linear theory, where configurations of the two barriers are described by the same shape function. By employing a simplified perturbational analysis together with appropriate applications of Green's integral theorem, first-order corrections to the reflection and transmission coefficients are obtained. As in the case of a single nearly vertical barrier, the first-order correction to the transmission coefficient is found to vanish identically, while the correction for the reflection coefficient is obtained in terms of a number of definite integrals involving the shape function describing the two barriers. The result for a single barrier is recovered when two barriers are merged into a single barrier.

Water wave scattering by two partially immersed nearly vertical barriers

Wave Motion ◽  
2005 ◽  
Vol 43 (2) ◽  
pp. 167-175 ◽  
Author(s):  
Soumen De ◽  
Rupanwita Gayen ◽  
B.N. Mandal
Keyword(s):  
Water Wave ◽  
Wave Scattering ◽  
Water Wave Scattering ◽  
Vertical Barriers

A new model for the three-dimensional propagation of water waves in the presence of vertically sheared, horizontally varying currents

2020 ◽  
Author(s):  
Julien Touboul ◽  
Kostas Belibassakis
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Wave Scattering ◽  
Three Dimensional ◽  
Water Wave Scattering ◽  
Coupled Equations ◽  
Coupled Mode ◽  
Constant Shear ◽  
Mild Slope Equation ◽  
Sea Bed

<p>In coastal areas, steep bathymetries and strong currents are often observed. Among several causes, the presence of cliffs, rocky beds, or human structures may cause strong variations of the sea bed, while oceanic circulation, tides, wind action or wave breaking can be responsible for the generation of strong currents. For both coastal safety and engineering purposes, there are many interests in providing efficient models predicting the nonlinear, phase resolved behavior of water waves in such areas. The difficulty is known to be important, and many models achieving that goal are described in the related literature.</p><p>Recently, it was established that beneath the influence of vertically uniform currents, the vorticity involved in depth varying mean flows could have significant impact on the propagation of water waves (Rey et al. 2014). This gave rise to new derivations of equations aimed to describe this interaction. First, an extended mild slope equation was obtained (Touboul et al. 2016). Then, the now classical coupled mode theory was introduced in the system to obtain a set of coupled equations, which could be compared to the system derived by Belibassakis et al (2011) but considering currents which may present constant shear with depth (Belibassakis et al. 2017, Belibassakis et al., 2019). In these works, the currents were assumed to vary linearly with depth, presenting a constant shear. However, this approach was recently extended to more general configurations (Belibassakis & Touboul, 2019; Touboul & Belibassakis, 2019).</p><p>In this work, we extend this model to three dimensional configurations. It is emphasized that the model is able to describe rotational waves, as expected, for example, when water waves propagate with a non-zero angle with respect to the current direction (see e.g. Ellingsen, 2016).</p><p>[1] Rey, V., Charland, J., Touboul, J., Wave – current interaction in the presence of a 3d bathymetry: deep water wave focusing in opposite current conditions. Phys. Fluids 26, 096601, 2014.</p><p>[2] Touboul J., Charland J., Rey V., Belibassakis K., Extended Mild-Slope equation for surface waves interacting with a vertically sheared current, Coastal Engineering, 116, 77–88, 2016.</p><p>[3] Belibassakis, K.A., Gerostathis, Th., Athanassoulis, G.A. A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography, Applied Ocean Res. 33, 384– 397, 2011.</p><p>[4] Belibassakis K.A., Simon B., Touboul J., Rey V., A coupled-mode model for water wave scattering by vertically sheared currents in variable bathymetry regions, Wave Motion, vol.74, 73-92, 2017.</p><p>[5] Belibassakis K., Touboul J., Laffitte E., Rey  V., A mild-slope system for Bragg scattering of water waves by sinusoidal bathymetry in the presence of vertically sheared currents,  J. Mar. Sci. Eng., Vol.7(1), 9, 2019.</p><p>[6] Belibassakis K.A., Touboul J. A nonlinear coupled-mode model for waves propagating in</p><p>vertically sheared currents in variable bathymetry-collinear waves and currents, Fluids, 4(2),</p><p>61, 2019.</p><p>[7] J. Touboul & K. Belibassakis, A novel method for water waves propagating in the presence of vortical mean flows over variable bathymetry, J. Ocean Eng. and Mar. Energy, https://doi.org/10.1007/s40722-019-00151-w, 2019.</p><p>[8] Ellingsen, S.A., Oblique waves on a vertically sheared current are rotational, Eur. J. Mech. B-Fluid 56, 156–160, 2016.</p>

scholarly journals Step Approximation for Water Wave Scattering by Multiple Thin Barriers over Undulated Bottoms

2021 ◽  
Vol 9 (6) ◽  
pp. 629
Author(s):  
Chang-Thi Tran ◽  
Jen-Yi Chang ◽  
Chia-Cheng Tsai
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Wave Scattering ◽  
Sparse Matrix ◽  
Linear Equations ◽  
System Of Linear Equations ◽  
Matching Method ◽  
Water Wave Scattering ◽  
Oblique Wave ◽  
Step Approximation

This paper investigates the scattering of oblique water waves by multiple thin barriers over undulation bottoms using the eigenfunction matching method (EMM). In the solution procedures of the EMM, the bottom topographies are sliced into shelves separated by steps. On each step, surface-piercing or/and bottom-standing barriers can be presented or not. For each shelf, the solution is composed of eigenfunctions with unknown coefficients representing the wave amplitudes. Then applying the conservations of mass and momentum, a system of linear equations is resulted and can be solved by a sparse-matrix solver. If no barriers are presented on the steps, the proposed EMM formulation degenerates to the water wave scattering over undulating bottoms. The effects on the barrier lengths, barrier positions and oblique wave incidences by different undulated bottoms are studied. In addition, the EMM is also applied to solve the Bragg reflections of normal and oblique water waves by periodic barrier over sinusoidal bottoms. The accuracy of the solution is demonstrated by comparing it with the results in the literature.

scholarly journals WATER WAVE SCATTERING BY A THIN VERTICAL SUBMERGED PERMEABLE PLATE

2021 ◽  
Vol 26 (2) ◽  
pp. 223-235
Author(s):  
Rupanwita Gayen ◽  
Sourav Gupta ◽  
Aloknath Chakrabarti
Keyword(s):  
Integral Equation ◽  
Water Waves ◽  
Water Wave ◽  
Wave Theory ◽  
Hypersingular Integral Equation ◽  
Cauchy Type ◽  
Hypersingular Integral ◽  
Permeable Plate ◽  
Linear Water Wave Theory ◽  
Surface Water Waves

An alternative approach is proposed here to investigate the problem of scattering of surface water waves by a vertical permeable plate submerged in deep water within the framework of linear water wave theory. Using Havelock’s expansion of water wave potential, the associated boundary value problem is reduced to a second kind hypersingular integral equation of order 2. The unknown function of the hypersingular integral equation is expressed as a product of a suitable weight function and an unknown polynomial. The associated hypersingular integral of order 2 is evaluated by representing it as the derivative of a singular integral of the Cauchy type which is computed by employing an idea explained in Gakhov’s book [7]. The values of the reflection coefficient computed with the help of present method match exactly with the previous results available in the literature. The energy identity is derived using the Havelock’s theorems.

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