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.     

Prediction of wave pressures on smooth impermeable seawalls

Simple prediction methods are proposed to estimate the wave induced pressures on smooth impermeable seawalls. Based on the physics of the wave structure interaction, the sloped seawall is divided into a total of five zones (zones 1, 2 and 3 during run-up (corresponding pressures are called as positive pressures) and zones 4 and 5 during run-down (corresponding pressures are called negative pressures)) (Fig. 1). Zone 1 (0<z<d−H_i/2), where the wave pressure is governed by the partial reflection and phase shift; Zone 2 (d−H_i/2<z<d), where the effect of wave breaking and turbulence is significant; Zone 3 (d<z<Run-up height), where the pressure is induced by the run-up water; Zone 4 (Run-down<z<d), where the wave pressure is caused by the run-down effect and Zone 5 (0<z<d-Run down), where the negative wave pressures are due to partial reflection and phase shift effects. Here d is the water depth at the toe of the seawall, H_i is the incident wave height and z is the vertical elevation with toe of the seawall as origin and z is positive upward. For wave pressure prediction in zones 1 and 5, the empirical formula proposed by Ahrens et al. (1993) to estimate the wave reflection and Sutherland and Donoghue's recommendations (1998) for the estimation of phase shift of the waves caused by the sloped structures are used. Multiple regression analysis is carried out on the measured pressure data and empirical formulas are proposed for zones 2, 3 and 4. The recommendations of Van der Meer and Breteler (1990) and Schüttrumpf et al. (1994) for the prediction of wave run-down are used for pressure prediction at zone 4. Comparison of the proposed prediction formulas with the experimental results reveal that the prediction methods are good enough for practical purposes. The present study also shows a strong relation between wave reflection, wave run-up, wave run-down and phase shift of waves on wave pressures on the seawalls.     

Prediction of geometrical properties of perfect breaking waves on composite breakwaters

Breaking wave loads on coastal structures depend primarily on the type of wave breaking at the instant of impact. When a wave breaks on a vertical wall with an almost vertical front face called the “perfect breaking”, the greatest impact forces are produced. The correct prediction of impact forces from perfect breaking of waves on seawalls and breakwaters is closely dependent on the accurate determination of their configurations at breaking. The present study is concerned with the determination of the geometrical properties of perfect breaking waves on composite-type breakwaters by employing artificial neural networks. Using a set of laboratory data, the breaker crest height, h_b, breaker height, H_b, and water depth in front of the wall, d_w, from perfect breaking of waves on composite breakwaters are predicted using the artificial neural network technique and the results are compared with those obtained from linear and multi-linear regression models. The comparisons of the predicted results from the present models with measured data show that the h_b, H_b and d_w values, which represent the geometry of waves breaking directly on composite breakwaters, can be predicted more accurately by artificial neural networks compared to linear and multi-linear regressions.     

Prediction of berm geometry using a set of laboratory tests combined with teaching–learning-based optimization and artificial bee colony algorithms

Understanding sediment movement in coastal areas is crucial in planning the stability of coastal structures, the recovery of coastal areas, and the formation of new coast. Accretion or erosion profiles form as a result of sediment movement. The characteristics of these profiles depend on the bed slope, wave conditions, and sediment properties. Here, experimental studies were performed in a wave flume with regular waves, considering different values for the wave height (H₀), wave period (T), bed slope (m), and mean sediment diameter (d₅₀). Accretion profiles developed in these experiments, and the geometric parameters of the resulting berms were determined. Teaching–learning-based optimization (TLBO) and artificial bee colony (ABC) algorithms were applied to regression functions of the data from the physical model. Dimensional and dimensionless equations were found for each parameter. These equations were compared to data from the physical model, to determine the best equation for each parameter and to evaluate the performances of the TLBO and ABC algorithms in the estimation of the berm parameters. Compared to the ABC algorithm, the TLBO algorithm provided better accuracy in estimating the berm parameters. Overall, the equations successfully determined the berm parameters.     

Sub aerial random wave pressure and layer thickness on impermeable sea walls

Investigations on sub aerial wave pressures and layer thickness on plane impermeable and non-overtopping seawallns were carried out by using physical model studies. Seawalls with slopes of 1:3, 1:4 and 1:6 were used. JONSWAP spectrum with significant wave height, H_s from 0.08 to 0.2 m and peak periods, T_p from 1.5 to 6.0 s and a constant water depth of 0.7 m is used. Based on extensive measurements, empirical formulas for practical applications are proposed to predict the maximum, significant and mean sub aerial random wave pressure and layer thickness (thickness of water layer perpendicular to the still water level on the run-up zone) by using the surf similarity parameter, significant wave height and elevation on the sub aerial region as inputs. It is found that the maximum layer thickness is 1.11 times the significant layer thickness and maximum sub Arial wave pressure is 1.06 times the significant wave pressures. The predictive equations based on extensive measurements can be used for the design of non-overtopping seawalls.     

