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

Single Gaussian wave groups with different initial wave steepness ε0 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 ε0 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 f1 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.


Introduction
Extreme waves appear frequently in the oceans and seas, and some extreme waves are called freak waves or abnormal waves. In ocean engineering, the extreme waves are dangerous for ships and platforms in the seas, which can induce ship accidents, human lives loss and catastrophe. The study for extreme wave is of great importance in deciding the criterion of the building in ocean (Kharif and Pelinovsky, 2003).
Extreme wave is the maximal wave locally, it appears by accompanying a group of several waves usually (Donelan et al., 1972). At present, the research of extreme wave in single wave group focuses on the studies referring to dispersion focus mainly (Rapp and Melville, 1990;Wu and Yao, 2004;Ma et al., 2015;Liang et al., 2017), however, studies related to other modes of single wave group are less relatively, yet studying the evolution process of a single wave group is one of the effective ways to reveal the mechanism for freak generation, and is also the requirement to forecast the freak wave and other giant waves.
Over the past years, much effort has been made on the evolution of wave groups. Many results have been obtained by experimental and numerical ways. A number of numerical simulations have been proceeded to learn the evolution of wave groups. Clamond et al. (2006) found that the interaction of envelope solitons split from a long wave group with small steepness could induce freak wave. Shemer and Sergeeva (2009) showed that large local spectrum width could support the maximal possibility to appear large wave. Adcock and Taylor (2009) used the conserved quantities of NLSE (cubic nonlinear Schrödinger equation) to model the evolution of Gaussian wave group, and the group dispersed more slowly as the initial nonlinearity increased for initial focus group.
For evolution of wave groups, vast experiments have been carried out for years. Wave elevation evolution is a usual investigated aspect for observation (Rapp and Melville, 1990;Liu et al., 2015). Geometric property is a di-rect research, and many experiments are also done (Bonmarin, 1989;Wu and Yao, 2004) in order to catch the physics of waves just prior to breaking, based on wave elevation, limiting steepness, horizontal asymmetry and vertical asymmetry and so on. Bonmarin (1989) showed that asymmetric characteristics depended on the breaking type, and plunging was more asymmetric than spilling. Mori and Janssen (2006) showed the growth of kurtosis could increase the occurrence of freak waves. Subsequently, Mori et al. (2007) studied the kurtosis evolution and maximal wave height distribution under four waves interaction and showed that the theoretical consequence of MER (modified ER, and ER distribution can refer to Mori and Yasuda (2002)) distribution predicted the extreme waves in nonlinear wave fields well. Subsequently, Tian et al. (2011) studied the skewness and kurtosis of the disperse focusing for wave groups, and found that the maximum skewness and kurtosis had dependence on the BFI (Benjamin-Feir Index). Liang et al. (2017) learned some parameters for wave groups and showed that small center frequency could generate larger local wave steepness and wave height. In this study, the evolution of some geometric parameters is also learned for the extreme waves. In the experimental studies, the variation of spectra (Rapp and Melville, 1990;Shemer and Sergeeva, 2009;Liu et al., 2015) and energy (Cherneva and Soares, 2012; are also the usual dynamics studied for the wave groups. However, studies referring to a single style wave group are still few, especially, the extreme wave generated in such a wave group.
The evolution of extreme wave is a strong non-linear and non-stationary process. In order to reveal the mechanism of generating extreme waves, non-stationary analyzing method is necessary. Up to now, the common method studying the non-stationary water waves is wavelet transform. However, wavelet transform is based on Fourier spectral analysis. Therefore, it is also linear essentially and not suitable to reveal the nonlinear process. Hence, HHT is adopted here. HHT is a method for analyzing non-linear and non-stationary signal. Wave packet is of short time duration, especially, from steep wave and then up to occurrence of breaking, HHT is made up of a "sifting" process and Hilbert transform.
