Wave Breaking Analysis with Laboratory Tests

Transcription

1 1 Wave Breaking Analysis with Laboratory Tests André José Figueira Martins Instituto Superior Técnico Technical University of Lisbon, Av. Rovisco Pais 1, Lisbon, Portugal Abstract Wave breaking is a phenomenon characterized by energy dissipation, turbulence effects and air emulsion. The importance of studying this phenomenon is due to the effects it may have, as the process of wave breaking is both one of the most visually dramatic, and one of the most important physically for the wave motion and for the development of near shore currents. Since the knowledge of the processes involved in this phenomenon is still far from complete, the experimental results play an important role in their clarifying. This study began with a brief literature review on the current status of the subject and, at the practical level, the presented work describes a range of wave channel tests performed at the National Laboratory of Civil Engineering (LNEC), with the main objective of introducing an extensive analysis of the waves, especially the analysis of wave propagation in conditions prone to wave breaking. Therefore, this paper shows the experimental setup, the incident wave conditions and the measurements of the free surface elevation along the wave channel. Based upon the time series of the wave data measurements, a statistical time domain analysis, a standard Fourier based spectral analysis and a wavelet analysis was performed and presented. This study also aims to compare the results, acquired in these tests with predictions obtained from the application of empirical formulations relating to geometries similar to the structure under study. Thus, the analysis of similar cases to the case study will contribute to a better understanding of these empirical formulations, specially their range of application, and represent a move towards the systematization of the knowledge we can gain using a combination of experimental results and simple theoretical approximations. Key-Words: Waves, Breaking, Trials, Time series analysis, Spectral analysis. 1. Introduction The determination of the wave breaking zone is essential for studies referring coastal hydrodynamics and sediment transport issues. Since wave breaking is a nonlinear complex phenomenon that occurs with different scales, research in this area, specifically the location and extension of wave breaking are two of the main factors for these studies, since they determine the coastal structures location and stability of the subsequent sediment transport. In order to obtain data of the free surface elevation and velocity field measured along the channel (Neves et al., 2011) conducted a series of tests in LNEC s waves channel, for 15 incident wave condition. The tests were conducted under the Project BRISA - Breaking waves and Induced Sand transport, financed by the Fundação para a Ciência e Tecnologia (PTDC/ECM/67411/2006 contract). The project s main objective is to contribute to the understanding and numerical modeling of the wave breaking phenomena and sediment transport in coastal areas. The bottom profile consisted in a series of ramps, with variable slopes, in which a depth of 10 cm water column was set on top of the second ramp, as a way of inducing wave breaking in that area. A great number of experimental data was obtained, and whose treatment was done by using classical analyses in time and frequency domains. 2. Main subjects 2.1. Time Domain Parameters The wave period is the time distance between two consecutive down crossings (or up crossings), whereas the wave height is the vertical distance from a trough to the next crest as it appears on the wave record. Another and more commonly used kind Figure 1 Sample of a wave record. of wave height is the zero-crossing wave height, being the vertical distance between the highest and the lowest value of the wave record between two zero-down crossings (or up crossings). A typical wave record is shown in Figure 1. When the wave record contains a great variety of wave periods, the number of crests becomes greater than the number of zero-down crossings. In that case, there will be some difference between the crest-to-trough wave height and. In this chapter, however, this difference will be neglected and will be used implicitly.

