Shu-xiu Liang,Zhao-chen Sun,Yan-ling Chang,2,Ying Shi

1.State Key Laboratory of Coastal and Offshore Engineering,Dalian University of Technology,Dalian 116024,China

2. Richway (Beijing)Technology Co., LTD, Beijing 100097,China

Abstract:The wave parameters (the wave height and period)are important environmental factors in the ocean engineering design.General numerical wave models,such as SWAN and WAVEWATCH, are widely employed to simulate the wave conditions. However,significant differences wereobserved between the measurement and the simulation for both the wave height and period, which asksfor wave model improvements.The differences are mainly due to theuncertainty of parameterizing various physical processes,including the wavebreaking.Theenergy transfer and lossduring thewave breaking involves an important physical mechanism,and the energy dissipation and the period changesare not well studied.This paper studies thedeep and shallow water wave breaking using thewave focusing and the slope platform random wave experiments.The characteristics of the wave periods under different conditions are studied in detail,including the period variation.The results show that the periods change during the wave propagation and breaking processes.The energy transfer caused by the strongly nonlinear interaction between the wave components,as well as the energy loss caused by the wave breaking,are the primary causes.The corresponding relationships are established by fitting the data.For the deep water breaking wavesinduced by the wave focusing, the spectrally averaged period (SAP) increases, and a positive correlation between the rate of change and the wave steepnessisfound.In the shallow water,the nonlinear interactionsarestronger than in the deep water,the wave periods are significantly reduced,and a negativecorrelation between therate of changeand a nonlinear parameter is found.Theinherent mechanism of the period variation is analyzed based on the energy spectrum distribution variations.Thecontributionsof the nonlinear interactionsand the wavebreaking to the SAP evolution are discussed.

Key words:Wave period,wave breaking,nonlinear interaction,experiment

Introduction

The wave parameters,such as the wave height and period,are important environmental factors in the ocean engineering design,especially the wave height and average period[1].Currently,numerical wave models,the SWAN and the WAVEWATCH III(WWIII),are commonly used to simulate the wind wave conditions in a large area.To obtain a fine wave field in the coastal area,the mild slope and the Boussinesq equations are often employed.The wave period is treated as constant during the propagation despite of nonlinear interactions during the wave propagation.In the models based on the energy balance(WAM,WWIII,SWAN),the wave breaking is considered.Many studies[2-4]show that the wave period is affected by the nonlinear influence of the four-wave and three-wave interactions with an energy distribution,and due to this shortcoming,the wave refraction-diffraction models cannot be used to simulate the period changes.Although the SWAN model has been successfully applied to the evolution of wind waves in coastal regions,lakes,and estuarine waters,the simulated wave periods are generally less accurate than the predicted wave height.

The wave periods predicted by the SWAN or the WWIII are based on the wave spectrum,including

0,1,T0,2,T10,T-pT etc..They are usually underestimated[5-6].Tosolve the problem of the underestimation of thewave periods by the SWAN,Rogerset al.[5]proposed an adjusting parameter,δ ,ranging from 0 to 1,to provide an improved prediction of the wave energy at lower frequencies.The proposed modes of Rogers[5],Banner and Young[7]were applied numerically by using the SWAN model by Wang[8],who proposed a corresponding improvement,with a saturation based dissipation above the frequency twice the peak frequency to improve the white capping dissipation at low frequencies.However,these improvements were not based on the wave dynamic mechanism.Babanin research group[9-10]developed some new dissipation expressions based on the dynamics characteristics and the spectral distribution of breaking waves obtained by experiment.As is well known,the wave energy transfer and dissipation caused by wave breaking is extremely complex,and is the least understood part in the source terms of the numerical models.There are many fundamental experiments on the wave breaking to study the energy transfer and dissipation.Rapp and Melville[11],Tulin and Waseda[12]and Meza et al.[13]described the evolution of the wave trains in great details,as references for many other breaking experiments.The characteristics of energy dissipation,transfer and nonlinear interaction between wave components during wave breaking[14-21]are paid more attention recently.However,the characteristic-wave-period based on the wave spectrum redistribution after the wave breaking is not treated as an important issue in their papers.In view of the shortcomings of the third-generation ocean model in predicting the wave period discussed above,this paper experimentally studies the evolution characteristics of the wave period for deep and shallow water breaking waves,with the emphasis on the following three aspects:(1)The kind of period suitable for describing the period change of breaking wavesgenerated by wave focusing.(2)The variation characteristics,the influencing factors and the quantification of the wave period caused by typical wave breaking both in deep and shallow waters.(3)The mechanism of the wave period change.This study is to provide some insight for further experiments and improvements for the wave models.