Some studies for extreme waves have obtained some results with HHT. Griffin et al.'s (1996) laboratorial study on wave group by using Hilbert transform directly showed that the Hilbert frequency appeared near constant for nonbreaking wave, slow growth for spilling and sharp increase for plunging during the evolution to the focusing point. Schlurmann (2002) made investigations on monochromatic wave and field wave group with extreme and freak waves, and the phase of the IMFs carrying the main energy was the same as original data, which could reflect the instantaneous characteristics of extreme wave and occurrence of the abnormal wave height. In Wu and Yao's (2004) experimental research for the freak wave, the small difference between the two instantaneous frequencies of the top IMFs was an indicator for large limit wave steepness of freak wave under a comparative frequency band. Veltcheva and Soares (2007) studied two kinds of abnormal waves and pointed out that the asymmetry of abnormal wave was related to the degree of intra-wave frequency modulation, and the rising instantaneous frequency accompanied the asymmetric single abnormal wave. Veltcheva and Soares (2016) analyzed the field abnormal wave and showed that the growth of the asymmetry for abnormal wave increased with the length of field wave groups. Parameter CDN was proposed and showed that large CDN corresponded to the asymmetrical abnormal wave.
To the authors' knowledge, study on the single Gaussian wave group in finite water depth is still rare. As the wave packet evolves during propagation with possible breaking process, the energy, the geometric parameters and energy transfer among the frequency components undergo significant changes. A systematic experimental investigation with the HHT technique on the nonlinear evolution of a single Gaussian wave packet propagating in a horizontal flume with both breaking and nonbreaking cases is presented in this paper. The study consists of five parts. Introduction is given above firstly, and then the experimental set-up is provided subsequently. The analysis method used here is shown in Section 3, and the analysis results and discussions are included in Section 4, respectively. In the last, the conclusions are presented.

Wave flume
The experiments were conducted in the wave flume located at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, China. A sketch layout of the experimental setup is shown in Fig. 1.
The flume is 65.0 m long, 2.0 m wide and 1.8 m deep. In the experiment, the water depth is set 1.0 m. The flume is equipped with a hydraulically driven piston-type wavemaker at one end, and the absorbing device was installed to mitigate wave reflection at the other end. Since the reflection coefficient is about 5% for waves at 1 Hz shown previously, the reflection effect can be ignored here. In this study, the mean position of the wavemaker is defined as x = 0 and the propagating direction is marked in Fig. 1.
To ensure the two-dimensionality of the wave field, a surface-ground concrete wall was employed to subdivide the flume into two sections with 0.8 m and 1.2 m widths from 4.9 m downstream, and the narrower 0.8 m was chosen to be the working section.
In the experiment, 34 resistance-type wave probes were used to measure the water surface elevations simultaneously. The interval between wave gauges is 2 m for x < 12.9 m and 1 m for x > 12.9 m. The absolute accuracy of each gauge is of the order ±1 mm. Before using the probes, each wave gauge was examined for soundness and then calibrated. The first wave gauge was installed at x = 4.9 m, and the initial condition of all cases comes from the data here. Extreme events were determined in this experiment artificially, and every case was repeated three times. For each case, the wave gauges can be removed according to the occurrence locations where the extreme event happens for the first time. Within the breaking zone, the smaller space interval of 0.2 m was employed here to obtain the time series for extreme waves.

Wave group parameters
The statistical parameters are related to the local spectral width and the initial steepness for Gaussian wave groups (Shemer and Sergeeva, 2009). Here, the wave groups with different initial wave structures are employed to study the nonlinear evolution of the wave groups.
In the present study, twelve cases are investigated. The Gaussian wave surface is considered the same as that in Shemer and Sergeeva (2009), which reads where a is the wave amplitude; f 0 = 1 Hz is the central frequency; N is the width of wave group. The initial parameters obtained at the wave gauge nearest to the wave generator for each case are listed in Table 1, where breaking stands for the wave breaking style; = k 0 ×A 0 is the initial wave steepness; k 0 is the initial wave number obtained by the dispersion relation in finite water depth corresponding to f 0 ; A 0 is the initial wave amplitude obtained at x = 4.9 m; is the initial dimensionless spectrum width, referring to the parameter in Shemer and Sergeeva (2009); BFI = is the Benjamin-Feir index.

