Wave diffraction through submerged flexible breakwaters

This paper presents an analytical, computationally efficient method for the wave reflection and dynamic displacement of a submerged flexible breakwater. The solution of the two-dimensional linearized hydrodynamic problem introduced is based on the eigenfunction expansion technique. The breakwater is assumed to be thin, impermeable, flexible, moored to the bed through tethers and kept in tension by means of a floating buoy at its tip. The beam structure is considered to be either clamped or hinged at the sea bed, situated in an arbitrary water depth and subjected to normal linear waves. Numerical examples presented by this method are compared with those obtained by the Boundary Integral Equation Method, presented by Williams et al. Comparisons show an excellent agreement over a wide range of parameters for the wave reflection and the dynamic displacement. Numerical results are presented, mainly to show the effect of the breakwater rigidity and the method of fixation on the wave reflection and the structural displacement over a wide range of wave frequencies.     

Wave diffraction on arc-shaped floating perforated breakwaters

An analytical method is developed to study the sheltering effects on arc-shaped floating perforated breakwaters. In the process of analysis, the floating breakwater is assumed to be rigid, thin, vertical, and immovable and located in water with constant depth. The fluid domain is divided into two regions by imaginary interface. The velocity potential in each region is expanded by eigenfunction in the context of linear theory. By satisfying continuity of pressure and normal velocity across the imaginary fluid interface, a set of linear algebraic equations can be obtained to determine the unknown coefficients for eigenfunction expansions. The accuracy of the present model was verified by a comparison with existing results for the case of arc-shaped floating breakwater. Numerical results, in the form of contour maps of the non-dimensional wave amplitude around the breakwater and diffracted wave amplitude at typical sections, are presented for a range of wave and breakwater parameters. Results show that the sheltering effects on the arc-shaped floating perforated breakwater are closely related to the incident wavelength, the draft and the porosity of the breakwater.     

Wave diffraction of breakwaters in the presence of a coastline

An integral method is described which is capable of computing the diffraction field produced by waves incident on a breakwater connected to or placed near a straight coastline. Some simple configurations are studied: a straight breakwater protruding normally from the coast, a straight breakwater parallel to the coast, an ‘elbow-shaped’ breakwater with one end connected to the coast, a pair of straight breakwaters protruding normally from the coast and a series of three equal straight breakwaters parallel to the coast. In all cases the water depth is assumed to be constant, while both the breakwater and the coastline walls are supposed to be perfectly reflective. Within the limits of these hypotheses the method is rather general, because breakwaters of arbitrary shape can be considered.     

Oblique wave diffraction by segmented offshore breakwaters

This paper presents a theoretical model to examine oblique wave diffraction by a detached breakwater system consisting of an infinite row of regularly-spaced thin, impermeable structures located in water of uniform depth. The fluid is assumed incompressible and inviscid and to undergo irrotational motion. Wave heights are assumed to be sufficiently small such that linear wave theory is applicable. The eigenfunction expansion solution of Dalrymple and Martin (1990) for normal wave incidence on this breakwater geometry is modified herein to study oblique wave effects. Numerical results, in the form of contour maps of the relative wave height behind the structure, or complex reflection coefficients, are presented for a range of wave and breakwater parameters. The accuracy of the present model is verified by a comparison with existing results for the limiting cases of an isolated detached breakwater, and a breakwater with a single gap. Also, for the multi-gap breakwater, the present solution is further verified for both normal and oblique wave incidence with results in the open literature.     

Wave height distribution in shallow water

Probability distribution of shallow water wave heights, obtained from a pressure type recorder, are examined. It is tested with the theoretical distributions of (a) Rayleigh, (b) Weibull, (c) Gluhovski, (d) Ibrageemov and (e) Goda. The best fit is shown by the Gluhovski probability density function with a correlation coefficient greater than 0.8. The functions of Weibull, Ibrageemov and Goda fit only half of the tested cases. The role of wave steepness in the wave height distribution is found to be negligible.     