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.     

Prediction of significant wave height using regressive support vector machines

Wave parameters prediction is an important issue in coastal and offshore engineering. In this literature, several models and methods are introduced. In the recent years, the well-known soft computing approaches, such as artificial neural networks, fuzzy and adaptive neuro-fuzzy inference systems and etc., have been known as novel methods to form intelligent systems, these approaches has also been used to predict wave parameters, as well. It is not a long time that support vector machine (SVM) is introduced as a strong machine learning and data mining tool. In this paper, it is used to predict significant wave height (H_s). The data set used in this study comprises wave wind data gathered from deep water locations in Lake Michigan. Current wind speed (u) and those belonging up to six previous hours are given as input variables, while the significant wave height is the output parameter. The SVM results are compared with those of artificial neural networks, multi-layer perceptron (MLP) and radial basis function (RBF) models. The results show that SVM can be successfully used for prediction of H_s. Furthermore, comparisons indicate that the error statistics of SVM model marginally outperforms ANN even with much less computational time required.     

Prediction of scour depth at breakwaters due to non-breaking waves using machine learning approaches

Coastal structures may cease to function properly due to seabed scouring. Hence, prediction of the maximum scour depth is of great importance for the protection of these structures. Since scour is the result of a complicated interaction between structure, sediment, and incoming waves, empirical equations are not as accurate as machine learning schemes, which are being widely employed for the coastal engineering modeling. In this paper, which can be regarded as an extension of Pourzangbar et al. (2016), two soft computing methods, a support vector regression (SVR), and a model tree algorithm (M5′), have been implemented to predict the maximum scour depth due to non-breaking waves. The models predict the relative scour depth (S_max/H₀) on the basis of the following variables: relative water depth at the toe of the breakwater (h_toe/L₀), Shields parameter (θ), non-breaking wave steepness (H₀/L₀), and reflection coefficient (Cr). 95 laboratory data points, extracted from dedicated experimental studies, have been used for developing the models, whose performances have been assessed on the basis of statistical parameters. The results suggest that all of the developed models predict the maximum scour depth with high precision, the M5′ model performed marginally better than the SVR model and also allowed to define a set of transparent and physically sound relationships. Such relationships, which are in good agreement with the existing empirical findings, show that the relative scour depth is mainly affected by wave reflection.     

Comparison between M5′ model tree and neural networks for prediction of significant wave height in Lake Superior

Prediction of wave height is of great importance in marine and coastal engineering. Soft computing tools such as artificial neural networks (ANNs) are recently used for prediction of significant wave height. However, ANNs are not as transparent as semi-empirical regression-based models. In addition, neural networks approach needs to find network parameters such as number of hidden layers and neurons by trial and error, which is time consuming. Therefore, in this work, model trees as a new soft computing method was invoked for prediction of significant wave height. The main advantage of model trees is that, compared to neural networks, they represent understandable rules. These rules can be readily expressed so that humans can understand them. The data set used for developing model trees comprises of wind and wave data gathered in Lake Superior from 6 April to 10 November 2000 and 19 April to 6 November 2001. M5′ algorithm was employed for building and evaluating model trees. Training and testing data include wind speed (U₁₀) as the input variable and the significant wave height (H_s) as the output variable. Results indicate that error statistics of model trees and feed-forward back propagation (FFBP) ANNs were similar, while model trees was marginally more accurate. In addition, model tree shows that for wind speed above 4.7 m/s, the wave height increases nonlinearly by the wind speed.     

Effects of high-order nonlinear interactions on unidirectional wave trains

Numerical simulations of gravity waves with high-order nonlinearities in two-dimensional domain are performed by using the pseudo spectral method. High-order nonlinearities more than third order excite apparently chaotic evolutions of the Fourier energy in deep water random waves. The high-order nonlinearities increase kurtosis, wave height distribution and H_max/H_1/3 in deep water and decrease these wave statistics in shallow water. Moreover, they can generate a single extreme high wave with an outstanding crest height in deep water. High-order nonlinearities (more than third order) can be regarded as one cause of freak waves in deep water.     

Wave run-up on bermed coastal structures

Reliable estimation of wave run-up is required for the effective and efficient design of coastal structures when flooding or wave overtopping volumes are an important consideration in the design process. In this study, a unified formula for the wave run-up on bermed structures has been developed using collected and existing data. As data on berm breakwaters was highly limited, physical model tests were conducted and the run-up was measured. Conventional governing parameters and influencing factors were then used to predict the dimensionless run-up level with 2% exceedance probability. The developed formula includes the effect of water depth which is required in understanding the influence of sea level rise and consequent changes of wave height to water depth ratio on the future hydraulic performance of the structures. The accuracy measures such as RMSE and Bias indicated that the developed formula is more accurate than the existing formulas. Additionally, the new formula was validated using field measurements and its superiority was observed when compared to the existing prediction formulas. Finally, the new design formula incorporating the partial safety factor was introduced as a design tool for engineers.     

