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.     

Particulate organic carbon fluxes to the ocean interior and factors controlling the biological pump: A synthesis of global sediment trap programs since 1983

Particulate organic carbon (POC) is vertically transported to the oceanic interior by aggregates and their ballasts, mainly CaCO₃ and biogenic opal, with a smaller role for lithogenic aerosols through the mesopelagic zone. Diel migrating zooplankton communities effect vertical transport and remineralization of POC in the upper layers of the ocean. Below 1.5 km, the presence of zooplankton is reduced and thus the aggregates travel mainly by gravitational transport. We normalized the fluxes of POC, CaCO₃, and biogenic opal from data published on samples collected at 134 globally distributed, bottom-tethered, time-series sediment trap (TS-trap) stations to annual mole fluxes at the mesopelagic/bathypelagic boundary (m/b) at 2 km and defined them as F_m/bC_org, F_m/bC_inorg, and F_m/bSi_bio. Using this global data set, we investigated (1) the geographic contrasts of POC export at m/b and (2) the supply rate of ∑CO₂ to the world mesopelagic water column. F_m/bC_org varies from 25 (Pacific Warm Pool) to 605 (divergent Arabian Sea) mmolC m⁻² yr⁻¹; F_m/bC_inorg varies from >8 (high latitude Polar Oceans) or 15 (Pacific Warm Pool) to 459 (divergent Arabian Sea) mmolC m⁻² yr⁻¹; and F_m/bSi_bio, the most spatially/temporally variable flux, ranges from 6 (North Atlantic Drift) to 1118 (Pacific Subarctic Gyre) mmolSi m⁻² yr⁻¹. The oceanic region exhibiting the highest POC flux over a significantly large region is the area of the North Pacific Boreal Gyres where the average F_m/bC_org = 213, F_m/bC_inorg = 126, and F_m/bSi_bio = 578 mmol m⁻² yr⁻¹. F_m/bC_org and F_m/bC_inorg are particularly high in large upwelling margins, including the divergent Arabian Sea and off Cape Verde. One of the data sets showing the lowest flux over a significant region/basin is F_m/bC_org = 39, F_m/bC_inorg = 69, and F_m/bSi_bio = 22 mmol m⁻² yr⁻¹ in the North Pacific subtropical/tropical gyres; Pan-Atlantic average fluxes are similar except F_m/bSi_bio fluxes are even lower. Where C_org/C_inorg and Si_bio/C_inorg are <1 defines the “Carbonate Ocean”, and where these ratios are ?1 defines the “Silica Ocean”. The Carbonate Ocean occupies about 80% of the present world pelagic ocean between the two major oceanographic fronts, the North Pacific Polar Front and the Antarctic Polar Front, and the Silica Ocean is found on the polar sides of these fronts. The total global annual fluxes of F_m/bC_org, F_m/bC_inorg, and F_m/bSi_bio at m/b calculated by parameterizations of the export flux data from 134 stations are surprisingly similar; 36.2, 33.8, and 34.6 teramol yr⁻¹ (120, 112, and 114 mmol m⁻² yr⁻¹), respectively, resulting in a near uniform binary ratio between the above three elements of about one. The global ternary % ratios estimated from 152 TS-trap samples of the three elements are 35:32:33. From our global F_m/bC_org and a published model estimate of the global export production, we estimate the regeneration rate of CO₂ through the mesopelagic zone by the biological pump is 441 teramolC yr⁻¹. Based on our global F_m/bC_inorg and recently estimated global primary production of PIC, 36-86 teramolC yr⁻¹ of PIC is assumed to be dissolved within the upper 2 km of the water column.     

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.     

Experimental study of unidirectional irregular wave slamming on the three-dimensional structure in the splash zone

The experimental investigation of unidirectional random wave slamming on the three-dimensional structure in the splash zone is presented. The experiment is conducted in the marine environment channel in the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The test wave is unidirectional irregular wave. The experiments are carried out with perpendicular random waves (β=0°) and oblique random waves (β=15°, 30°, 45°), the significant wave heights H_1/3 ranging from 7.5 to 20 cm with 2.5 cm increment, the peak wave periods T_p ranging from 0.75 to 2.0 s with 0.25 s increment, and the clearance of the model with respect to the significant wave height s/H_1/3 ranging from 0.0 to 0.5 with 0.1 increment. The statistical analysis results of different test cases are presented. The statistical distribution characteristics of the perpendicular irregular wave impact pressures are compared with that of the oblique irregular wave on the underside of the structure. The effect of the wave direction β on the wave impact forces on the underside of the structure is determined. The relation between the impact forces and the parameters such as the significant wave height, the relative structure width and the relative clearance of the structure is also discussed.     

