Geometrical properties of perfect breaking waves on composite breakwaters

The experimental results have so far shown that when a wave breaks on a vertical wall with an almost vertical front face at the instant of impact that is called perfect breaking or perfect impact, the greatest impact forces are produced on the wall. Therefore, the configuration of breaking waves is important in the design considerations of coastal structures. The present study is concerned with determining the geometrical properties of oscillatory waves that break perfectly on the vertical wall of composite-type breakwaters. The laboratory tests for perfect breaking waves on composite breakwaters are conducted with base slopes of 1/2, 1/4 and 1/6, and with berm widths of 0.00, 0.10, 0.20, 0.30 and 0.40 m. The shape and the dimensions of waves at the instant of perfect breaking on the wall are determined using a video camera. The experimental results for the geometrical properties of the breakers are presented non-dimensionally. Within the range of present experimental conditions, it is found that the dimensionless breaker crest height, h_b/d_w, and dimensionless breaker height, H_b/d_w, decrease; and, dimensionless breaker depth, d_w/H₀, increases with increasing relative berm width, B/D. The breaker height index, H_b/H₀, is almost unaffected by B/D. The deep-water wave steepness and the base slope of the breakwater do not seem to influence the geometrical properties of the breakers at wall systematically.     

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.     

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.     

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.     

A new predictive formula for inception of regular wave breaking

Efforts are made to enhance the predictive formula for the inception of wave breaking. To achieve success, the existing formulas are extensively reviewed. They are categorized into four types, i.e., the McCowan type, the Miche type, the Goda type and the Munk type. The inherent relations among the different types are then exploited. The differences among each formula within a group are also discussed. Four representative formulas from the different types are chosen to compare with the measured data for a total number of 1193 cases reported in literatures. It is shown that Goda's and Ostendorf and Madsen's formulas are advantageous in general among the selected ones. Goda's formula, however, is found to be inaccurate as the beach slope becomes steeper than 1/10. Ostendorf and Madsen's formula is fairly good even for cases of very steep slopes, but its accuracy for the cases of ordinary slopes is not as good as Goda's. A new predictive formula for the inception of wave breaking is proposed. The unique index, defined by ψ′_b = (1.21 − 3.30λ_b)(1.48 − 0.54γ_b)ψ_b, where ψ_b = gH_b/C_b², H_b is the breaking wave height, C_b is the breaking wave celerity, λ_b is the breaking wave steepness, γ_b is the relative breaking wave height, and g is the gravity acceleration, is introduced. The incipient condition of wave breaking is then given by ψ′_b = 0.69. This formula is a significant improvement to the existing ones in terms of the accuracy. In addition, it is a local relation. Further verification shows that the proposed formula performs similarly well when applied to the field and to the waves over permeable bed.     

Medium-term frequency distributions of cross-shore suspended sediment transport rates in water depths of 3 to 9 m

A data set of several thousands of hours of near-bed flow, obtained at three cross-shore positions in 3- to 9-m water depth in the multiple bar system of Terschelling (Netherlands), was used to estimate the medium-term (≈years) frequency distribution of the cross-shore suspended sediment transport rates induced by short waves, infragravity waves and cross-shore mean flows. Predictions of an energetics-based transport model were categorised into groups of the local height-over-depth ratio with a width of 0.02 and were subsequently coupled to the discrete medium-scale probability distribution of this ratio. At all depths, the estimated medium-scale sediment transport rate by the short waves, q_ms,h, and mean flows, q_ms,mf, were of approximately equal magnitude and were about three times as large as that of the infragravity waves. In general, the medium-term sediment transport rates were dominated neither by the most extreme conditions nor by day-to-day situations. This was related to the infrequent recurrence of the most energetic events and by the predicted negligible transport rates under daily conditions. In 9-m depth, breaking conditions contributed to about 90% of both q_ms,h and q_ms,mf. In shallower water (3–5 m), non-breaking conditions became increasingly important for q_ms,h, whereas q_ms,mf remained fully dominated by surf zone conditions. This observation as well as literature findings for water depths less than about 3 m suggest that the range of small-scale conditions that contribute most to q_ms,h and q_ms,mf changes in the onshore direction from mainly breaking conditions at depths in excess of 5–7 m towards prolonged non-breaking periods for q_ms,h and short breaking events for q_ms,mf on the beach.     