1.Experimental methodsand setups

In experiments,the breaking waves in both deep and shallow waters are studied.The most likely wave breaking mechanisms to generate them are chosen,that is,the wave focusing to generate the deepwater breaking wave and the water depth shoaling to generate the shallow water breaking wave.In deep water breaking wave experiments,each frequency component of the incident wave in every case is made to satisfy the deep water condition,kh≥3.140,with k being the wave number and h being the water depth.In shallow water experiment,if the effective wave period is adopted,the waves develop from finite water depth waves to shallow water ones.It is impossible to fully meet the requirement of the shallow water wave of kh≤0.314 for all irregular wave components.The deep and shallow water breaking waves are generated based on different theoriesin different wave flumes.

1.1 Deep water breaking waves

1.1.1 Experimental method and parameters

In the laboratory flume,the wave maker is programmed to generate breaking waves in deep water using the wave energy focusing. This technique was proposed by Longuet-Higgins[14],and the breaking mechanism can be summarized as the fast propagation of long waves and the slow propagation of short waves.In the two-dimensional case with a given water depth,focusing position,and focusing time,the amplitude of each wave component can be expressed as

Table 1 Experimental deep water wave parameters

1.1.2 Experimental setups

The experiments are performed in a large wavecurrent flume at the State Key Laboratory of Coastal and Offshore Engineering,Dalian University of Technology.The dimension of the flume is 69.0 m×4.0 m×2.5 m (length×width×depth)and it is filled to a working depth of 1.5 m.The flume is computer controlled,and equipped with an irregular wave maker and a data acquisition system.To reduce the influence of the wave reflection,absorbing devicesare arranged at the end of the tank.

To record the wave surface height during the wave propagation,23 resistance wire wave gauges are arranged along the wave propagation direction of the wave tank,as shown in Fig.1.The center of the wave maker is located at =0 m x.The resistance wire wave gauges were developed by the Tianjin Institute of Hydraulic Research.The wave surface elevations controlled by computer. To ensure the measurement accuracy,strict calibrations are performed before the experiments.The wave gauges record the data at 100 Hz,with the acquisition interval of 0.01 s,and the acquisition time of 163.84 s.Each experiment takes 8 min-10 min to ensure that the tank water will return to a calm state.Each case is repeated three times,and the average of the measured data is used for subsequent analysis.

1.2 Shallow water breaking waves

1.2.1 Experimental setups

The shallow water experiments are performed in a marine environmental water flume at the State Key Laboratory of Coastal and Offshore Engineering,Dalian University of Technology.The dimension of the flume is 50 m×3 m×1 m (length×width×depth),and it is equipped with an irregular wave maker on one end and absorbing devicesat the other end.

Fig.1 Experimental design for deep water wave generation.Locations of the 23 wave probes along the deepwater flume with water depth of 1.5 m.Thedistance x(m)is from theequilibrium position of the wave maker.Breaking zoneisfrom 16.1 m to 21.55 m

Fig.2 Experimental design for shallow water wave generation locationsof the 28 wave probes and the 1:15 slope in the shallow water wave flume with water depth of 0.45 m.The distance x(m)is from the equilibrium position of the wave maker

Figure 2 shows the experimental terrain layout.The center of the wave maker is at x=0 m.A 1:15 slope is set at x=22.50 m-26.25 m.In the constant water depth region before the slope,the water depth is h=0.45 m,and at the top of the slope there is a shallow water area (with the water depth of 0.2 m,the length of 6 m).This terrain is designed for two purposes:to obtain the wave data after the wave breaking on the platform,and to reduce the reflection of waves on the platform.To record the wave surface height in the flume,28 resistance wire wave gauges along the wave propagation direction of the wave tank are used.Specific locations of wave gauges are shown in Fig.2.The first wave gauge is located at the place of x=14 m,used to measure the initial incident waves.

1.2.2 Experimental method and parameters

The water depth is h=0.45 m before the slope and d =0.20 m on the platform.The significant periods of irregular waves are set to 1.0 s,1.2 s,1.4 s,1.6 s,1.8 s, 2.0 sand 2.2 s.