Analysis method
3.1 Parameters of extreme waves In this study, the geometric analysis for extreme waves is also employed. Some parameters are defined here. The concrete information for the wave group is marked in Fig. 2, where C stands for the direction of the wave propagation.
, and are the amplitude of the crest, the depth of front and rear troughs; t m , t f and t r are the corresponding time points respectively. , and are the time duration corresponding to crest and troughs, respectively. For wave group, the wave height is defined by zerodown crossing. The wave number k is derived from the wave period T according to the dispersion relation. In this study, the geometric effects derived from both the front and rear trough are considered here. The period T and two kinds of the horizontal asymmetry parameters are defined as follows in Eqs. (2), (3) and (4): which illustrates the asymmetry referring to the horizontal axis for the crest and front trough, and which illustrates the asymmetry referring to the horizontal axis for the crest and rear trough. Similarly, the vertical asymmetry parameters are which illustrates the asymmetry referring to the vertical axis for the double time durations corresponding to the wave crest and leading trough, and likewise, which illustrates the asymmetry referring to the vertical axis for the two time durations corresponding to the wave crest and rear trough. Additionally, the skewness S k is defined the same as Babanin et al. (2007) in Eq. (7), which shows the wave crest is sharp and high when the value is positive, and it also illustrates that twice of the amplitude of the crest is larger than the summation of the depth of the two troughs.
3.2 Hilbert-Huang transform (HHT) HHT is a method of analyzing non-linear and non-stable signal. The wave packet is of short time duration. Especially, from steep wave and then up to occurrence of breaking, it is instantaneous and nonlinear considerably. Therefore, it is reasonable to apply HHT to studying the Gaussian wave packet. About the introduction and computation of HHT, one can refer to the details by Huang et al. (1998). Only a brief introduction is given here.
HHT consists mainly of two parts, the EMD (empirical mode decomposition) and Hilbert transform. Before giving the brief introduction of EMD, it is necessary to show the definition of the IMF (intrinsic mode function). IMF needs to satisfy two conditions: (1) the number of extreme and zero-crossing points must be identical or differ by no more than one in the whole-time series; (2) At any point, the mean value of the upper and lower envelopes derived from the local maxima and minima is zero. The process of EMD is described briefly below.
Given arbitrary time series x(t), finding all the extrema, the upper and lower envelopes are formed by means of connecting the maximum and minimum extreme points by a cubic spline curves, respectively. The mean value of the upper and lower envelopes is designated m 1 , and the difference between the original data x(t) and the mean value m 1 is denoted as h 1 , If h 1 satisfies the definition of IMF, h 1 = IMF 1 , and the process of EMD finishes, one component IMF 1 is obtained. But if h 1 fails to satisfy the definition of IMF, for the next sifting process, h 1 is treated as the original data, and then the upper and lower envelopes for h 1 can be obtained just like obtaining that of x(t). And the mean of the upper and lower envelopes is m 11 , the difference between h 1 and m 11 is h 11 : Repeat the process as many times as necessary to eliminate the multi-extrema between two zero crossings. And finally, the sifting progress has been proceeded up to k times, and the residual h 1k is satisfied as an IMF, denoted as IMF 1 , After obtaining IMF 1 , the difference between the original data x(t) and IMF 1 is treated as the original data. By employing the same sifting process, all IMFs can be obtained.
After the EMD process, Hilbert transform is applied to each IMF, and the instantaneous frequency and amplitude (amplitude varying with time) for each IMF can be obtained. By adding all the instantaneous amplitude at the same frequency and time, the time-frequency-amplitude distribution, i.e., the Hilbert spectra, can be acquired finally. According to Hilbert spectra, the dynamic characteristics for the nonlinear and non-stationary data can be learned clearly.