Placement Effect on the Stability of Tetrapod Armor Unit on Breakwaters in Irregular Waves

Tetrapod, one of the well-known artificial concrete units, is frequently used as an armor unit on breakwaters. Two layers of tetrapod units are normmaly placed on the breakwaters with different placement methods. In this study, the stability of tetrapod units with two different regularly placement methods are investigated experimentally in irregular waves. Stability coefficients of tetrapod units for both placement methods are obtained. The important characteristic wave parameters of irregular waves causing the same damage ratio as those of the regular waves are also determined. It reveals that the average of one-tenth highest wave heights within the wave train (H_1/10) causes the similar damage as regular waves.     

Hydrodynamic efficiency of partially immersed caissons supported on piles

The efficiency of the breakwater, which consists of caissons supported on two or three rows of piles, was studied using physical models. The efficiency of the breakwater is presented as a function of the transmission, reflection and the wave energy dissipation coefficients. Regular waves with wide ranges of wave heights and periods and constant water depth were used. Different characteristics of the caisson structure and the supporting pile system were also tested. It was found that, the transmission coefficient (k_t) decreases with increasing the relative breakwater draft D/L, increasing the relative breakwater width B/h, and decreasing the piles gap-diameter ratio G/d. It is possible to achieve k_t values less than 0.25 when D/L≥0.1. The reflection coefficient takes the opposite trend especially when D/L≤0.15. The proposed breakwater dissipates about 10-25% of the incident wave energy. Also, simple empirical equations are developed for estimating the wave transmission and reflection. In addition, the proposed breakwater model is efficient compared with other floating breakwaters.     

Shoreline response to a single shore-parallel submerged breakwater

Submerged breakwaters (SBWs) are becoming a popular option for coastal protection, mainly due to their low aesthetic impact on the natural environment. However, SBWs have rarely been employed for coastal protection in the past and therefore, their efficacy remains largely unknown. The main objective of the present study was to investigate the structural and environmental conditions that govern the mode of shoreline response (i.e shoreline erosion vs shoreline accretion) to SBWs. The relative importance of the key structural and environmental parameters governing the response mode to a single shore parallel SBW is investigated through a combination of theoretical analysis and numerical modelling. Using physical considerations, a theoretical response-function model is derived under several simplifying assumptions including parallel depth contours, linear wave theory, shore normal waves, and no wave–current interaction. Numerical modelling is undertaken with the Mike21 model suite to simulate the depth averaged velocity fields (without morphological updating) due to waves acting on a single shore-parallel SBW located on a schematised beach with parallel depth contours. In total 92 coupled wave–current simulations were undertaken. The results indicate that the mode of shoreline response to the SBW can be expressed in terms of the two non-dimensional parameters h_B/H₀ and (s_B/h_B)^3/2(L_B/h_B)²(A³/h_B)^1/2 (variables defined in the text).     

A Probabilistic Method for Motion Analysis of Caisson Breakwaters

1 .IntroductionInrecent years ,especiallyfrom1994 when aninternational conference on breakwater design washeldinJapan,muchresearchonthe dynamic analysis of caisson breakwaters has been done ,and manyimportant progresses have been made .Oumeraci and Korten…     

Performance of solid and perforated U-type breakwaters under regular and irregular waves

An experimental investigation of U-type breakwaters was carried out in a laboratory channel. Both regular and irregular waves were used during testing. Two types of breakwaters such as solid and perforated were studied to analyse the porosity effect of structures. In order to investigate performance of these breakwaters for different immersion depths, four depths of immersions of the solid and perforated breakwaters were selected. Different wave groups were generated over these breakwaters, and the transmission, reflection and energy dissipation characteristics of each breakwater were determined. Three coefficients such as transmission, reflection and energy dissipation coefficients, which were named as C_t, C_r, and C_l, respectively, were used during the evaluation of the test results. The most important parameters governing performance of these breakwaters were determined by using earlier investigations and experimental results. These parameters were expressed as a dimensionless group by using π theory. Based on the test results, empirical expressions were formulated to describe the C_t, C_r, and C_l for different immersion depths of solid and perforated breakwaters under regular and irregular waves.     