Irregular waves are generated according to the JONSWAP spectrum with the spectral peak factor γ=3.3.For a given wave period,the input wave height is changed to generate non-breaking waves,by spilling to intense plunging on the platform.Wave gauges record the data at 100 Hz,with the acquisition time of 163.84 s.We also observe and record the wave breaking type and the numbers of the irregular waves in the surf zone,to compare with the wave gauge results.Each case is repeated three times,with the average being used for subsequent analysis.

2.Analysismethods

For the focusing wave with limited wave components,it is not certain which kind of wave period is suitable for describing its change during the wave breaking processes.The wave periods based on both time domain and frequency domain are analyzed.The fast Fourier transform (FFT)is employed to provide the spectrally-averaged wave period.Some characteristic wave parameters are considered as follows.

2.1 Characteristic wave period

The wave periods calculated from the spectrum can be calculated in a variety of ways.The spectrally averaged periods0,1T and0,2T are usually calculated in wavenumerical models,e.g.,the SWAN

(3)Peak period

The peak period,pT ,is an important parameter in the engineering design,which is defined as the corresponding period of the maximum spectral value of the wave spectrum.

2.2 Global wave steepness

wherena ,nf are the amplitude and the frequency of the th n component, respectively,calculated from the amplitude spectra,and ( )nfΔ represents the frequency difference between components of the wave group,which is assumed to be constant.To ensure stability, the surface elevation measured by the third wave gauges located at x=8.91m (approximately 3Lmax-4Lmax)downstream the mean position of the wave maker is used to compute S.Compared with the center or peak frequency,the spectrally weighted wave frequency is more representative of the dominant wave frequency.Thus,the wave steepness can better characterize the wave breaking intensity.

In calculating the focusing wave steepness and periods,the maximum wave height is taken as the center point and the intercept wave surface data length is taken to be 40.96 s.This data length ensures both the requirement of FFT,including all the wave components after some tests and avoids the effect of the reflected waves.The global wave steepness and the SAP are insensitive to the signal duration,provided that all non-zero surface elevations of the mechanically generated wave packet are included.

2.3 Wave nonlinear parameter

The wave nonlinearity may be enhanced in the wave propagation due to the wave-wave interaction or the water depth shoaling.Higher and steeper wave crests and shallower wave troughs will appear,along with significantly increased wave height and strengthened wave nonlinearity.The nonlinear parameter proposed by Goda[2]is adopted to describe the wave nonlinear characteristics in the shoaling region and period,calculated based on the wave gaugedata on the platform before thesurf zone.

3.Rusults

3.1 Wave period evolution for deep water breaking waves

Table 2 shows the input wave amplitude at the focal point A,the global wave steepness,S,and the experimental observations of wave breaking type for each case of waves in group D1(P-M spectrum,fc=1.019 Hz ,Δf / fc=0.46).As A increases,S increases,and the types of non-breaking and breaking wavesranging from gently spilling to intense plunging are observed.

Figure 3 shows that for the development of various wave periods during the wave focusing and breaking, the following two observations apply.

(1)Wave periods by zero-up-crossing

T and TH1/3vary in a rather unstable way before the wave focusing, then remain stable after the wave focusing or breaking,which has little effect on their values.The period instability,calculated by the zero-up crossing method,is mainly related to the small number of wave components.For a valid statistical analysis,more than 100 continuous wave components should be included in a standard wave series,but N =64 does not satisfy these statistical requirements.Thus,T and TH1/3are not chosen for further study.

(2)Spectrally averaged wave periods

In the following sections,only0,1T is employed as the SAP in view of the fact that0,2T might be contaminated greatly if high order wave components appear.Before the wave focusing or breaking,0,1T is stable and takes a same value under different wave steepness,and thewave period variation isirregular inthe wave focusing area (or the wave breaking zone).After the wave breaking(x ＞21m),T0,1remains stable too.Compared with non-breaking waves( S =0.277),the periods of breaking waves are larger,and the period increases with the increase of the wave breaking intensity.

Table 2 Experimental parameters and phenomena for deep water breaking waves

Fig.3 Evolution of wave periodsalong the flume for different wave steepness

(3)Peak period

The peak period TPremains almost unchanged during the wave focusing and breaking in all cases,which means that the main frequency energy remains the largest during the whole wave breaking process.

Figure 4 shows the variation of T0,2for the different spectral distributions(P-M, CWSand CWA).The groups D1-D3 correspond to the spectral distributions,respectively,for the center frequency fc=1.019 Hz and the relative frequency bandwidth Δf / fc=0.46.For the same wave steepness,Δ T0,2is the largest for the CWA distribution waves and the smallest for the P-M.For the plunging type wave breaking with the CWA,T0,2increases by approximately 7.0%after the wave breaking.