2 2 A measured wave record never repeats itself exactly, due to the random appearance of the sea surface. But if the sea state is stationary, the statistical properties of the distribution of periods and heights will be similar from one record to another. The most appropriate parameters to describe the sea state from a measured wave record are therefore statistical (Laing et al., 1998). The following are frequently used: Average wave height; Maximum wave height occurring in a record; Average zero-crossing wave period; the time obtained by dividing the record length by the number of down crossings (or up crossings) in the record; The average height of the 1/n highest waves (i.e. if all wave heights measured from the record are arranged in descending order, the one-nth part, containing the highest waves, should be taken and is then computed as the average height of this part); The average period of the 1/n highest waves. A commonly used value for n is 3; Significant wave height (its value roughly approximates to visually observed wave height); Significant wave period (approximately equal to the wave period associated with the spectral maximum) Spectral Parameters A wave spectrum is the distribution of wave energy (or variance of the sea surface) over frequency (or wavelength or frequency and direction, etc.). Thus, as a statistical distribution, many of the parameters derived from the spectrum parallel similar parameters from any statistical distribution. Hence, the form of a wave spectrum is usually expressed in terms of the moments of the distribution (spectrum). The nth-order moment, spectrum is defined by: In this formula,, of the ( ) (1.1) ( ) denotes the variance density at frequency,, as in Figure 2, so that ( ) represents the variance contained in the ith interval between and. In practice, the integration in Equation 1.1 is approximated by a finite sum, with = : it follows that the moment of zeroorder, From the definition of form this is:, represents the area under the spectral curve. In finite which is the total variance of the wave record obtained by the sum of the variances of the individual spectral components. The area under the spectral curve therefore has a physical meaning which is used in practical applications for the definition of wave-height parameters derived from the spectrum. Recalling that for a simple wave the wave energy (per unit area),, was related to the wave height by: then, if one replaces the actual sea state by a single sinusoidal wave having the same energy, its equivalent height would be given by: the so-called root-mean-square wave height. total energy (per unit area) of the sea state. now represents the We would like a parameter derived from the spectrum and corresponding as closely as possible to the significant wave height (as derived directly from the wave record). It has been shown that should be multiplied by the factor in order to arrive at the required value. Thus, the spectral wave height parameter commonly used can be calculated from the measured area,, under the spectral curve as follows: state ( Note that we sometimes refer to the total variance of the sea ) as the total energy, but we must be mindful here that the total energy is really. In theory, the correspondence between and is valid only for very narrow spectra which do not occur often in nature. However, the difference is relatively small in most cases, with =5. on average. The significant wave height is also frequently denoted by. In that case, it must be indicated which quantity (4 or ) is being used (Laing et al., 1998). Wave spectra systems typically have a form like that shown in Figure 2. The derivation of parameters for wave period is a more complicated matter, owing to the great variety of spectral shapes related to various combinations of sea and swell. There is some similarity with the problem about defining a wave period from statistical analysis. Spectral wave frequency and wave period parameters commonly used are: Wave frequency corresponding to the peak of the spectrum (modal or peak frequency); Figure 2 - Typical wave-variance spectrum for a single system of wind waves. By transformation of the vertical axis into units of ( ), a wave energy ( ) spectrum is obtained. Wave period corresponding to, i.e., = ;