Calculation of the added mass of elliptical cylinders in shallow water

In order to carry out any studies of ship motions, concerning either seakeeping or manoeuvring, it is usually necessary to have knowledge of the added mass of the hull section shapes. In deep water, the added mass can be found using conformal mapping techniques combined with residue calculus, or by means of surface singularity distributions. In shallow water, the need to utilise an infinite number of mirror images, to represent the effects of the seabed and the free surface, precludes the use of the deep-water methods in this case. In a previous paper, the author presented methods to evaluate the added mass of semi-circular sections. In this paper, a quite different but more general technique using Schwarz–Christoffel methods is developed. This technique gives entirely new results for the added mass of elliptical body sections and compares the special case of semi-circular sections with those from the previous paper.     

Wave interaction with T-type breakwaters

The wave transmission, reflection and energy dissipation characteristics of partially submerged ‘T'-type breakwaters (Fig. 1) were studied using physical models. Regular and random waves, with wide ranges of wave heights and periods and a constant water depth were used. Five different depths of immersions of the ‘T'-type breakwater were selected. The coefficient of transmission, K_t, coefficient reflection, K_r, were obtained from the measurements and the coefficient of energy loss, K_l is calculated using the law of conservation of energy. It is found that the coefficient of transmission generally reduces with increased wave steepness and increased relative water depth, d/L. This breakwater is found to be effective closer to deep-water conditions. K_t values less than 0.35 is obtained for both normal and high input wave energy levels, when the horizontal barrier of the T type breakwater is immersed to about 7% of the water depth. This breakwater is also found to be very efficient in dissipating the incident wave energy to an extent of about 65% (i.e. K_l>0.8), especially for high input wave energy levels. The wave climate in front of the breakwater is also measured and studied.

Full-size image (12K)

Fig. 1. Schematic view of the T-type breakwater.     

Wave reflection from perforated-wall caisson breakwaters

Analytical models for predicting wave reflection from a perforated-wall caisson breakwater have been developed. Most of the existing models deal with the case in which the waves are normally incident to the caisson lying on a flat sea bottom. In the present paper, using the Galerkin-eigenfunction method, an analytical model is developed that can predict the reflection coefficient of a perforatedwall caisson mounted on a rubble mound foundation when waves are obliquely incident to the breakwater at an arbitrary angle. The developed model is compared with other theoretical results and hydraulic experimental data.     

Wave interactions with multiple-row curtainwall-pile breakwaters

In this study, a mathematical model has been developed that can compute various hydrodynamic characteristics of a multiple-row curtainwall-pile breakwater. To examine the validity of the developed model, laboratory experiments have been conducted for double- and triple-row breakwaters with various combinations of drafts of curtain walls, porosities between piles, and distances between rows. Comparisons between measurement and prediction show that the mathematical model adequately reproduces most of the important features of the experimental results. As a whole, the transmission coefficient decreases with an increase in relative water depth, whereas the reflection coefficient, normalized run-up and force exhibit an opposite trend in their variations. With fixed values of the draft of the curtain wall and the porosity of lower perforated part of the first row of a double-row breakwater, as these values of the second row increase and decrease, respectively, the transmission coefficient decreases, as expected. On the other hand, their effects on wave reflection, run-up, and wave force change with the relative depth. As for the distance between the rows, the transmission coefficient becomes a maximum when it is about one half of the wave length, suggesting that this condition should be avoided to achieve the advantage of the breakwater in reducing wave transmission. It is shown that for prototype breakwaters, on an average, the transmission coefficient would be smaller than 0.3 for wave periods less than 6.0 s, and it would be about 0.45 even for the wave period of 9.0 s, although there would be a variation depending on the geometry of the breakwater. It is also shown that wave transmission is significantly reduced by multiple-row breakwaters compared with a single-row breakwater, while the difference between double-row and triple-row breakwaters is marginal. Finally, engineering monograms are provided for double-row breakwaters to be used in practical engineering applications of the breakwaters.     