Megacusps on rip channel bathymetry: Observations and modeling

The formation of beach megacusps along the shoreline of southern Monterey Bay, CA, is investigated using time-averaged video and simulated with XBeach, a recently developed coastal sediment transport model. Investigations focus on the hydrodynamic role played by the bay's ever-present rip channels. A review of four years of video and wave data from Sand City, CA, indicates that megacusps most often form shoreward of rip channels under larger waves (significant wave height (H_s) = 1.5–2.0 m). However, they also occasionally appear shoreward of shoals when waves are smaller (H_s ~ 1 m) and the mean water level is higher on the beach. After calibration to the Sand City site, XBeach is shown to hindcast measured shoreline change moderately well (skill = 0.41) but to overpredict the erosion of the swash region and beach face. Simulations with small to moderate waves (H_s = 0.5–1.2 m) suggest, similar to field data, that megacusps will form shoreward of either rip channels or shoals, depending on mean daily water level and pre-existing beach shape. A frequency-based analysis of sediment transport forcing is performed, decomposing transport processes to the mean, infragravity, and very-low-frequency (VLF) contributions for two highlighted cases. Results indicate that the mean flow plays the dominant role in both types of megacusp formation, but that VLF oscillations in sediment concentration and advective flow are also significant.     

Application of a fuzzy inference system for the prediction of longshore sediment transport

A fuzzy inference system (FIS) and a hybrid adaptive network-based fuzzy inference system (ANFIS), which combines a fuzzy inference system and a neural network, are used to predict and model longshore sediment transport (LST). The measurement data (field and experimental data) obtained from Kamphuis [1] and Smith et al. [2] were used to develop the model. The FIS and ANFIS models employ five inputs (breaking wave height, breaking wave angle, slope at the breaking point, peak wave period and median grain size) and one output (longshore sediment transport rate). The criteria used to measure the performances of the models include the bias, the root mean square error, the scatter index and the coefficients of determination and correlation. The results indicate that the ANFIS model is superior to the FIS model for predicting LST rates. To verify the ANFIS model, the model was applied to the Karaburun coastal region, which is located along the southwestern coast of the Black Sea. The LST rates obtained from the ANFIS model were compared with the field measurements, the CERC [3] formula, the Kamphuis [1] formula and the numerical model (LITPACK). The percentages of error between the measured rates and the calculated LST rates based on the ANFIS method, the CERC formula (K_sig = 0.39), the calibrated CERC formula (K_sig = 0.08), the Kamphuis [1] formula and the numerical model (LITPACK) are 6.5%, 413.9%, 6.9%, 15.3% and 18.1%, respectively. The comparison of the results suggests that the ANFIS model is superior to the FIS model for predicting LST rates and performs significantly better than the tested empirical formulas and the numerical model.     

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.     

Experimental Investigation on the Extreme Waves Induced by Single Wave Packets in Finite Water Depth

Single Gaussian wave groups with different initial wave steepness ε_0 and width N are produced in laboratory in finite depth to study the nonlinear evolution, the extreme events and breaking. The results show that wave groups with larger ε_0 will evolve to be several envelope solitons(short wave groups). By analyzing geometric parameters, a break in the evolution of the wave elevation and asymmetric parameters after extreme wave may be an indicator for the inception of refocus and the maximal wave moving to the middle, namely, wave down-shift occurs. The analysis of the surface elevations with HHT(Hilbert-Huang Transform), which presents the concrete local variation of energy in time and frequency can be exhibited clearly, reveals that the higher frequency components play a major role in forming the extreme event and the contribution to the nonlinearity. Instantaneous energy and frequency in the vicinity of the extreme wave are also examined locally. For spilling breakers, the energy residing in the whole wave front dissipates much more due to breaking, while the energy in the rear of wave crest loses little, and the intra-wave frequency modulation increases as focus. It illustrates that the maximal first order instantaneous frequency f_1 and the largest crest tend to emerge at the same time after extreme wave when significant energy dissipation happens, and vice versa. In addition, it shows that there is no obvious relation of the CDN(combined degree of nonlinearity) to the wave breaking for the single Gaussian wave group in finite water depth.     