Surface elevations
Wave evolution is the basic characteristics of wave study. For non-breaking and breaking cases, two wave groups are shown in Fig. 3. Evolution of wave groups with different initial wave group widths N is studied here. The maximal elevation appears at k 0 x = 108.4 ( = 8.48 cm) and 80.2 ( = 9.49 cm) for Figs. 3a and 3b, respectively. It can be observed that the maximal surface wave occurs at the front of the group, which results from the fast group velocity in the peak of the wave packet envelope attributed to the high amplitude, and forms the accompanied chasing process for lowfrequency waves and the bound waves. After breaking, the bound waves derived from strong wave-wave interaction are released to be free waves. The amplitude decreases are due to the failure of superposition. Meanwhile, the decrease of some amplitudes partly resulted from breaking. Therefore, the amplitude of the peak wave dissipates much more, and the velocity of the peak wave for the envelope becomes slow. As the lower-frequency wave continues to overtake the packet front, the local maximal wave moves forward to the middle gradually and far surpasses higher-frequency waves. Then some envelope solitons start to separate gradually and some envelope solitons are separated after a distance.
For wave groups including obvious extreme waves, the 378 HE Yan-li et al. China Ocean Eng., 2018, Vol. 32, No. 4, P. 375-387 η max distance L for the envelope solitons to separate and the maximal wave elevation as a function of BFI have been shown in Fig. 4. From Fig. 4a, it can be seen that the amplitude of the extreme wave of breaking wave groups is larger than that of non-breaking cases. However, the amplification extent for them is almost identical as shown in Fig. 4b. In Figs. 4a and 4b, for waves with the same initial wave steepness, the maximal crest decreases with the growth of instability (increasing BFI), and waves with large initial steepness (denoted by the gray squares) have higher wave elevation but lower amplification coefficient. With the evolution of some wave groups, some envelop solitons start to appear, and when these envelope solitons are separated totally, the distance from the wavemaker L and the distance difference ΔL between the initiation and separation of envelope solitons completely as a function of BFI are shown in Figs. 4c and 4d. Generally, L is in the range of 30-40 m, suggesting that the location for separation of single wave groups are included well in the experiment (the channel is 65 m long). From the distance difference ΔL, two regions can be observed obviously in Fig. 4d. For non-breaking cases, when the envelope soliton starts to emerge, they need longer evolution distance to separate envelope solitons completely. Nevertheless, much shorter distance is needed for the breaking cases.
The initial parameters have obvious effect on the evolution of Gaussian wave groups. Fig. 5 shows the evolution of nonbreaking wave groups. Owing to the chase of low-frequency waves, the leading larger wave runs gradually from the front to the middle of wave group. The envelope solitons start to appear at k 0 x = 140.6 and 144.7, and separate at k 0 x = 176.9 and 185 in Figs. 5a and 5b, respectively, which suggests that envelope solitons tend to appear later for the wider wave group (N = 4). η max η max Likewise, evolution of breaking wave groups is shown in Fig. 6, whose initial width is the same with different initial wave steepness. The large wave can be seen obviously in the leading of the two wave groups. The maximal wave elevation appears at k 0 x = 84.2 ( = 9.61 cm) and 60 ( = 9.63 cm) in Figs. 6a and 6b and then wave breaks, which exhibits that the wave tends to break earlier for wave group with large initial wave steepness. During the evolution, it was observed artificially that the number of breakers is 3 and 6 in Figs. 6a and 6b, respectively. Furthermore, during the evolution, the envelope soliton initiates to rise at k 0 x = 136.6 and 120.5 for the two cases. The reason may be that the low-frequency components have more energy due to the large energy input and then run faster to exceed the highfrequency ones under the large initial wave steepness. By  HE Yan-li et al. China Ocean Eng., 2018, Vol. 32, No. 4, P. 375-387 comparing the results in Fig. 5 and Fig. 6, it illustrates that the initial width of wave group and wave steepness have important effects on the evolution of single Gaussian wave group in finite water depth. Large wave steepness and small width of wave group can induce the extreme waves appear earlier. In addition, the larger initial steepness can induce more breakers in the wave packet.