3 3 spectrum: Wave period corresponding to the mean frequency of the Wave period theoretically equivalent with mean zero-down crossing period : Note that the wave period is sensitive to the high frequency cut-off in the integration (Equation 1.1) which is used in practice. Therefore this cut-off should be noted when presenting and, in particular, when comparing different data sets. For buoy data, the cut-off frequency is typically 0.5 Hz as most buoys do not accurately measure the wave spectrum above this frequency. (Goda, 1978) has shown that, for a variety of cases, average wave periods of the higher waves in a record, e.g., remain within a range of to Wave breaking The surf zone is the name of shallow water areas where waves break, for example on a beach. The wave-breaking is associated with a conversion of the energy from ordered wave energy to turbulence and to heat. The surf zone is the area with the most intense sediment transport because of the high intensity of the turbulence and the shallow water which makes agitation of sediment from the bottom easy. As waves propagate into shallower water, the process of shoaling leads to increasing wave heights. This process cannot continue, and at a certain location the wave breaks. The wavebreaking will typically take place when the wave height is about 0.8 times the local water depth. The waves break because their steepness becomes very large as the depth becomes shallower. The forward wave orbital velocity at the crest becomes large, and the crest topples because it is unstable. While the shoaling process is characterized by a very small energy loss, the wave breaking is associated with a very large loss of wave energy. The surf zone along the beach is where the wave energy flux from offshore is dissipated to turbulence and heat. Due to the strong energy dissipation, the wave height decreases towards the shore in the surf zone. The breaking waves can be divided into several different types, the three most important are (Fredsøe & Deigaard, 1992): Spilling breakers; Plunging breakers; Surging breakers. where Figure 3 presents the comparison of the 3 breaking waves types. Spilling breakers are characterized by the forward slope of the wave top becoming unstable. A plume of water and air bubbles slides down the slope from the crest. The volume of the plume increases, and it travels with the wave as a surface roller. Spilling breakers are also found among waves in deep water, where their energy dissipation is an important part of the energy budget for wind generated waves. For a plunging breaker the crest of the wave moves forward and falls down at the trough in front of it as a single structured mass of water or a jet. The impact of the jet generates a splash-up of water which continues the breaking process and creates large coherent vortices, which can reach the bottom and stir up considerable amounts of sediment. The flow caused by entrained air further spreads the sediment over the vertical, and clouds of suspended sediment are often observed at a location of plunging breakers. In surging breakers it is not the crest of the wave that becomes unstable. It is the foot of the steep front that rushes forward, causing the wave crest to decrease and disappear. The occurrence of the different types of breakers depends on the character of the incoming waxes and of the beach. The most important factor is the slope of the beach and the steepness of the incoming waves. Spilling breakers occur at very gentle beach slopes and relatively steep incoming waves, while plunging breakers are found for steeper bed slopes and less steep waves. Surging breakers are found on very steep beaches. (Galvin, 1968) found a relationship between the wave geometry and the breaker type. The waves can be characterized by the surf similarity parameter (Battjes, 1974), which is the ratio between the beach slope and the square root of the wave steepness. The wave steepness can be calculated from the deep water wave height or the wave height at breaking wave length is used in the expressions for : where. In both cases the deep water is the beach slope. Table 1 shows the domains from Galvin s experimental data. Figure 3 Comparison between the main types of wave breaking and four distinct moments in their evolution (Derived from Dally et al., 1985).

4 4 Table 1 Breaker type criteria. Zone Breaker type Deep water Breaking point Spilling < 0.5 < 0.4 Plunging 0.5 < < < < 2.0 Surging > 3.3 > Experimental settings The wave tank experiments were conducted in a wave channel with 32.4 m length and m width. A beach profile, with different bottom slopes, was constructed as shown in Figure 5. The slope angle of the front face of the bar and the beach section was fixed with 1:20 and the slope of the lee side of the bar was inexistent. Water depth was measured to be 0.1 m at the crest of the 1:20 bar. Figure 4 presents a plant of the wave channel. Figure 4 Wave channel plant and positions along the longitudinal (x) axis (derived from Conde, 2012). Along the wave channel, several equipments were installed: - The measurement equipment (wave resistive gauges and an Acoustic Doppler Velocimeter [ADV]) to measure free surface elevations and particle velocities; - The National Instruments acquisition system to forward the wave generation signal to the wave maker ( piston type); - A computer named CPU2 to connect and customize the ADV sensor. wave maker by the National Instruments