Phytoplankton community adaptation to changing light levels in the southern Beaufort Sea,Canadian Arctic

The chlorophyll a specific absorption coefficient of phytoplankton, a_φ^∗(λ) is an important parameter to determine for primary production models and for the estimation of phytoplankton physiological condition. Knowledge of this parameter at high latitudes where nutrient rich cold water submitted to low incident light is a common environment is almost nonexistent. To address this issue, we investigated the light absorption properties of phytoplankton as a function of irradiance, temperature, and nutrients using a large data set in the southern Beaufort Sea during the open water to ice cover transition period. The a_φ^∗(λ) tended to increase from autumn when open water still existed to early winter when sea ice cover was formed, resulting from a biological selection of smaller-size phytoplankton more efficient to absorb light. There was no significant correlation between a_φ^∗(λ) and irradiance or temperature for both seasons. However, a_φ^∗(λ) showed a significant positive correlation with NO₃ + NO₂. Implications of the results for phytoplankton community adaptation to changing light levels are discussed.     

Infra-gravity wave generation by the shoaling wave groups over beaches

A physical parameter, μb, which was used to meet the forcing of primary short waves to be off-resonant before wave breaking, has been considered as an applicable parameter in the infra-gravity wave generation. Since a series of modulating wave groups for different wave conditions are performed to proceed with the resonant mechanism of infra-gravity waves prior to wave breaking, the amplitude growth of incident bound long wave is assumed to be simply controlled by the normalized bed slope, βb. The results appear a large dependence of the growth rate, α, of incident bound long wave, separated by the three-array method, on the normalized bed slope, βb. High spatial resolution of wave records enables identification of the cross-correlation between squared short-wave envelopes and infra-gravity waves. The cross-shore structure of infra-gravity waves over beaches presents the mechanics of incident bound- and outgoing free long waves with the formation of free standing long waves in the nearshore region. The wave run-up and amplification of infra-gravity waves in the swash zone appear that the additional long waves generated by the breaking process would modify the cross-shore structure of free standing long waves. Finally, this paper would further discuss the contribution of long wave breaking and bottom friction to the energy dissipation of infra-gravity waves based on different slope conditions.     

Effect of wave amplitude on turbulent flow in a wavy channel by direct numerical simulation

The present study numerically investigates the characteristics of three-dimensional turbulent flow in a wavy channel. For the purpose of a careful observation of the effect of the wave amplitude on the turbulent flow, numerical simulations are performed at a various range of the wave amplitude to wavelength ratio (0.01?α/λ?0.05), where the wavelength is fixed with the same value of the mean channel height (H). The immersed boundary method is used to handle the wavy surface in a rectangular grid system, using the finite volume method. The Reynolds number (Re=U_bH/ν) based on the bulk velocity (U_b) is fixed at 6760. The present computational results for a wavy surface are well compared with those of references. When α/λ=0.02, the small recirculating flow occurs near the trough at the instant, but the mean reverse flow is not observed. In the mean flow field, the reverse flow appears from α/λ=0.03 among the wave amplitude considered in this study. The domain of the mean reverse flow defined by the locations of separation and reattachment depends strongly on the wave amplitude. The pressure drag coefficient augments with increasing the wave amplitude. The friction drag coefficient shows the increase and decrease behavior according to the wave amplitude. The quantitative information about the flow variables such as the distribution of pressure and shear stress on the wavy surface is highlighted.     

Effect of bed roughness on turbulent boundary layer and net sediment transport under asymmetric waves

This paper presents an investigation of the roughness effects in the turbulent boundary layer for asymmetric waves by using the baseline (BSL) k–ω model. This model is validated by a set of the experimental data with different wave non-linearity index, N_i (namely, N_i = 0.67, N_i = 0.60 and N_i = 0.58). It is further used to simulate asymmetric wave velocity flows with several values of the roughness parameter (a_m/k_s) which increase gradually, namely from a_m/k_s = 35 to a_m/k_s = 963. The effect of the roughness tends to increase the turbulent kinetic energy and to decrease the mean velocity distribution in the inner boundary layer, whereas in the outer boundary layer, the roughness alters the turbulent kinetic energy and the mean velocity distribution is relatively unaffected. A new simple calculation method of bottom shear stress based on incorporating velocity and acceleration terms is proposed and applied into the calculation of the rate of bed-load transport induced by asymmetric waves. And further, the effect of bed roughness on the bottom shear stress and bed-load sediment transport under asymmetric waves is examined with the turbulent model, the newly proposed method, and the existing calculation method. It is found that the higher roughness elements increase the magnitude of bottom shear stress along a wave cycle and consequently, the potential net sediment transport rate. Moreover, the wave non-linearity also shows a big impact on the bottom shear stress and the net sediment transport.     