Wave modelling in the vicinity of submerged breakwaters

This paper describes a simple method for modelling wave breaking over submerged structures, with the view of using such modelling approach in a coastal area morphodynamic modelling system.A dominant mechanism for dissipating wave energy over a submerged breakwater is depth-limited wave breaking. Available models for energy dissipation due to wave breaking are developed for beaches (gentle slopes) and require further modifications to model wave breaking over submerged breakwaters.In this paper, wave breaking is split into two parts, namely: 1) depth-limited breaking modelled using Battjes and Janssen's (1978) theory [Battjes, J.A. and Jannsen, J.P.F.M. (1978). Energy loss and setup due to breaking of random waves. Proceedings of the 16th Int. Conf. Coast. Eng., Hamburg, Germany, pp. 569-587.] and 2) steepness limited breaking modelled using an integrated form of the Hasselmann's whitecapping dissipation term, commonly used in fully spectral wind–wave models. The parameter γ₂, governing the maximum wave height at incipient breaking (H_max = γ₂d) is used as calibration factor to tune numerical model results to selected laboratory measurements. It is found that γ₂ varies mainly with the relative submergence depth (ratio of submergence depth at breakwater crest to significant wave height), and a simple relationship is proposed. It is shown that the transmission coefficients obtained using this approach compare favourably with those calculated using published empirical expressions.     

Calculation of the added mass of elliptical cylinders with vertical fins in shallow water

In order to carry out any studies of ship motions, concerning either seakeeping or manoeuvring, it is usually necessary to have knowledge of the added mass of the hull section shapes. In deep water, the added mass can be found using conformal mapping techniques combined with residue calculus, or by means of surface singularity distributions. In shallow water, the need to utilise an infinite number of mirror images, to represent the effects of the seabed and the free surface, precludes the use of the deep-water methods in this case. In previous papers, the author presented methods to evaluate the added mass of semi-circular and elliptical body sections. Now, using a similar Schwarz–Christoffel method, the added mass of elliptical body sections with vertical fins in shallow water is evaluated.     

2D Wave setup behind submerged breakwaters

In addition to reducing the incoming wave energy, submerged breakwaters also cause a setup of the sea level in the protected area, which is relevant to the whole shadow zone circulation, including alongshore currents and seaward flows through the gaps. This study examines such a leading hydraulic parameter under the simplified hypothesis of 2D motion and presents a prediction model that has been validated by a wide ensemble of experimental data. Starting from an approach originally proposed by Dalrymple and Dean [(1971). Piling-up behind low and submerged permeable breakwaters. Discussion note on Diskin et al. (1970). Journal of Waterways and Harbors Division WW2, 423–427], the model splits the rise of the mean water level into two contributions: one is due to the momentum flux release forced by wave breaking on the structure, and the other is associated with the mass transport process. For the first time, the case of random wave trains has been explicitly considered.     

Wave force on vertically submerged circular thin plate in shallow water

The paper presented is a solution of shallow water wave force, using small amplitude linear wave theory on two-dimensional vertically submerged circular thin plates under three different configurations: (1) a surface-piercing circular thin plate, (2) a submerged circular thin plate, and (3) a bottom-standing circular thin plate. Finally Morison's equation is used for the determination of wave force which is based on the linear wave theory. The plate is submerged in water near the shore on uniformly sloping bottom. The solution method is confined in a finite domain, which contains both the region of different depth of water and the plate. Laplace's equation and boundary value problems are solved in a finite domain, by the method of separation of variables and the small amplitude linear wave theory. The variation of horizontal force by single particle, total horizontal force and moment with respect to the wave amplitude are obtained at different depth of water and at different wave period. It is observed that the force and moment are converging with the increase of wave period and the gradients of force and moment with respect to the wave amplitude are extremely high for lower wave period.     

Wave reflection and transmission by porous breakwaters: A new analytical solution

This study gives a new analytical solution for wave reflection and transmission by a surface-piercing porous breakwater. Velocity potential decompositions in the breakwater are used to obtain the solution. Different from traditional solutions, the new solution needs no complex dispersion relations (complex wave numbers) for wave motion through porous media. Thus, difficult procedures in traditional solutions for solving complex dispersion relations and handling non-self-adjoint eigenvalue problem are avoided. The calculated results of the new solution are in very good agreement with those of traditional solutions and multi-domain boundary element method solutions.     

Bragg reflection of water waves by multiple submerged semi-circular breakwaters

This study examines the Bragg reflection of water waves by multiple submerged semi-circular breakwaters. The multipole expansions combined with the shift of polar coordinates are used to develop full linear potential solutions of the problem. In the full solutions, the obliquely and normally incident waves are independently considered. Experimental tests are carried out to measure the reflection and transmission coefficients of the breakwaters at different wave periods and body spacings. The analytical results are in reasonable agreement with the experimental data. The peak reflection coefficient of multiple submerged semi-circular breakwaters and the bandwidth of Bragg reflection are carefully examined by numerical examples. Some significant results for practical application are discussed.     