To demonstrate the characteristics of breaking extreme wave more clearly, more wave probes are added and the spatial interval is 0.2 m in the vicinity of the first breaker. The evolution of wave elevation for the first breakers of wave groups in Figs. 6a and 6b are shown in Figs. 7a and 7b, respectively. It can be seen that the extreme wave occurs at the leading of the wave group with chasing low-frequency waves. After wave breaking, it can be observed obviously that the location of the extreme wave occurrence moves forward to the middle of wave group. The variation is marked by arrows in Figs. 7a and 7b.

Geometric parameters
Besides the evolution of wave elevation, the evolution of the geometric parameters for extreme waves within the η max detailed measured part is also analyzed here. The detailed results for breaking Case 7 versus the non-dimensional distance are shown in Fig. 8, where , and denote the maximal crest, the leading and following trough depths, and are nondimensionalized by A 0 . Owing to the nonlinear wave-wave interaction, wave surface evolves from initial symmetric profile to be an asymmetric one, which is exhibited with the horizontal and vertical asymmetric parameters shown in Fig. 8, and the dash vertical line in Figs. 8b-8d means the location where the maximal wave elevation appears. It is also the location to be denoted by in Fig. 8a. In Fig. 8, the maximal crest is about 1.7 times the initial amplitude which appears at k 0 x = 91.5 (Fig. 8a). Close to the focusing point, the nonlinearity grows strongly because the bound waves become pronounced, which results in more asymmetric waveform about the horizontal axis, the rapid growth of the crest, and the decrease of the leading trough depth. After breaking, it is interesting that the variation of the front and rear trough maintains the continuous trend to shoal and deepen until k 0 x = 94 around, and then, a break variation appears between k 0 x = 93.9 and 94.7, which just  η f −N corresponds the maximal crest moving to the middle of wave group (it can also be observed in Fig. 7a), and the bound waves are regenerated again due to the wave-wave interaction when the wave refocus again. It is necessary to describe the sharp change of the depth of the two troughs more detailed around k 0 x = 94. After wave breaking at k 0 x = 91.5 (which is marked in both Fig. 7a and Fig. 8a), the local maximal amplitude declines due to breaking. However, as waves evolve, the low-frequency waves with large velocity overtake non-stop the short waves and move to the middle, which can be seen in Fig. 7a at k 0 x = 94.7 and more obviously at k 0 x = 95.5, the local maximal wave elevation appears at the next crest. By this time, the front trough now is just the rear one before the maximum moving into the middle, therefore, the depth of the decreasing front trough appears a sharp growth after the maximal crest chan-η r−N ging which can be seen in Fig. 8a. In Fig. 8a, the rearing trough before the maximal crest change increases all the time, and after the local maximal wave surface shifting into the next crest, the rearing trough is the high-frequency wave with small amplitude and declines suddenly.