Prediction model for occurrence of impact wave force

This study investigates the applicability of neural networks to predict whether impact wave force will act on the upright section of a composite breakwater. We employ a three-layered neural network whose units of input layer are h/L, H/h, d/h and B_M/h (h: the total water depth; L: the wavelength; H: the wave height; d: the water depth above the mound; B_M: the horizontal distance from the shoulder of mound to the caisson). Teach signals are 0.99 and 0.01 according to the cases of occurrence and absence of impact wave force, respectively. The neural network whose parameters are determined through self-learning can accurately predict whether impact wave force occurs.     

Depth inversion in shallow water based on nonlinear properties of shoaling periodic waves

Two depth inversion algorithms (DIA) applicable to coastal waters are developed, calibrated, and validated based on results of computations of periodic waves shoaling over mild slopes, in a two-dimensional numerical wave tank based on fully nonlinear potential flow (FNPF) theory. In actual field situations, these algorithms would be used to predict the cross-shore depth variation h based on sets of values of wave celerity c and length L, and either wave height H or left–right asymmetry s₂/s₁, simultaneously measured at a number of locations in the direction of wave propagation, e.g., using video or radar remote sensing techniques. In these DIAs, an empirical relationship, calibrated for a series of computations in the numerical wave tank, is used to express c as a function of relative depth k_oh and deep water steepness k_oH_o. To carry out depth inversion, wave period is first predicted as the mean of observed L/c values, and H_o is then predicted, either based on observed H or s₂/s₁ values. The celerity relationship is finally inverted to predict depth h. The algorithms are validated by applying them to results of computations for cases with more complex bottom topography and different incident waves than in the original calibration computations. In all cases, root-mean-square (rms)-errors for the depth predictions are found to be less than a few percent, whereas depth predictions based on the linear dispersion relationship—which is still the basis for many state-of-the-art DIAs—have rms-errors 5 to 10 times larger.     

Sediment transport and beach morphodynamics induced by free long waves,bound long waves and wave groups

New laboratory data are presented on the influence of free long waves, bound long waves and wave groups on sediment transport in the surf and swash zones. As a result of the very significant difficulties in isolating and identifying the morphodynamic influences of long waves and wave groups in field conditions, a laboratory study was designed specifically to enable measurements of sediment transport that resolve these influences. The evolution of model sand beaches, each with the same initial plane slope, was measured for a range of wave conditions, firstly using monochromatic short waves. Subsequently, the monochromatic conditions were perturbed with free long waves and then substituted with bichromatic wave groups with the same mean energy flux. The beach profile changes and net cross-shore transport rates were extracted and compared for the different wave conditions, with and without long waves and wave groups. The experiments include a range of wave conditions, e.g. high-energy, moderate-energy, low-energy waves, which induce both spilling and plunging breakers and different turbulent intensities, and the beaches evolve to form classical accretive, erosive, and intermediate beach states. The data clearly demonstrate that free long waves influence surf zone morphodynamics and promote increased onshore sediment transport during accretive conditions and decreased offshore transport under erosive conditions. In contrast, wave groups, which can generate both forced and free long waves, generally reduce onshore transport during accretive conditions and increase offshore transport under erosive conditions. The influence of the free long waves and wave groups is consistent with the concept of the relative fall velocity, H/w_sT, as a dominant parameter controlling net beach erosion or accretion. Free long waves tend to reduce H/w_sT, promoting accretion, while wave groups tend to increase the effective H/w_sT, promoting erosion.     