For the focusing waves with different spectral distributions,the energy loss in the breaking process is mainly from the high frequency part.The waves with the CWA energy distribution are mostly uniform.Therefore,the unstable high frequency energy occurs the most,and its loss is most significant.For the waves with a P-M spectrum,the high frequency energy occurs the least and the spectrum shape is the most stable along the flume.

3.2 Wave period evolution for shallow water breaking waves

Figure 5 shows the wave period evolution for irregular waves with the significant period T0=1.4 s and T0=2.0 s along the flume,where the icon in descending order corresponds to the non-breaking wave(Hs=0.067 m,T0=1.4 s ),the plunging wave (Hs=0.011m,T0=1.4 s),the non-breaking wave (Hs=0.065 m,T0=2.0 s),and the plunging wave (Hs=0.011m, T0=2.0 s).The surf zone is located at x =26.75 m-31.05 m.

(1)Wave periods by zero up-crossing

Fig.4 Frequency spectrum dependent changesof 0,2T

(2)Spectrally averaged wave periods

T0,1changes little along the flume for nonbreaking waves.However,as the incident wave height increases,the waves break on the platform.0,1T significantly decreases in the shoaling region before the wave breaking,as is consistent with Yu[1]for the effect of nonlinearity on the wave periods in the shallow water.The period decreases significantly with the increasing wave height.After the wave breaking,T0,1increases slightly,but remains less than the incident wave periods.Thus,for propagating waves,the effect of the strong nonlinear interactions on the SAPin the shoaling process are more significant than during thewave breaking process.

(3)Peak period

There is no significant change of the peak period Tpunder different wave heights as shown in Fig.5.By contrast,the change of Tpfor the waves with shorter period and larger height is more significant.

Figures 6 and 7 show the evolution characteristics of the significant wave period Tsand the SAP,T0,1,for different incident periods at a fixed relative wave height H / d =0.5.Irregular waves in each case break on the platform,the surf zone is located at x=27.00 m-31.25 m,and the plunging type wave breaking prevails.Figure 6 shows that the wave periods based on zero up-crossing are not affected by the nonlinear interaction when the water becomes shallower before the wave breaking.In the breaking process,with the increase of the incident wave period,one sees intensified wave deformation and attenuation.and Tsare significantly reduced for the incident waveswith longer periods(T0=1.8 s-2.2 s).

Figure 7 shows that the effect of the incident wave period on the change of the SAP is significant.As the incident wave period increases,T0,1decreases more significantly,e.g.,T0,1decreases by about 40.5%for Ts=2.2 s while it decreases by less than 10%for Ts=1.0 s . In addition,all the peak periods of waves with different incident periods increase due to the nonlinear interaction between the wave components before the wave breaking.The change is irregular after the wave breaking,but the rate of change is relatively small.

3.3 Quantitative analysis of period variation

The above analysis shows that the breaking intensity of deep water waves is related to the wave steepness.Therefore,the relationship between the period change rate and the incident wave steepness should be analyzed.For the breaking waves induced by the water depth,the SAP is greatly influenced by the nonlinearity.The relationship between the period variation and the wave nonlinear parameter,∏,can be established.

Fig.5 Evolution of wave periods

Fig.6 Evolution characteristics for the significant wave period at different incident periods

Fig.7 Evolution characteristics for spectrally averaged period,0,1 T ,at different incident periods

Fig.8 (Color online)Period change rate,0,1T ,and wave steepness,S ,for different frequency distributions

The wave steepness reflects the breaking intensity.With the increase of S,the focusing waves are in the range from the non-breaking type to the intense plunging type,and the rate of change of0,1T increases,with a positive correlation.The most rapid increase of the change rate comes from the CWA distributed waves,and the slowest one comes from the P-M distributed waves.The following relations are obtained by the least square fitting.

Figure 9 shows the relationship between the change rate of the SAP and the wave nonlinear parameter ∏ for irregular breaking waves in shallow water.For the calculation of the period change rate,an initial incident wave period,T0,is adopted to be the average measured values of the wave gauges 2 and 3,and the wave period after the wave breaking,T′,is adopted to be the averaging measured values of the wave gauges 26 and 27,on the end of the platform.The period change rate is then ΔT / T0=(T ′-T0)/T0.The wave nonlinear parameter,∏,is adopted to bethe measured values of the wave gauge18,before the surf zone.crease more quickly.Thecorresponding incident wave period is mostly in the range of 1.8 s-2.2 s.For the waves with a longer period (T0≥1.8 s),T and Tsare reduced significantly in the wave breaking process.With the increase of the incident wave height,∏increases,and the reduction of T and Tsbecomes more significant for the waves ranging from the non-breaking type to the intense plunging type. When ∏=0.6,T and Tsdecrease by 26.1%and 41.2%,respectively.