High-resolution observations of aggregate flux during a sub-polar North Atlantic spring bloom

An aggregate flux event was observed by ship and by four underwater gliders during the 2008 sub-polar North Atlantic spring bloom experiment (NAB08). At the height of the diatom bloom, aggregates were observed as spikes in measurements of both particulate backscattering coefficient (b_bp) and chlorophyll a fluorescence. Optical sensors on the ship and gliders were cross-calibrated through a series of simultaneous profiles, and b_bp was converted to particulate organic carbon. The aggregates sank as a discrete pulse, with an average sinking rate of ∼75 m d⁻¹; 65% of aggregate backscattering and 90% of chlorophyll fluorescence content was lost between 100 m and 900 m. Mean aggregate organic carbon flux at 100 m in mid-May was estimated at 514 mg C m⁻² d⁻¹, consistent with independent flux estimates. The use of optical spikes observed from gliders provides unprecedented coupled vertical and temporal resolution measurements of an aggregate flux event.     

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.     

A ten-year data set for fetch- and depth-limited wave growth

This paper presents the key results from a ten-year data set for Lake IJssel and Lake Sloten in The Netherlands, containing information on wind, storm surges and waves, supplemented with SWAN 40.51 wave model results. The wind speeds U₁₀, effective fetches x and water depths d for the data set ranged from 0–24 m s^− 1, 0.8–25 km and 1.2–6 m respectively. For locations with non-sloping bottoms, the range in non-dimensional fetch x? ( = gxU₁₀^− 2) was about 25–80,000, while the range in dimensionless depth d? ( = g d U₁₀^− 2) was about 0.03–1.7. Land–water wind speed differences were much smaller than the roughness differences would suggest. Part of this seems due to thermal stability effects, which even play a role during near-gale force winds. For storm surges, a spectral response analysis showed that Lake IJssel has several resonant peaks at time scales of order 1 h. As for the waves, wave steepnesses and dimensionless wave heights H? ( = gH_m0U₁₀^− 2) agreed reasonably well with parametric growth curves, although there is no single curve to which the present data fit best for all cases. For strongly depth-limited waves, the extreme values of d? (0.03) and H_m0 / d (0.44) at the 1.7 m deep Lake Sloten were very close to the extremes found in Lake George, Australia. For the 5 m deep Lake IJssel, values of H_m0 / d were higher than the depth-limited asymptotes of parametric wave growth curves. The wave model test cases of this study demonstrated that SWAN underestimates H_m0 for depth-limited waves and that spectral details (enhanced peak, secondary humps) were not well reproduced from H_m0 / d = 0.2–0.3 on. SWAN also underestimated the quick wave response (within 0.3–1 h) to sudden wind increases. For the remaining cases, the new [Van der Westhuysen, A.J., Zijlema, M., and Battjes, J.A., 2007. Nonlinear saturation-based whitecapping dissipation in SWAN for deep and shallow water, Coast. Eng., 54, 151–170] SWAN physics yielded better results than the standard physics of Komen, G.J., Hasselmann, S., Hasselmann, K., 1984. On the existence of a fully developed wind-sea spectrum. J. Phys. Oceanogr. 14, 1271–1285, except for persistent overestimations that were found for short fetches. The present data set contains many interesting cases for detailed model validation and for further studies into the evolution of wind waves in shallow lakes.     

