# Reflection and transmission from porous structures under oblique wave attack

TL;DR: The linear theory for water waves impinging obliquely on a vertically sided porous structure is examined in this article, where the reflection and transmission coefficients are significantly altered and they are calculated using a plane-wave assumption.

Abstract: The linear theory for water waves impinging obliquely on a vertically sided porous structure is examined. For normal wave incidence, the reflection and transmission from a porous breakwater has been studied many times using eigenfunction expansions in the water region in front of the structure, within the porous medium, and behind the structure in the down-wave water region. For oblique wave incidence, the reflection and transmission coefficients are significantly altered and they are calculated here. Using a plane-wave assumption, which involves neglecting the evanescent eigenmodes that exist near the structure boundaries (to satisfy matching conditions), the problem can be reduced from a matrix problem to one which is analytic. The plane-wave approximation provides an adequate solution for the case where the damping within the structure is not too great. An important parameter in this problem is Γ 2 = ω 2 h ( s - i f )/ g , where ω is the wave angular frequency, h the constant water depth, g the acceleration due to gravity, and s and f are parameters describing the porous medium. As the friction in the porous medium, f , becomes non-zero, the eigenfunctions differ from those in the fluid regions, largely owing to the change in the modal wavenumbers, which depend on Γ 2 . For an infinite number of values of ΓF 2 , there are no eigenfunction expansions in the porous medium, owing to the coalescence of two of the wavenumbers. These cases are shown to result in a non-separable mathematical problem and the appropriate wave modes are determined. As the two wavenumbers approach the critical value of Γ 2 , it is shown that the wave modes can swap their identity.

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TL;DR: In this article, the authors derived a relation for the fluid motion through thin porous structures in addition to the conventional governing equation and boundary conditions for small-amplitude waves in ideal fluids.

Abstract: Diffraction of water waves by porous breakwaters is studied based on the linear potential wave theory. The formulation of the problem includes a newly derived relation for the fluid motion through thin porous structures in addition to the conventional governing equation and boundary conditions for small-amplitude waves in ideal fluids. The porous boundary condition, indirectly verified by collected experimental data, is obtained by assuming that the flow within the porous medium is governed by a convection-neglected and porous-effect-modeled Euler equation. A vertically two-dimensional problem with long-crested waves propagating in the normal direction of an infinite porous wall is first solved and the solution is compared with available experimental data. The wave diffraction by a semiinfinite porous wall is then studied by the boundary-layer method, in which the outer approximation is formulated by virtue of the reduced two-dimensional solution. It is demonstrated that neglect of the inertial effect of the porous medium leads to an overestimate of the functional performance of a porous breakwater.

280 citations

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TL;DR: In this article, Liu et al. proposed a numerical model named COrnell BReaking waves And Structures (COBRAS) based on the Reynolds Averaged Navier-Stokes (RANS) equations to simulate the most relevant hydrodynamic near field processes that take place in the interaction between waves and low-crested breakwaters.

Abstract: This paper describes the capability of a numerical model named COrnell BReaking waves And Structures (COBRAS) [Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. Journal of Fluid Mechanics 359, 239–264; Liu, P.L.-F., Lin, P., Chang, K.A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. Journal of Waterway, Port, Coastal and Ocean Engineering 125, 322–330, Liu, P.L.-F., Lin, P., Hsu, T., Chang, K., Losada, I.J., Vidal, C., Sakakiyama, T., 2000. A Reynolds averaged Navier–Stokes equation model for nonlinear water wave and structure interactions. Proc. Coastal Structures '99, 169–174] based on the Reynolds Averaged Navier–Stokes (RANS) equations to simulate the most relevant hydrodynamic near-field processes that take place in the interaction between waves and low-crested breakwaters. The model considers wave reflection, transmission, overtopping and breaking due to transient nonlinear waves including turbulence in the fluid domain and in the permeable regions for any kind of geometry and number of layers. Small-scale laboratory tests were conducted in order to validate the model, with different wave conditions and breakwater configurations. In the present study, regular waves of different heights and periods impinging on a wide-crested structure are considered. Three different water depths were tested in order to examine the influence of the structure freeboard. The experimental set-up includes a flow recirculation system aimed at preventing water piling-up at the lee of the breakwater due to overtopping. The applicability and validity of the model are examined by comparing the results of the numerical computations with experimental data. The model is proved to simulate with a high degree of agreement all the studied magnitudes, free surface displacement, pressure inside the porous structure and velocity field. The results obtained show that this model represents a substantial improvement in the numerical modelling of low-crested structures (LCS) since it includes many processes neglected previously by existing models. The information provided by the model can be useful to analyse structure functionality, structure stability, scour and many other hydrodynamic processes of interest.