acquisition system. For the data acquisition, the sensors signals elapsed through the Wave Probe Monitor in order to convert the analogical signal in a digital format. Therefore the signal was recorded at the CPU1 computer Incident wave characteristics and experimental tests A piston-type wave maker generated a combination of regular waves combining four wave periods ( =1.1, 1.5, 2.0 and 2.5 s) with four wave heights ( =12, 14, 16, and 18 cm). The wave with =1.1 s and =18 cm presented a very steep wave that broke in front of the wave maker, therefore, this incident wave condition was excluded from the experimental tests. Thus, in total, for each position along the canal, fifteen combinations of waves were tested, as indicated in Table 2. Table 2 Incident wave conditions. The total set of experiments was divided in two phases: I. Phase I the objective of the first phase was to measure II. [cm] [s] x x x x 14 x x x x 16 x x x x 18 - x x x free surface elevations along the channel with an 8 wave gauges mobile structure; Phase II the objective of the second phase was to measure the particle velocity along the channel in the middle of the water column, using the ADV sensor. At the same time, a resistive gauge was located near the ADV sensor to measure the free surface elevation. This work focused on the Phase I experiments results Equipment and experimental procedures Figure 5 Wave channel profile and positions along the longitudinal (x) axis (Derived from Conde, 2012). At the monitoring office, the computer named CPU1 was installed to generate the incident wave signal and to acquire and record the data from both the wave resistive gauges and the ADV sensor. The CPU1 computer received the sensor signals through intermediate devices, like the Wave Probe Monitor and the National Instruments acquisition system. The connection between the computer CPU1 and the wave maker was made through the Signal Express software generating the signal to be sent to the In Phase I, a mobile structure with eight wave gauges was placed along the channel to measure the free surface elevations. The mobile structure provided an easy transport and allocation of the wave gauges along the channel. The covered length of the wave channel was from the beginning of the first ramp (x=-1000 cm) till x=560 cm when the wave breaking is shown to be completely over for every incident wave condition. To calibrate the input wave height, and since the bottom profile was not flat, a wave gauge, named AØ, was installed at the toe of the front face of the first slope (x=-1080 cm). Each gauge in the mobile structure was separated by a fixed distance (20 cm) and measurements separated by 10 cm were taken along the covered area. The sampling frequency of AØ was 25 Hz, whereas the other eight gauges were 100 Hz. It is important to note that some positions (only two) are repeated due to limitations of the channel, which contain a set of

5 Hs (cm) 5 transverse metal bars that sometimes prevent the placement of the eight gauges structure. Each experimental test (incident wave) had the duration of 490 s. 4. Analysis of results In this chapter, we will analyze the results obtained in the incident wave tests. The wave breaking section will be the basis for comparisons with some previous studies made by other authors. point, The main parameter to define the wave height in the breaking, was to consider the maximum wave height registered for each wave incident condition. Using a similar reasoning, the water depth at breaking point,, was considered to be the minimum depth in which the maximum wave height was measured. For each of the 15 incident wave conditions, the time series of the free surface elevation was obtained in the various positions, from x=-1000 cm to x=560 cm. Based on those data, different types of data analysis were considered, but as in this study only the Phase I experiments were relevant, the results obtained were mainly focused on time, spectral, and statistical analysis of the free surface elevations. Also the relative wave height ( ) along the surf zone was calculated for each incident wave condition. In the next sections, the experimental location values for the beginning and ending of the wave breaking zone are presented. Then some samples are presented of the obtained data and the performed analysis, with varying incident wave conditions for each presented sample Time, breaking, statistical and spectral analysis (i.a) Fourier, with Sam Mod 7 software (Capitão, 2002); (i.b) Wavelet, with Matlab software (Mori, 2009). - Breaking analysis: (i) Relative wave height; (ii) Wave breaking type; (iii) Limiting breaker height; (iv) Wave transformation on the internal part of surf zone Wave breaking zone The wave breaking section was defined for each one of the 15 incident waves. Table 3 presents the values of the initial and final location