Fig.9 (Color online)The relationship between the change rate of period 0,1T and nonlinear parameter ∏

Fig.10 (Color online) Change rates for periods T ,sT and nonlinear parameter,∏ ,for different incident wave periods,0T

4.Discussions

Fig.11 Frequency spectra at selected flume locations

4.1 Mechanism of period variation for deep water waves

Table 3 Spectrum energy for different frequency bands and different wave heights

4.2 Mechanism of period variation for shallow water waves

Figure 12 shows the evolution of the spectral density for irregular waves with a dimensionless spectrum to allow a comparative analysis.The dimensionless frequency is defined as f*= f / fp,and the dimensionless spectral density is S*( f*)= S ( f ) fp/m0. In the shoaling area on the slope,the spectrum energy increases at the end of the slope.The energy transfer among the dominate frequencies is significant,which downshifts the spectrum peak.Simultaneously,a small amount of energy transfersto both the low and high frequency bands as a result of the nonlinear interaction.With the increase of the wave height,the nonlinear interaction becomes stronger and the high frequency energy increment increases.

Fig.12 Evolution of dimensionless spectral density

In the wave breaking process,there is a significant energy loss in the dominant frequency band,and more energy transfers to both the low and high frequency bands as compared with that during the shoaling.However,the proportion of S ( f )f and S ( f )f2for high frequency bands is larger than that for low frequency bands.The oppositetendency of the wave period variation by the dominant frequency energy loss and the high frequency energy gain makes the period variation more complicated,and there is no definite conclusion for the influence of the wave breaking against the waveheight.

From the variation of the wave period indifferent evolution processes,as shown in Fig.7,and the above discussion,we conclude that the period reduction caused by the shoaling is significantly greater than the period increase induced by the wave breaking.

5.Conclusions

The evolutionary characteristics of the wave periods of deep and shallow water breaking wavesare experimentally studied in this paper.The main conclusions are as follows:

(1)The wave periods by the zero up-crossing T and1/3HT are not suitable for describing the focusing waves.The spectrally averaged wave periods0,1T can reflect the energy transfer among parts of different frequencies during the wave propagation.pT remains almost unchanged in deepwater breaking waves generated by linear focusing,but with a notable change for shallow water breaking waves although it issmall.

(2)For deep-water breaking waves generated by linear focusing,the SAP increases after the wave breaking and a positive linear correlation is identified between the change rates of T0,1,and the wave steepness S.A critical value for S ≈0.270 is obtained,when S >0.270,the SAP T0,1increases after the wave breaking, and otherwise it decreases after the wave breaking.For the plunging type wave breaking with the CWA,themaximum0,1T change is measured,which increases by approximately 6.5%after the wave breaking.With the increase of the wave steepness,the increments of the wave breaking intensity and the SAPall increase.The corresponding relationship expressions are obtained through the least squaresfitting.

(3)In the shallow water,0,1T is significantly reduced as compared with the incident waves,and a negative correlation is identified between0,1T change rates and the nonlinear parameter,∏.The energy loss in the first harmonic frequency band is significant when the wave breaks,and the proportion of high frequency energy increases.With the increase of ∏,The wave nonlinearity strengthens,and T0,1decreases significantly.The corresponding relationship expressionsare obtained through the least squares fitting.The wave periods based on zero up-crossing are not affected by the nonlinear interaction when the water becomes shallower before the wave breaking when T0≤1.4 s .However,T and Tsare significantly reduced for incident wave periods T0≥1.8 s in the breaking process.

Thus,the wave periods change during the wave propagation and the breaking processes both in deep and shallow waters.The primary causesare theenergy transfer between the wave components caused by strong nonlinear interaction,and the energy loss caused by the wave breaking.The energy transfer caused by nonlinear interactions is more important for the change in the shallow water.To improve the simulation accuracy for wave periods of ocean wave numerical models,the nonlinear interaction term and the energy dissipation term need to be improved further.

Acknowledgement

The authors thank Dr.Yi Wang from National Marine Environmental Forecasting Center for some constructive suggestions.