The stability of the antifer units used on breakwaters in case of irregular placement

The primary aim of the study is to experimentally investigate the stability performance of antifer units on the trunk section of breakwaters under the effect of regular and irregular waves in case of irregular placement. The stability performance tests were conducted for different slopes, i.e. cot α=1.25, 1.5, 2.0, 2.5, under irregular waves and for cot α=2.5 under regular waves. Hudson’s formula was employed in order to characterize the stability performance of antifer units for the irregular placement technique. Different representative wave height parameters, i.e. H_s, H_1/10 and H_max, were examined to determine the one best characterizing breakwater stability. Furthermore, the effects of wave period and wave steepness on the stability of the breakwater were explored.     

Estimation of the wave-related ripple characteristics and induced bed shear stress

A large data set on ripples was collected and examined. A set of new formulas for the prediction of the ripple characteristics is proposed with an emphasis on the disappearance of the ripples. The ripple wavelength was observed to be proportional to the bottom wave excursion but also to be a function of the grain-related Shields parameter and wave period parameter introduced by Mogridge et al. (1994). The ripple steepness was found to be nearly constant for orbital ripples, and with a sharp decrease for suborbital ripples. Two empirical functions are added including the effects of the critical Shields parameters (inception of transport and inception of sheet flow), i.e. giving the boundaries for the ripple existence's domain. The proposed formulas yield better prediction capabilities compared to the previously published formulas, especially when ripples are washed out. The effect of the ripple characteristics on the roughness height and the calculation of the bed shear stress is also discussed. It appeared that the bed shear stress calculation is more sensitive to the empirical coefficient a_r introduced in the estimation of the ripple-induced roughness height or to the limits of existence of the ripples than the ripple characteristics themselves.     

Improvement of sand activation depth prediction under conditions of oblique wave breaking

This study presents sand activation depth (SAD) measurements recently obtained on two contrasting beaches located along the Atlantic coast of France: the gently sloping, high-energy St Trojan beach where wave incidence is usually weak, and the steep, low-energy Arçay Sandspit beach where waves break at highly oblique angles. Comparisons between field measurements and predictions from existing formulae show good agreement for St Trojan beach but underestimate the SAD on the Arçay Sandspit beach by 40–60%. Such differences suggest a strong influence of wave obliquity on SAD. To verify this hypothesis, the relative influence of wave parameters was investigated by means of numerical modelling. A quasi-linear increase of SAD with wave height was confirmed for shore-normal and slightly oblique wave conditions, and a quasi-linear increase in SAD with wave obliquity was also revealed. Combining the numerical results with previously published relations, both a new semi-empirical and an empirical formula for the prediction of SAD were developed which showed good SAD predictions under conditions of oblique wave breaking. The new empirical formula for the prediction of SAD (Z ₀) takes into account the significant wave height (H _s), the beach face slope (β) and the wave angle at breaking (α), and is of the form $ Z_{0} = 1.6\tan {\left( \beta \right)}H^{{0.5}}_{{\text{s}}} {\sqrt {1 + \sin {\left( {2\alpha } \right)}} } This study presents sand activation depth (SAD) measurements recently obtained on two contrasting beaches located along the Atlantic coast of France: the gently sloping, high-energy St Trojan beach where wave incidence is usually weak, and the steep, low-energy Ar?ay Sandspit beach where waves break at highly oblique angles. Comparisons between field measurements and predictions from existing formulae show good agreement for St Trojan beach but underestimate the SAD on the Ar?ay Sandspit beach by 40–60%. Such differences suggest a strong influence of wave obliquity on SAD. To verify this hypothesis, the relative influence of wave parameters was investigated by means of numerical modelling. A quasi-linear increase of SAD with wave height was confirmed for shore-normal and slightly oblique wave conditions, and a quasi-linear increase in SAD with wave obliquity was also revealed. Combining the numerical results with previously published relations, both a new semi-empirical and an empirical formula for the prediction of SAD were developed which showed good SAD predictions under conditions of oblique wave breaking. The new empirical formula for the prediction of SAD (Z ₀) takes into account the significant wave height (H _s), the beach face slope (β) and the wave angle at breaking (α), and is of the form . The use of a dataset from the literature demonstrates the predictive skill of these new formulae for a wide range of wave heights, wave incidence and beach gradients.     