Wave height parameter for damage description of rubble-mound breakwaters

In this paper it will be shown that the wave height parameter H₅₀, defined as the average wave height of the 50 highest waves reaching a rubble-mound breakwater in its useful life, can describe the effect of the wave height on the history of the armor damage caused by the wave climate during the structure's usable life.Using Thompson and Shuttler (Thompson, D.M., Shuttler, R.M., 1975. Riprap design for wind wave attack: A laboratory study on random waves. HRS Wallingford, Report 61, UK) data it will be shown that H₅₀ is the wave parameter that best represents the damage evolution with the number of waves in a sea state. Using this H₅₀ parameter, formulae as van der Meer (van der Meer, J.W., 1988. Rock slopes and gravel beaches under wave attack. PhD Thesis. Technical University of Delft) and Losada and Giménez-Curto (Losada, M.A., Gimenez–Curto, L.A., 1979. The joint effect of the wave height and period on the stability of rubble mound breakwaters using Iribarren's number. Coastal Engineering, 3, 77–96) are transformed into sea-state damage evolution formulae. Using these H₅₀-transformed formulae for regular and irregular sea states it will be shown how damage predictions are independent of the sea state wave height distribution.To check the capability of these H₅₀-formulae to predict damage evolution of succession of sea states with different wave height distributions, some stability tests with regular and irregular waves have been carried out. After analysing the experimental results, it will be shown how H₅₀-formulae can predict the observed damage independently of the sea state wave height distribution or the succession of sea states.     

Wave reflection and transmission by curtainwall–pile breakwaters using circular piles

This paper presents a mathematical model which computes the hydrodynamic characteristics of a curtainwall–pile breakwater (CPB) using circular piles, by modifying the model developed for rectangular piles by Suh et al. [2006. Hydrodynamic characteristics of pile-supported vertical wall breakwaters. Journal of Waterway, Port, Coastal and Ocean Engineering 132(2), 83–96]. To examine the validity of the model, laboratory experiments have been conducted for CPB with various values of draft of curtain wall, spacing between piles, and wave height and period. Comparisons between measurement and prediction show that the mathematical model adequately reproduces most of the important features of the experimental results. The mathematical model based on linear wave theory tends to over-predict the reflection coefficient as the wave height increases. As the draft of the curtain wall increases and the porosity between piles decreases, the reflection and transmission coefficient increases and decreases, respectively, as expected. As the relative water depth increases, however, the effect of porosity disappears because the wave motion is minimal in the lower part of a water column for short waves.     

Wave kinematics for simulated shallow water storm waves - Analysis and experiments

Computation of wave kinematics at or near offshore structures is a vitally important consideration in the design of offshore structures. Design waves often include breaking and near-breaking storm waves in the presence of currents. It is important to predict the kinematics of these steep waves. Experiments were carried out in a wave tank with simulated steep waves with and without in-line current in which the wave profiles and the corresponding kinematics were simultaneously measured. The simulated waves represent shallow-water Gulf of Mexico storm waves. Many of these waves broke at or near the measuring instruments. Irregular stream-function theory was used to compute the wave kinematics and was found to generally predict the measured wave-current kinematics well. The differences found between the two are noted. Some of the noteworthy features of the breaking waves are also discussed.     

Scattering of obliquely incident water waves by partially reflecting non-transmitting breakwaters

In the present paper, analytic solutions are derived for scattering of water waves obliquely incident to a partially reflecting semi-infinite breakwater or breakwater gap. In order to examine the correctness of the derived solutions, they are compared with the solutions derived by McIver (1999) and Bowen and McIver (2002) for a semi-infinite breakwater and a breakwater gap, respectively, in the case of perfect reflection. The derived analytic solutions are used to investigate the effect of reflection coefficient of the breakwater and wave incident angle upon the tranquility at harbor entrance. The tranquility is deteriorated by the reflected waves as the reflection coefficient increases and as waves are incident more obliquely.