scholarly journals Reflection from a water waves from a vertical vortex sheet in water of finite depth

1987 ◽  
Vol 29 (2) ◽  
pp. 127-141 ◽  
Author(s):  
W. D. McKee ◽  
F. Tesoriero
Keyword(s):  
Integral Equation ◽  
Variational Method ◽  
Water Waves ◽  
Water Wave ◽  
Finite Depth ◽  
Vortex Sheet ◽  
Transmission Properties ◽  
Obliquely Incident ◽  
Galerkin Solution ◽  
Wave Problems

AbstractThe reflection-transmission properties of water waves obliquely incident upon a vortex sheet in water of finite depth are studied. The problem is reduced to that of solving two integral equation. An accurate Galerkin solution is obtained which supports the use of the “variational method” in water wave problems that has recently been questioned by Kirby and Dalyrmple.

Reflexion of water waves by a permeable barrier

1979 ◽  
Vol 95 (1) ◽  
pp. 141-157 ◽  
Author(s):  
C. Macaskill
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Finite Depth ◽  
Infinite Depth ◽  
Permeable Barrier ◽  
Impermeable Barrier ◽  
Linearized Problem ◽  
Special Case ◽  
Thin Barrier ◽  
Standard Techniques

The linearized problem of water-wave reflexion by a thin barrier of arbitrary permeability is considered with the restriction that the flow be two-dimensional. The formulation includes the special case of transmission through one or more gaps in an otherwise impermeable barrier. The general problem is reduced to a set of integral equations using standard techniques. These equations are then solved using a special decomposition of the finite depth source potential which allows accurate solutions to be obtained economically. A representative range of solutions is obtained numerically for both finite and infinite depth problems.

Scattering of surface waves obliquely incident on a fixed half immersed circular cylinder

1984 ◽  
Vol 96 (2) ◽  
pp. 359-369 ◽  
Author(s):  
B. N. Mandal ◽  
S. K. Goswami
Keyword(s):  
Integral Equation ◽  
Surface Water ◽  
Circular Cylinder ◽  
Water Waves ◽  
Wave Number ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Obliquely Incident ◽  
Surface Water Waves

AbstractThe problem of scattering of surface water waves obliquely incident on a fixed half immersed circular cylinder is solved approximately by reducing it to the solution of an integral equation and also by the method of multipoles. For different values of the angle of incidence and the wave number the reflection and transmission coefficients obtained by both methods are evaluated numerically and represented graphically to compare the results obtained by the respective methods.

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 Step Approximation for Roseau's Analytical Solution of Water Waves

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Chia-Cheng Tsai ◽  
Tai-Wen Hsu ◽  
Yueh-Ting Lin
Keyword(s):  
Transfer Matrix ◽  
Water Waves ◽  
Water Wave ◽  
Solution Method ◽  
Linear Equations ◽  
Solution Procedure ◽  
Water Wave Scattering ◽  
Scattering Effects ◽  
Wave Problems ◽  
Marching Method

An indirect eigenfunction marching method (IEMM) is developed to provide step approximations for water wave problems. The bottom profile is in terms of successive flat shelves separated by abrupt steps. The marching conditions are represented by the horizontal velocities at the steps in the solution procedure. The approximated wave field can be obtained by solving a system of linear equations with unknown coefficients which represents the horizontal velocities under a proper basis. It is also demonstrated that this solution method can be exactly reduced to the transfer-matrix method (TM method) for a specific setting. The combined scattering effects of a series of steps can be described by a single two-by-two transfer matrix for connecting the far-field behaviors of both sides for this method. The solutions obtained by the IEMM are basicallyexactfor water wave problems considering step-like bottoms. Numerical simulations were performed to validate the present and commonly used methods. Furthermore, it also shows that the solutions obtained by the IEMM converge very well to Roseau's analytical solutions for both mild and steep curved bottom profiles. The present method improves the converges of the TM method for solving water wave scattering over steep bathymetry.

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.