of the wave breaking section for all the incident waves in the tests described above. Table 3 Breaking waves locations, wave height and water depth at breaking point for the 15 incident waves. Breaking positions (x) Incident Initial Final wave [cm] [cm] [cm] [cm] T11H T11H T11H T15H T15H T15H T15H T20H T20H T20H T20H T25H T25H T25H T25H Time domain analysis Based on the free surface records along the channel, the following parameters were calculated: Significant Wave Height (Constant Period T=2.0 s) - Time analysis of the free surface elevation values: (i) (Maximum wave height); (ii) (Significant wave height); (iii) (Average wave height); (iv) (Significant wave period); (v) (Average wave period), based on the up-crossing method; - Statistical analysis of the free surface elevation values: (i) Average; (ii) Standard deviation; (iii) Skewness; (iv) Kurtosis. - Spectral analysis of the free surface elevation values: (i) Spectral density variance was calculated in order to define the energy distribution in the frequency spectrum, with help of two different sets of analysis: Figure 6 Significant wave height throughout the channel, for incident waves with 2.0 s period. Figure 6 shows the significant wave heights, for incident waves with =2.0 s. We can see an initial decrease of the followed by a significant increase throughout the section close to x=-700 cm due to the shoaling effect. Some oscillations on the wave height values are also observed. Subsequently, the wave breaking effect produces a significant reduction in the wave height preceded by a section with a constant value for the wave heights at the lee side of the channel due to the ending of the wave breaking. H12 H14 H16 H18

6 Hs/h (-) H/h (-) Tm (s) Figure 7 Average period throughout the channel, for incident waves with 16 cm wave height. Figure 7 shows the average wave periods along the channel, for incident waves with =16 cm. Tests show that the wave period remains approximately constant throughout the channel, only to be disturbed at the wave breaking zone. In this last section we observe major changes in the wave period, particularly the ones with bigger periods Relative wave height The relative wave height ( ) is often used as an index for the wave breaking section in shallow water conditions. Dally et al. (1985) considers this index as reference for a stabilized wave condition. Unlike the beginning of the wave breaking, there is no standard or commonly used value for the end of the wave breaking section. (Dally et al., 1985), indicates multiple values ( =0.35 to 0.47) for different bottom slopes in order to get a curve that fits best the experimental results. It is expected that at the end of the surf section the relative wave height reveals values close to =0.35 to 0.40, which is the value for an horizontal bottom case Figure 8 Relative have height throughout the channel, for incident waves with 12 cm wave height. Figure 8 shows, for the 4 incident wave conditions with =12 cm, the relative wave height evolution from the top of the bar until the end of the surf section. It can be concluded that the behavior of has two phases. Before the wave breaking zone the curve is steeper, increasing rapidly, while after the wave breaking, there is a decrease of and the slope presents a smoother curve. After breaking, this value steadily decreases, with the exception of the x=0 cm area (bottom slope changing zone), where it momentarily increases. After that, it decreases until it remains almost constant in the end of the breaking zone. For these tested incident waves, the rate of Average Period (Constant Wave Height H=16 cm) Relative Wave Height (H=12 cm) at the end of the wave breaking section is between T1.1 T1.5 T2.0 T2.5 T11H12 T15H12 T20H12 T25H and 0, which can be considered a bit high for the values we were comparing. Another comparison was made with data from another study (Fredsøe & Deigaard, 1992). After a wave has broken as a spilling or plunging breaker, a transition occurs. For the spilling breaker the surface roller grows and the wave height decreases rapidly. For the plunging breaker the jet of water that plunges down pushes up a very turbulent mass of water which continues the wavebreaking process. In both cases the wave is transformed into a bore-like broken wave, and this inner part of the surf zone can be described as a series of periodic bores (Svendsen et al., 1978). The ratio between the local wave height and mean water depth decreases from the value of about 0.8 at the point of wave-breaking to become almost constant at about 0.5 in the inner zone. Figure 9 shows the experimental results together with the empirical relation by (Andersen & Fredsøe, 1983): Relative Wave Height (T15H16) after breaking, with constant bottom slope Δx/h b (-) Figure 9 Comparison between empirical expression and experimental data for incident wave with 