Testing and calibrating parametric wave transformation models on natural beaches

To provide coastal engineers and scientists with a detailed inter-comparison of widely used parametric wave transformation models, several models are tested and calibrated with extensive observations from six field experiments on barred and unbarred beaches. Using previously calibrated (“default”) values of a free parameter γ, all models predict the observations reasonably well (median root-mean-square wave height errors are between 10% and 20%) at all field sites. Model errors can be reduced by roughly 50% by tuning γ for each data record. No tuned or default model provides the best predictions for all data records or at all experiments. Tuned γ differ for the different models and experiments, but in all cases γ increases as the hyperbolic tangent of the deep-water wave height, H_o. Data from two experiments are used to estimate empirical, universal curves for γ based on H_o. Using the new parameterization, all models have similar accuracy, and usually show increased skill relative to using default γ.     

随机波浪中畸形波在三维岛礁地形上的试验研究

在试验水池中,开展了波浪在岛礁地形上演化问题的研究。首先在实验水池中建立了西太平洋某岛礁地形的模型,然后采用改进的JONSWAP谱,由造波机产生不同周期、波高的随机波浪。试验中观察到了不同类型畸形波生成的过程及不同波面形态的畸形波。对偏度、峰度及水深与畸形波要素H_m/H_s（H_m表示波列中的最大波高, H_s为有效波高）的关系进行了详细的分析,同时,对畸形波波高H_fr与偏度的关也进行了分析。通过对试验结果分析,发现峰度与畸形波要素i>H_m/H_s呈正相关, H_fr增大时相应的偏度也会呈现增大的趋势。此外,水深的变化剧烈时（如斜坡、海山位置）有助于畸形波的发生。     

Profiles of fully developed (Airy) waves in different water depths

Airy waves have a sinusoidal profile in deep water that can be modeled by a time series at any point x and time t, given by η(x,t) = (H_o/2) cos[2πx/L_o − 2πt/T_w], where H_o is the deepwater height, L_o is the deepwater wavelength, and T_w is the wave period. However, as these waves approach the shore they change in form and dimension so that this equation becomes invalid. A method is presented to reconstruct the wave profile showing the correct wavelength, wave height, wave shape, and displacement of the water surface with respect to the still water level for any water depth.     

Wave damping over artificial Posidonia oceanica meadow: A large-scale experimental study

An experimental study, conducted in the large wave flume of CIEM in Barcelona, is presented to evaluate the effects of Posidonia oceanica meadows on the wave height damping and on the wave induced velocities. The experiments were performed for irregular waves from intermediate to shallow waters with the dispersion parameter h/λ ranging from 0.09 to 0.29. Various configurations of the artificial P. oceanica meadow were tested for two stem density patterns (360 and 180 stems/m²) and for plant's height ranging from 1/3 to 1/2 of the water depth.The results for wave height attenuation are in good agreement with the analytical expressions found in literature, based on the assumption that the energy loss over the vegetated field is due to the drag forces. Based on this hypothesis, an empirical relationship for the drag coefficient related to the Reynolds number, Re, is proposed. The Reynolds number, calculated using the artificial P. oceanica leaf width as the length scale and the maximum orbital velocity over the meadow edge as the characteristic velocity scale, ranges from 1000 to 3500 and the drag coefficient C_d ranges from 0.75 to 2.0.The calculated wave heights, using the analytical expression from literature and the proposed relationship for the estimation of C_d, are in satisfactory agreement with those measured. Wave orbital velocities are shown to be significantly attenuated inside the meadow and just above the flume bed as indicated by the calculation of an attenuation parameter. Near the meadow edge, energy transfer is found in spectral wave velocities from the longer to the shorter wave period components. From the analysis it is shown that the submerged vegetation attenuates mostly longer waves.     

Formation of beach cusps in a wave tank

Beach cusps with a longshore spacing of 20 to 150 cm have been built by the continuous action of incident waves on a steep laboratory beach floor covered uniformly with a thin bed of glass beads. Breaking of incident waves was observed to induce vortices on the bed by interacting with swash motion along the beach face. Beach cusps formed when the value of a dimensionless parameter H_b/sgT_i² became smaller than 0.042; H_b is the breaking height of the incident waves, T_i their period, s the beach slope and g the acceleration due to gravity. This critical value occurred at a nearly central part of the generation region 0.003 < H_b/sgT_i² < 0.068 for plunging breakers presented by Galvin (1968). Breaking-wave-induced vortices rather than breaker types controlled the movement of bed material in the nearshore zone. Most of the measured spacings of beach cusps, including previous observations, were in good agreement with half a wavelength of the zero-mode subharmonic edge wave, which is generated on the beach by the refraction of incident waves and has twice the period of the waves. The role of edge waves at each stage of cusp formation still remains as an important problem to be clarified.