It also can be seen from Fig. 8b that around the focusing point (where occurs), the first horizontal asymmetric parameter reaches up to 3.5 and follows a sharp drop at the end of breaking, while the second horizontal asymmetry parameter shows a mild decline and follows a step up at the end of the breaking. The horizontal asymmetry shows a sharp break here in Fig. 8b, as the maximal crest moving to the middle, recovers 1 nearly and then appears a little growth, it illustrates as it is a crossing point of the two troughs (adjacent to the crest) or of the two horizontal asymmetry parameters defined by down and up-zero HE Yan-li et al. China Ocean Eng., 2018, Vol. 32, No. 4, P. 375-387 381 crossing which predicts that wave may refocus again. Seen from Fig. 8c, it is interesting that the two vertical asymmetries defined by different methods show a very similar tendency, which is opposite to the variation of the extreme crest, that is, when the wave group achieves the focus point, the level of narrowing for crest is larger than that for the two troughs. The skewness is referred to that in Babanin et al. (2007), which has similar evolution to the crest shown in Fig. 8a. By analyzing the skewness in these cases, it shows that S k increases gradually as wave focusing. It has similar but larger range of variation to the local maximal crest, which illustrates that the wave crest sharpens quickly as approaching to extreme wave due to the strong nonlinearity and turns to recover back relatively fast after breaking. It also can be seen in Fig. 8c that the high and sharp crest turns to be mild, and wave group varies in symmetry comparatively after spilling. In this study, the geometric characteristics of extreme wave are shown. With the evolution of amplitude of the maximal crest and depth of the two troughs adjacent to it, the two troughs have no sudden change before and after breaking occurrence and keep the tendency for a short distance. In the study of Bonmarin (1989), the depth of trough increases at the beginning and then starts to decrease before breaking occurrence. However, the depth of the front trough keeps the declining tendency before and after spilling in the present study. The reason for the difference between this result and Bonmarin's study may be the different type and mechanism of the breaking, i.e., the plunging breaker in Bonmarin (1989) is induced by B-F instability naturally and spilling here is resulted from natural self-focusing. It also maybe the scale in Bonmarin (1989) is larger and we can capture the part after the growth of wave before the plunger appearance.
For the skewness S k , we employed the same definition as Babanin et al. (2007). For the extreme wave here, the maximal skewness exceeds the limiting value of 1.0 shown in Babanin et al. (2007) for the two-dimensional deep-water wave trains and the maximal S k is about 1.03 in Fig. 8d. The reason may be the different depth of water (finite depth here), or the different results of breaking for wave trains and wave packets, i.e., the modulation instability of nonlinear evolution in Babanin et al. (2007) vs. coalescing focus here.

HHT analysis
For the non-linear and non-stationary evolution of the Gaussian wave group, the HHT is employed here to study extreme wave further. Fig. 9 is the sample of 3D Hilbert spectrum, the colorbar shows the amplitude level, the unit is centimeter, the lowest amplitude is exhibited white, and the highest amplitude is denoted dark red, and the spectrum shows the time-frequency-amplitude distribution. The black solid line denotes the wave surface corresponding to the spectrum and all the wave surface elevations below are standardized by the initial wave amplitude A 0 at x = 4.9 m. From the figure, it can be observed the energy focusing on the main part of the wave group, where there are some high waves, and at the wave crest around, the instantaneous frequency can reach the local extreme and then decreases sharply, which shows the intra-wave frequency modulation well. In the next analysis, the Hilbert spectrum is shown in the vertical view, and the unit of the color bar for all Hilbert spectra here is centimeter.
η max Fig. 10 shows the evolution of non-breaking wave group Case 3 (a case with strong nonlinearity but without breaking). It can be seen that the energy spreads to a wider time period and frequency modulation appears gradually. Its maximal wave elevation occurs at k 0 x = 108.4 ( = 8.5 cm). At initial state, the primary energy mainly focuses on wave at the frequency of 1 Hz. With the evolution, partial energy spreads to higher and lower frequency wave contents. When extreme wave occurs, the waves at higher frequencies possess more energy. A strong frequency modulation is observed at k 0 x = 108.4 due to the nonlinearity. After the extreme wave, the level of intra-wave frequency modulation recovers the initial state gradually, and the energy is divided into two parts corresponding to the two envelope solitons, but the main energy is still kept in the main wave group. η max For the breaking group in Case 12, the variation of the Hilbert spectrum is shown in Fig. 11. The maximal wave elevation occurs at k 0 x = 64.1 ( = 9.4 cm), and then wave breaking occurs. It can be observed that the intra-wave frequency modulation is stronger than that of Case 3 in Fig. 10. Meanwhile, the energy of higher-frequency components increases, which illustrates that higher-frequency waves affect intra-wave frequency modulation strongly, and play an important role in the formation of extreme wave. From k 0 x = 64.1 to 112.4, there are several breakers. The nonlinearity is quite pronounced and the obvious intra-wave frequency modulation exhibits within the breaking region. Out of the breaking region downstream (for k 0 x > 112.4), the intrawave frequency modulation alleviates, and the energy residing in higher frequency waves dissipates more (the maximal instantaneous frequency decreases) and achieves a relat- ively stable state. Some bound waves are also released to be free waves due to breaking, and these free waves travel slowly and are surpassed by other lower frequency long waves. Finally, large waves in the front of the wave group transform fast and higher frequency short waves propagate slowly in tail. The envelope solitons are formed at k 0 x = 144.7, where energy also focuses on two parts mainly corresponding to the two envelope solitons and then lessens due to breaking.