1.5 s period and 16 cm wave height. The obtained results show that, in spite of a certain degree of dispersion of the acquired data, it clearly shows a similar tendency to the theoretical expression, represented by the orange line. Also, it is important to note that the blue data dots are comprehended between the breaking zone point and the end of the 1:20 bottom slope. The subsequent points, in the horizontal slope area, were not used in this analysis Breaking analysis Table 4 Wave breaking type for the fifteen different incident waves. Incident wave Breaking [-] [m] [m] [-] [-] type T11H Spilling T11H Spilling T11H Spilling T15H Spilling T15H Spilling T15H Spilling T15H Spilling T20H Spilling T20H Spilling T20H Spilling T20H Spilling T25H Spilling T25H Spilling T25H Spilling T25H Spilling

7 H/H b (-) H b /h b (-) H b /h b (-) 7 As expected, all the incident waves had a surf similarity parameter inside the range of spilling wave breaking, essentially characterized by very low slopes, as was the case of this particular wave channel (1:20). According to an author (Goda, 1985), we can easily observe that a train of regular waves in a laboratory flume undergoes shoaling over a sloping bottom and breaks at a certain depth. The location at which waves break is almost fixed for regular waves, and there is a distinct difference between the oscillatory wave motion before breaking and the turbulent wakes with air entrainment after breaking. The terminology wave breaking point, depth and height is employed do denote the location, water depth, and height of wave breaking, respectively. The expression limiting breaker height is sometimes also used, in the sense of the upper limit of progressive waves physically possible at a certain water depth for a given wave period. The ratio of limiting breaker height to water depth depends on the bottom slope and the relative water depth. Compilation of a number of laboratory results has yielded the design diagram of Figure 10 as an average relation, although a scatter in the data of more than 10% must be mentioned. Using our experimental results and putting them together with Goda s data, we reached the results presented in Figure 10. Relative breking height vs h b /L T T1.5 T T h b /L 0 (-) base. This data was compared to our own values and Figure 11 shows the comparison. Figure 11 Relative breaking height (Derived from Corps of Engineers, 2003). Again we have a tendency pretty much the same as the wave channel slope (1:20, i.e., =5). However, to a certain extent, we still have a discrepancy in the larger periods, in this case, bigger or equal to 2.0 s. Wanting to make an analysis of the wave transformations occurring after the breaking point, (Horikawa & Kuo, 1966) made a stud with the objective of presenting an approach to this problem, having analytical and experimental treatments as a backup. Having some results for the same slope was we had, 1:20, we tried to see the differences and similarities, as illustrated in Figure Relative breking height vs H b /g.t H b /(g.t 2 ) (-) Normalized wave heigh vs normalized depth T1.1 T1.5 T2.0 T2.5 Figure 10 Relative breaking height (Derived from Goda, 1985). The obtained results say that, in relative terms, our data had a similar arrangement to the author s results, in the sense that, for bigger periods and respective wave heights, it exist a curve with an approximate tendency to his values. However, only for the incident waves with the smallest period, 1.1 s, subsists a great synchronism with the existing slope (1:20). The other incident waves, with bigger periods, had the tendency to move away h/h b (-) T2.0H12 T2.0H14 T2.0H16 T2.0H18 Weggel (1972) presented a reevaluation of some breaking waves studies, in a way to establish some kind of norms for monochromatic waves, with gentle slopes. His studies implied a great number of theoretical and experimental data, therefore having a great range of results which, by not being exactly the same, enabled a broader vision of the phenomenon and of the parameters he wished to find. Later on, a revision of is results was made (Corps of Engineers, 2003), having Weggel s studies as a Figure 12 Normalized wave height and depth, for a bottom slope of 1:20 (Derived from Horikawa & Kuo, 1966). For this analysis, we had a slight difference in the wave periods, 2.2 versus 2.0 s, which had to be count on in the beginning. Despite this, it is visible that the data had a very good correlation with the author s data. Being a graphic that evolves form the right to the left, we can see in the first phase a concavity leaning down, to a concavity leaning up, with an inflexion point in the middle of them, a kind of transition area.