Experimental study of wave–wave nonlinear interactions using the wavelet-based bicoherence

The wavelet-based bicoherence, which is a new and powerful tool in the analysis of nonlinear phase coupling, is used to study the nonlinear wave–wave interactions of breaking and non-breaking gravity waves propagating over a sill. Two cases of mechanically generated random waves based on Jonswap spectra are used for this purpose. Values of relative depth, k_ph (k_p is the wave number of the spectral peak and h is the water depth) for this study range between 0.38 and 1.22. The variations of wavelet-based total bicoherence for the test cases indicate that the degree of quadratic phase coupling increases in the shoaling region consistent with a wave profile that is pitched shoreward, relative to a vertical axis as seen in the experiments, but decreases in the de-shoaling region. For the non-breaking case, the degree of quadratic phase coupling continues to increase until waves reach the top of the sill. Breaking waves, however, achieve their highest level of quadratic phase coupling immediately before incipient breaking and the degree of phase coupling decreases sharply following breaking. In addition the wavelet-based bicoherence spectra provide evidence of the harmonics' growth which is reflected in the energy spectra. The bicoherence spectra also show that quadratic phase coupling between modes within the peak frequency as well as between modes of the peak frequency and its higher harmonics are dominant in the shoaling region, even though there are relatively high levels of quadratic phase coupling occurring between other frequencies. Furthermore, using the temporal resolution property of the wavelet-based bicoherence, we find that the quadratic wave interactions occur more readily during segments of time with large change of wave amplitude, rather than those segments having large wave amplitudes, but small gradients in amplitude.     

An extension of the Airy theory for linear waves into shallow water

As a fully developed (Airy) wave propagates from deep into shallow water, its crest becomes more peaked while the trough flattens out. The median crest diameter MCD, defined as the distance between the wave flanks under the crest at a level halfway between the crest and trough, therefore decreases relative to the similarly defined median trough diameter MTD, which remains constant up to the breaking point. The MCD is directly related to other wave characteristics, which enables water particle velocities to be calculated for any water depth without having to recur to more complex, higher-order Stokes, cnoidal or Fenton theories. Over a nearly horizontal bottom, most fully developed wave characteristics can be expressed as functions of the wave period T_w. It is shown that the horizontal particle velocity at the bottom under the breaker crest is at least 9 times faster than under the breaker trough, which explains why sediment is transported landward under fair weather conditions. The proposed equations also shed new light on the formation of spilling, plunging and surging/collapsing breakers.     

Equilibrium beach profile model for perched beaches

A beach profile equilibrium model for perched beaches is presented. The model assumes that wave reflection at the seaward and leeward sides of the breakwater is the most important process that modifies Dean's equilibrium profile model for non-perched beaches. The influence of wave breaking over the submerged structure is also discussed. Several laboratory data sets are used to analyze the merit of the proposed model for describing the equilibrium condition of a perched beach. A good comparison is obtained. Results show that if the ratio between the water depth above the submerged structure, d, and the water depth at the toe of the structure, h_e, is large, d/h_e>0.5, only minor advance of the shoreline is achieved with the construction of a toe structure. A considerable advance is obtained for d/h_e less that 0.1. In these situations, however, resonant effects may result in an inefficient structure. The proposed model is used to provide an estimation for the required sand volume and the associated beach advance for the case of narrow breakwaters.     

Unsteady air bubble entrainment and detrainment at a plunging breaker: dominant time scales and similarity of water level variations

At plunging breakers, air bubbles are entrained at the impingement of the water jet, formed at the top of the wave, with the water free surface in front. During the present study, air bubble entrainment at a pseudo-plunging breaker was investigated at near full-scale and further experimental work studied the bubble detrainment process. Experimental observations included the generation and propagation of waves downstream of the plunge point. Experimental results highlighted a number of unsteady air–water flow patterns and emphasise high levels of aeration: i.e., depth-averaged void fraction of more than 10% next to jet impact in shallow waters. Unsteady bubble injection experiments showed a strong vortical motion induced by the rising bubbles. Altogether, the results suggest that a dominant time scale is the bubble rise time d₁/u_r, which cannot be scaled properly with an undistorted Froude model. The study contributes to a better understanding of unsteady bubble entrainment at a pseudo-plunging breaker and the associated vortical circulation.