187 citations

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TL;DR: In this article, the transformation and interaction of regular wave trains with submerged permeable structures is modeled using an eigenfunction expansion 3D model and a 2D model based on a mild-slope equation for porous media to account for breakwater slope.

Abstract: Modelling of the transformation and interaction of regular wave trains with submerged permeable structures is carried out. The existing literature, is summarized relevant theories presented, and theoretical results are compared with existing laboratory data. Special attention is paid to wave reflection. The influence of wave characteristics including oblique incidence, structure geometry and porous material properties on the kinematics and dynamics over and inside the breakwater is considered. Two different models are presented: an eigenfunction expansion 3-D model and a 2-D model based on a mild-slope equation for porous media to account for breakwater slope.

173 citations

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TL;DR: In this article, the authors presented a numerical model of wave interactions with a thin vertical slotted barrier extending from the water surface to some distance above the seabed, and described laboratory tests undertaken to assess the numerical model.

Abstract: The present paper outlines the numerical calculation of wave interactions with a thin vertical slotted barrier extending from the water surface to some distance above the seabed, and describes laboratory tests undertaken to assess the numerical model. The numerical model is based on an eigenfunction expansion method and utilizes a boundary condition at the barrier surface that accounts for energy dissipation within the barrier. Numerical results compare well with previous predictions for the limiting cases of an impermeable barrier and a permeable barrier extending down to the seabed. Comparisons with experimental measurements of the transmission, reflection, and energy dissipation coefficients for a partially submerged slotted barrier show good agreement provided certain empirical coefficients of the model are suitably chosen, and indicate that the numerical method is able to account adequately for the energy dissipation by the barrier. The effects of porosity, relative wave length, wave steepness, and irregular waves are discussed and the choice of suitable parameters needed to model the permeability of the breakwater is described.

163 citations

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TL;DR: In this paper, the authors reviewed the use of Darcy's law for analyzing waves moving past a porous structure, and extended the theories to a porous barrier structure as a breakwater in a two-dimensional harbor.

Abstract: ▪ Abstract This article reviews the use of Darcy's law for analyzing waves moving past a porous structure. The engineering applications of these analyses are emph asized. The first part of the article studies theories on the effect of a porous structure on incoming wave trains. It also reviews the movement of waves past a plate with regular gaps in it in an attempt to compare the results of potential theory with those of Darcy's law. The second part reviews the use of a porous structure as a wavemaker when the structure is subject to horizontal oscillation. The third part extends the theories to the use of a porous structure as a breakwater in a two-dimensional harbor. The effect of the structure on reducing waves and suppressing harbor resonance is investigated.

132 citations

##### References

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01 Jan 1988

TL;DR: In this article, the Electromagnetic Field and its interaction with Matter are discussed, and a matrix formulation for Isotropic Layered Media is proposed. But it is not shown how to apply it to a single homogeneous and isotropic layer.

Abstract: Chapter 1. The Electromagnetic Field. Chapter 2. Interaction of Electromagnetic Radiation with Matter. Chapter 3. Reflection and Refraction of Plane Waves. Chapter 4. Optics of A Single Homogeneous and Isotropic Layer. Chapter 5. Matrix Formulation for Isotropic Layered Media. Chapter 6. Optics of Periodic Layered Media. Chapter 7. Some Applications of Isotropic Layered Media. Chapter 8. Inhomogeneous Layers. Chapter 9. Optics of Anisotropic Layered Media. Chapter 10. Some Applications of Anisotropic Layered Media. Chapter 11. Guided Waves in Layered Media. Chapter 12. Optics of Semiconductor Quantum Wells and Superlattice Structures. Appendix: Zeros of Mode Dispersion RElation. Author Index. Subject Index.

2,324 citations

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01 Jan 1988

TL;DR: Optics of Semiconductor Quantum Wells and Superlattice Structures: Optics of A Single Homogeneous and Isotropic Layer and some Applications of Isotropic Layered Media.