8 Average (cm) Kurtosis (-) Skewness (-) H/h (-) Standard deviation (cm) 8 Relative wave heigth vs normalized depth 5.0 Standard deviation (Incident wave T=1.5 s e H=18 cm) T2.0H 12 T2.0H 14 T2.0H Figure 15 - Standard deviation for an incident wave with 1.5 s period and 18 cm wave height h/h b (-) Figure 13 Relative wave height, for a bottom slope of 1:20 (Derived from Horikawa & Kuo, 1966). In the case of the analysis in Figure 13, we had contradictory data against the author s results. It is clear our data has a tendency to descend, instead of going up. The great difference lies is in the vertical axis. While in the horizontal axis we have the normalized depth, which necessarily must come down as we move forward in the channel, with lower depths,, in the vertical axis our values have a tendency to decrease. This is because the wave height decreases at a bigger rate than the local depth Statistical analysis The performed statistical analysis of the free surface elevation records corresponds to the average, standard deviation, skewness and kurtosis. The next figures present the evolution throughout the channel of the above mentioned parameters for an incident regular wave of =1.5 s and =18 cm. Average (Incident wave T=1.5 s e H=18 cm) Figure 14 Average for an incident wave with 1.5 s period and 18 cm wave height. Figure 14 presents the average values of the free surface elevation related to the Mean Water Level (MWL). A slight setdown is observed between the x=-900 cm and x=-800 cm, before increasing steadily until the wave breaking begins. After the breaking point, the average free surface starts to decrease until x=200 cm, where it starts to stabilize. Around x=400 cm the average free surface begins to increase. The standard deviation (Figure 15) essentially reproduces the same as the wave significant heights: there is slightly increase related with the depth decreasing (shoaling effect) until x=-470 cm, more or less. During the wave breaking section, the standard deviation decreases significantly until right around x=100 cm, where the standard deviation is almost constant Figure 16 Skewness for an incident wave with 1.5 s period and 18 cm wave height. Figure 17 Kurtosis for an incident wave with 1.5 s period and 18 cm wave height. In Figures 16 and 17, skewness and kurtosis values behave similarly. Initially, both present small values and begin to increase until the wave breaking point. During the surf section, the skewness and the kurtosis have great variability. After that, they begin to rise significantly (approximately at x=0 cm) and fall again returning to values closer to the initial range (at x=400 cm), most probably due to the end of the generated turbulence of the wave breaking process. Skewness (Incident wave T=1.5 s e H=18 cm) Kurtosis (Incident wave T=1.5 s e H=18 cm)

9 Spectral analysis The next set of Figures, ranging from Fig. 18 to Fig. 25, shows some of the calculated spectra for the incident wave condition of =2.5 s and =18 cm, at the positions, x=-1000 cm, x=-500 cm, x=0 cm, and x=400 cm. They were the selected positions because they represent 4 distinct moments in the wave evolution: (i) generated wave with few interferences, (ii) wave before breaking zone, (iii) wave after breaking zone and (iv) wave outside of breaking zone, near to the end of its cycle. It is important to note an important fact: to ensure that the wavelet transforms at each scale are directly comparable to each other and to the transforms of other time series, i.e., Fourier, the wavelet function at each scale is normalized to have unit energy. To make it easier for the wavelet spectra comparison, it is desirable to find a common normalization. For white noise time series, the expected value of the wavelet transform is the variance, i.e., (Torrence & Compo, 1998). Figure 21 Fourier analysis of incident wave with 2.5 s period and 18 cm wave height. Position x=-500 cm. Figure 22 Wavelet analysis of incident wave with 2.5 s period and 18 cm wave height. Position x=-150 cm. Figure 23 Fourier analysis of incident wave with 2.5 s period and 18 cm wave height. Position x=-150 cm. Figure 18 Wavelet analysis of incident wave with 2.5 s period and 18 cm wave height. Position x=-1000 cm. Figure 19 Fourier analysis of incident wave with 2.5 s period and 18 cm wave height. Position x=-1000 cm. Figure 204 Wavelet analysis of incident wave with 2.5 s period and 18 cm wave height. Position x=400 cm. Figure 25 Fourier analysis of incident wave with 2.5 s period and 18 cm wave height. Position x=400 cm. Figure 20 Wavelet analysis of incident wave with 2.5 s period and 18 cm wave height. Position x=-500 cm. The results for the spectral analysis show that as the wave propagates along the channel, there is an increasingly number of harmonics and a strong reduction in the amplitude of the main frequency. Comparatively, it is possible to observe that after the wave breaking, the spectrum energy of the main frequency decreases to the point of being more similar to the other generated frequencies.