Laboratory flume studies on monochromatic wave-fine sandy bed interactions Part 2. Sediment suspensions

As reported in preceding paper (Part 1. Soil Fluidization), the observed phenomena of sediment suspensions above a fluidized sandy bed of Sand II (d₅₀ = 0.092 mm) under monochromatic wave actions are quantitatively investigated. The suspended sediment concentration (SSC) at a single point within 5 cm above the bed was synchronously measured with water waves and bed soil's pore pressures with an intrusive optical sediment-concentration probe. The measurements show that SSC initiates several wave cycles after initiation of bed soil's fluidized response and grows to a peak value mainly in the post-fluidization phase. Under similar wave loadings in the same test series, SSC is usually higher over a resonantly fluidized (RF) bed than over a non-resonantly fluidized (NRF) bed. On the contrary, only relatively low SCC can be identified above an unfluidized bed. The analyses illustrate that to certain extent, peak values of SSC are directly proportional to the thickness of fluidized soil layer d_f. Values of d_f usually decrease with repeated fluidized response, longer consolidation periods, and in deeper water depths. Once the fluidized responses initiate, pore pressures are generally much significantly amplified in both shallow fluidized soil layers and near below the fluidized layer, especially during the resonance event. The resulting depth gradients of dynamic pore pressure amplitudes in shallow layers are likely to have caused higher initial rises of SSC in a RF bed than in the subsequent NRF bed. Those in deeper layer should have contributed to sustain the fluidization state for further SSC increments. Immediately after termination of wave loading, re-deposited suspended sediments always result in a typical flat bed form. For a pre-fluidized bed, wave-induced drastic sediment suspensions are still obtainable very near above the bed with even a rather thin fluidized surface soil layer.     

Effective fetch and non-linear four-wave interactions during wave growth in slanting fetch conditions

Wave growth in slanting fetch (with wind blowing obliquely off a coast) is investigated with 7 years worth of routine wave measurements in Lake IJssel in The Netherlands and with the SWAN wave model. Two aspects are considered in particular for this case: the validity of the concept of effective fetch and the role of the non-linear four-wave interactions. For slanting and parallel fetch conditions, we found some significant deviations from the effective fetch assumption, leading to 20–35% mismatch in either the peak period T_p or the significant wave height H_m0 respectively. However, the effect of discrepancies between various widely accepted wave growth formulas turned out to be even more important. The wave directions during slanting fetch are significantly ‘steered’ by the coastline, especially in the first kilometre(s) off the coast. The role of the non-linear four-wave interactions is investigated by running the SWAN (version 40.41) wave model with three different quadruplet formulations. Exact quadruplet methods (Xnl) yielded relatively strong wave steering, despite the four-wave interactions being relatively weak. Application of Xnl did not lead to better overall agreement with measurements — improvements for the mean wave period T_m01 were offset by some deterioration for the wave height H_m0.     

Variability of wave-induced ripple migration in wave-flume experiments and its implications for sediment transport

A thorough discussion of results from laboratory experiments with regular waves sheds light on the gap that lies between the sediment transport associated with ripple migration and the performance of a standard bedload transport formula in terms of bed shear concept. It is found that the extent of deviations of the bedload transport formula by Ribberink (1998) from the measured rate of sediment transport associated with ripple migration becomes systematically apparent under conditions of increasing settling time factor Ω_s (= η/(w₀T); η is the ripple height, w₀ the settling velocity and T the wave period). Re-examination of previous two field studies demonstrates a further reinforcement for phase-lag argument addressed in this paper.     

Characteristic friction coefficient and scale effects in oscillatory porous flow

This work presents a simple method to evaluate the performance of a porous breakwater when it is impinged with normal incidence by a non-breaking monochromatic wave train. It is based on: 1) a potential flow model for wave interaction with permeable structures and 2) a set of experimental tests on a rectangular porous structure with uniform granular distribution. A characteristic friction diagram is obtained considering wave energy balance in a control volume, minimising the error between the numerical model and the experimental results for the wave transmission coefficient. Results show that, for large breakwater widths, the reflection process reaches a saturation regime before the waves exit the structure at a distance from the seaside between the interval 0.2 < x/L < 0.45. For larger breakwater widths, the reflection coefficient is almost constant (except for “resonant” conditions) and wave transmission decreases exponentially. Under such conditions, the wave propagation through the porous medium depends on the relative diameter D/L and the porosity of the material; the dependence on the relative breakwater width B/L and the ratio diameter wave height D/H is weak. This diagram intends to be useful for preliminary engineering studies of breakwater's efficiency and performance and as an adequate selection criteria of the experimental stone diameter to minimize scale effects in laboratory studies.     

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.