Abstract: Chapter 1. The Electromagnetic Field. Chapter 2. Interaction of Electromagnetic Radiation with Matter. Chapter 3. Reflection and Refraction of Plane Waves. Chapter 4. Optics of A Single Homogeneous and Isotropic Layer. Chapter 5. Matrix Formulation for Isotropic Layered Media. Chapter 6. Optics of Periodic Layered Media. Chapter 7. Some Applications of Isotropic Layered Media. Chapter 8. Inhomogeneous Layers. Chapter 9. Optics of Anisotropic Layered Media. Chapter 10. Some Applications of Anisotropic Layered Media. Chapter 11. Guided Waves in Layered Media. Chapter 12. Optics of Semiconductor Quantum Wells and Superlattice Structures. Appendix: Zeros of Mode Dispersion RElation. Author Index. Subject Index.

2,294 citations

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31 Jan 1986

TL;DR: In this paper, a generalized Lagrangian mean (GLM) formulation is proposed for nonlinear wave-train evolution and three-wave resonance is used to derive the evolution equations.

Abstract: Part I. Introduction: 1. Introduction Part II. Linear Wave Interactions: 2. Flows with piecewise-constant density and velocity 3. Flows with constant density and continuous velocity profile 4. Flows with density stratification and piecewise-constant velocity 5. Flows with continuous profiles of density and velocity 6. Models of mode coupling 7. Eigenvalue spectra and localized disturbances Part III. Introduction to Nonlinear Theory: 8. Introduction to nonlinear theory Part IV. Waves and Mean Flows: 9. Spatially-periodic waves in channel flows 10. Spatially-periodic waves on deformable boundaries 11. Modulated wave-packets 12. Generalized Lagrangian mean (GLM) formulation 13. Spatially-periodic means flows Part V. Three-wave Resonance: 14. Conservative wave interactions 15. Solutions of the conservative interaction equations 16. Linearly damped waves 17. Non-conservative wave interactions Part VI. Evolution of a Nonlinear Wave-Train: 18. Heuristic derivation of the evolution equations 19. Weakly nonlinear waves in inviscid fluids 20. Weakly nonlinear waves in shear flows 21. Properties of the evolution equations 22. Waves of larger amplitude Part VII. Cubic Three- and Four-wave Interactions: 23. Conservative four-wave interactions 24. Mode interactions in Taylor-Couette flow 25. Rayleigh-Benard convection 26. Wave interactions in planar shear flows Part VIII. Strong Interactions, Local Instabilities and Turbulence: A Postscript: 27. Strong interactions, local instabilities and turbulence: A postscript References Index.

522 citations

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29 Jan 1972TL;DR: In this article, an approximate solution to conventional rubble mound breakwater designs is formulated in terms of an equivalent rectangular breakwater with an additional consideration for wave breaking, and experimental and theoretical results are compared and evaluated.

Abstract: A theory is derived to predict ocean wave reflection and transmission at a permeable breakwater of rectangular cross section. The theory solves for a damped wave component within the breakwater and matches boundary conditions at the windward and leeward breakwater faces to predict the reflected and transmitted wave components. An approximate solution to conventional rubble mound breakwater designs is formulated in terms of an equivalent rectangular breakwater with an additional consideration for wave breaking. Experimental and theoretical results are compared and evaluated.

477 citations

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TL;DR: In this article, the authors consider the case of a ship lying dead in the water and assume that the body does not disturb the water much during its forward motion, for example, slenderness or thinness.

Abstract: We shall restrict ourselves here to floating bodies without any means of propelling themselves. The body may, of course, be a ship lying dead in the water, but there is no real limitation to practical shapes of any particular sort except that we shall suppose the body to be hydrostatically stable. This will restrict the extent of this survey in an important way: we are able to slough off all effects associated with an average velocity of the body. Since mathematical solution of problems almost inevitably proceeds by way of linearization of the boundary conditions, this means that we may avoid introducing a linearization parameter whose smallness expresses the fact that the body doesn't disturb the water much during its forward motion, for example, slenderness or thinness. If we do introduce such a geometrical assumption, it will be an additional approximation, not one forced upon us by the physical situation. Fortunately, Newman's (1970) article treats, among other things, the recent advances in the theory of motion of slender ships under way. More can be found in a paper by Ogilvie (1964) . We shall assume from the beginning that motions are small and take this into account in formulating equations and boundary conditions. Further more, we shaH assume the fluid inviscid, and without surface tension. It is not difficult to write down equations and boundary conditions for a less restricted problem. However, since most results are for the case of small motions and since the perturbation expansions associated with the deriva tion of the linearized problem from the more exact one do not present any special points of interest, it seems more efficient to start with the simpler problem. Even so, some account will be given of recent attempts to consider nonlinear problems.

419 citations