10 10 5. Conclusions In this paper, recent physical modeling tests on a wave channel from the National Laboratory of Civil Engineering in Lisbon (LNEC), Lisbon, Portugal, were presented. The tests aimed mainly to introduce a new analysis based on the directional spread of the wave on a wave channel to study the wave breaking hydrodynamics on complex bathymetries, namely bathymetries with variable bottom slopes. This work represented a step forward to understand and better define the wave breaking process throughout the surf section since the beginning till the very end, considering incident wave conditions. The physical modelling was performed on a wave channel, built for wave propagation studies, with a bottom characterized by a ramp with 1:20 and then by a zone with horizontal slope, respectively. The tested waves resulted from a combination of 1.1 s, 1.5 s, 2.0 s and 2.5 s periods with wave heights of 12 cm, 14 cm, 16 cm and 18 cm. The measured data from the resistive gauges (free surface elevation) enabled a time, spectral and statistical analysis and the calculation of several parameters, such the relative wave height. From the physical model (wave channel), a wide set of wave data (free surface elevation and particle velocity) along the channel and especially in the wave breaking section is available. Along the channel, as the wave begins to break, the observed parameters show a considerable dispersion. This aspect is related with major values of turbulence intensity and air emulsion of the wave breaking phenomena. Thus, any problems in positioning the resistive gauges and the occurrence of air bubbles in the water appear as important parcels, stacked to the results. These problems mostly occurred in the wave breaking region, where the resistive wave gauges, were eventually, briefly out of the water. However, the results show that the measured data present enough quality to draw relevant conclusions for the proposed study. Future work, concerning a more complete analysis of the breaking criteria here presented and the comparison with other parameters for the wave breaking definition will be a sure plus. A further analysis on the wave breaking physical process is also needed. If possible with irregular incident waves, in order to have also more interesting spectral analysis. References Andersen, O., Fredsøe, J., Transport of Suspended Sediment Along the Coast. Progress Report Nº 59, Institute of Hydrodynamics and Hydraulic Engineering, ISVA, Technical University of Denmark, Copenhagen, Denmark. pp Battjes, J., Computation of Set-up, Longshore currents, Runup and Over-overtopping due to Wind-generated Waves. Report Nº 74-2, Department of Civil Engineering, Delft University of Technology, Delft, Netherlands. 241 pp. Capitão, R., Sea Wave Stochastic Numerical and Physical Modeling. PhD Thesis em Engenharia Civil. Instituto Superior Técnico, Lisbon, Portugal. 434 pp. Conde, J., Metodologia de Uso do Canal de Ondas Irregulares COI3. Laboratório Nacional de Engenharia Civil, Lisbon, Portugal. 47 pp. Corps of Engineers, U.S. Army, Coastal Engineering Manual. Chapter II-4 - Surf Zone Hydrodynamics. U.S. Government Printing Office, Washington D.C., United States of America. pp Dally, W.R., Dean, R.G., Dalrymple, R.A., Wave Height Variation Across Beaches of Arbitrary Profile. Journal og Geophysical Research, Volume 90, Nº C6. pp Fredsøe, J., Deigaard, R., Mechanics of Coastal Sediment Transport. World Scientific, Singapore. 369 pp. Galvin, C., Breaker Type Classification on Three Laboratory Beaches. Journal of Geophysical Research, Volume 73 (12). pp Goda, Y., The Observed Joint Distribution of Periods and Heights of Sea Waves. Proceedings 16 th Coastal Engineering Conference, Chapter 11, Hamburg, Germany. pp Goda, Y., Random Seas and Design of Maritime Structures. World Scientific, Singapore. 443 pp. Horikawa, K., Kuo, C., A Study on Wave Transformation Inside Surf Zone. Proceedings 10 th Coastal Engineering Conference, Chapter 15, Tokyo, Japan. pp Laing, A., Gemmill, W., Magnusson, A.K., Burroughs, L., Reistad, M., Khandekar, M., Holthuijsen, L., Ewing, J.A., Carter, D.J.T., Guide to Wave Analysis and Forecasting. World Meteorological Organization, Geneva, Switzerland. 159 pp. Mori, N., MACE Toolbox. Consulted in June 2012 on website: Neves, D.R.C.B., Endres, L.A.M., Fortes, C.J.E.M., Okamoto, T., Rebentação das Ondas. Análise e Tratamento de Dados Obtidos em Ensaios em Modelo Físico. Relatório BRISA. Laboratório Nacional de Engenharia Civil, Lisbon, Portugal. 235 pp. Svendsen, I.A., Madsen, P., Hansen, J.B., Wave Characteristics In the Surf Zone. Proceedings 16 th Coastal Engineering Conference, Chapter 29, Hamburg, Germany. pp Torrence, C., Compo, G.P., A Practical Guide to Wavelet Analysis. Bulletin of the American Meteorological Society, Volume 79, January. pp Weggel, J.R., Maximum Breaker Height. Journal of the Waterways, Harbours and Coastal Enginnering Division, Volume 98, Nº 4, November. pp