Spectral Theory of Water Waves

1981 ◽  
Vol 25 (01) ◽  
pp. 1-7
Author(s):  
Allen H. Magnuson
Keyword(s):  
Analytic Continuation ◽  
Spectral Theory ◽  
Water Waves ◽  
Water Wave ◽  
Spectral Representation ◽  
Compressible Liquid ◽  
Spectral Space ◽  
Spectral Integration ◽  
Liquid Compressibility ◽  
Wave Problems

Spectral theory is applied to elemental water wave problems for a compressible liquid. Compressibility is introduced primarily to facilitate the spectral representation. Two basic representations are developed. The first (Havelock) expansion is derived from a vertical (z) spectral integration while the second modified Fourier form or Fourier-Bessel form comes from the lateral (x or r) spectrum. Both representations are interrelated by analytic continuation in the spectral space, and are shown to be consistent with existing results when compressibility is suppressed.

Limiting gravity waves in water of finite depth

1981 ◽  
Vol 302 (1466) ◽  
pp. 139-188 ◽  
Keyword(s):  
Integral Equation ◽  
Solitary Wave ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Wave ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Wave Crest ◽  
Wave Form ◽  
Deep Water Wave

Progressive, irrotational gravity waves of constant form exist as a two-parameter family. The first parameter, the ratio of mean depth to wavelength, varies from zero (the solitary wave) to infinity (the deep-water wave). The second parameter, the wave height or amplitude, varies from zero (the infinitesimal wave) to a limiting value dependent on the first parameter. For limiting waves the wave crest ceases to be rounded and becomes angled, with an included angle of 120°. Most methods of calculating finite-amplitude waves use either a form of series expansion or the solution of an integral equation. For waves nearing the limiting amplitude many terms (or nodal points) are needed to describe the wave form accurately. Consequently the accuracy even of recent solutions on modern computers can be improved upon, except at the deep-water end of the range. The present work extends an integral equation technique used previously in which the angled crest of the limiting wave is included as a specific term, derived from the well known Stokes corner flow. This term is now supplemented by a second term, proposed by Grant in a study of the flow near the crest. Solutions comprising 80 terms at the shallow-water end of the range, reducing to 20 at the deep-water end, have defined many field and integral properties of the flow to within 1 to 2 parts in 106. It is shown that without the new crest term this level of accuracy would have demanded some hundreds of terms while without either crest term many thousands of terms would have been needed. The practical limits of the computing range are shown to correspond, to working accuracy, with the theoretical extremes of the solitary wave and the deep-water wave. In each case the results agree well with several previous accurate solutions and it is considered that the accuracy has been improved. For example, the height: depth ratio of the solitary wave is now estimated to be 0.833 197 and the height: wavelength ratio of the deep-water wave to be 0.141063. The results are presented in detail to facilitate further theoretical study and early practical application. The coefficients defining the wave motion are given for 22 cases, five of which, including the two extremes, are fully documented with tables of displacement, velocity, acceleration, pressure and time. Examples of particle orbits and drift profiles are presented graphically and are shown for the extreme waves to agree very closely with simplified calculations by Longuet-Higgins. Finally, the opportunity has been taken to calculate to greater accuracy the long-term Lagrangian-mean angular momentum of the maximum deep-water wave, according to the recent method proposed by Longuet-Higgins, with the conclusion that the level of action is slightly above the crest.

scholarly journals Water Wave Scattering by an Infinite Step in the Presence of an Ice-Cover

2019 ◽  
Vol 24 (4) ◽  
pp. 157-168
Author(s):  
S. Ray ◽  
S. De ◽  
B.N. Mandal
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Water Wave ◽  
Wave Scattering ◽  
Fluid Velocity ◽  
Galerkin Approximation ◽  
Classical Problem ◽  
Finite Depth ◽  
Ice Cover ◽  
Water Wave Scattering

Abstract The classical problem of water wave scattering by an infinite step in deep water with a free surface is extended here with an ice-cover modelled as a thin uniform elastic plate. The step exists between regions of finite and infinite depths and waves are incident either from the infinite or from the finite depth water region. Each problem is reduced to an integral equation involving the horizontal component of velocity across the cut above the step. The integral equation is solved numerically using the Galerkin approximation in terms of simple polynomial multiplied by an appropriate weight function whose form is dictated by the behaviour of the fluid velocity near the edge of the step. The reflection and transmission coefficients are obtained approximately and their numerical estimates are seen to satisfy the energy identity. These are also depicted graphically against thenon-dimensional frequency parameter for various ice-cover parameters in a number of figures. In the absence of ice-cover, the results for the free surface are recovered.

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