With the region being added more wave probes, the detailed evolution of the Hilbert spectrum around the first breaker for two cases is shown in Fig. 12 and Fig. 13, respectively. From the two figures, it exhibits that the instantaneous frequency becomes narrow and sharp within one wave period, which shows a strong nonlinearity induced by higher frequency waves derived from wave-wave interaction when approaching the focusing point. Furthermore, the instantaneous frequency and the maximal crest have the similar focusing trend. From the Hilbert spectrum, the energy resides in the maximal whole crest and the two troughs closed to it main part before breaking, as wave approaching the focusing point, energy spreading to the high-frequency waves becomes more and more, and the range of frequency turns to be wider due to the resonant effect, combined with the dispersion and nonlinearity due to bound waves, the extreme wave is formed and the strong intra-wave frequency modulation is shown at k 0 x = 56 in Fig. 12 and 91.5 in Fig.  13. The energy dissipation from the wave front after breaking can be observed clearly (k 0 x = 56.8 in Fig. 12 and 92.3 in Fig. 13), which indicates that the energy loss due to spilling in the finite depth water for single Gaussian group is mainly derived from the whole wave front of wave group, and the local variation of amplitude-frequency-time can be observed from the spectrum. Many studies have shown that HE Yan-li et al. China Ocean Eng., 2018, Vol. 32, No. 4, P. 375-387 energy at higher frequency components was dissipated due to breaking with the classical FFT (fast Fourier transform), such as Rapp and Melville (1990), Tian et al. (2011) andLiu et al. (2015). However, compared with other data processing methods like FFT, the Hilbert spectrum features can show the detailed energy variation locally. Meanwhile, the energy in the whole wave front loses a lot due to spilling, while the energy in the back of the crest changes little. It indicates that breaking has very little effect on the energy at the rear trough.
In Fig. 12 and Fig. 13, at the front trough, the instantaneous frequency has a little growth corresponding to the minimal depth (around t = 43.5 s in Fig. 12 and 51.5 s Fig. 13), a little growth in instantaneous frequency shows energy residing in waves with these frequencies increases, and it illustrates that in wave focus process waves with a little higher frequency gain some energy, and nonlinearity here is comparatively strong due to the increasing decline of the front trough depth.
By EMD, the main energy focuses on the first component IMF 1 , and the first instantaneous frequency f 1 is studied here. In order to show the detailed variation, the instantaneous frequency f 1 corresponding to IMF 1 of four breaking cases at various locations as a function of time t subtracting t m is shown in Fig. 14, where t m is the time point corresponding to the maximal crest. In Fig. 14, there is significant variation for f 1 as wave evolving to the focusing point. It modulates and increases intensely, and f 1 reaches above 2 Hz at the maximal wave crest, meaning that the intra-wave frequency modulation becomes very strong here. At the initial station, the initial condition for instantaneous frequency is around 1 Hz. As the wave evolves to the extreme wave, the intra-wave frequency modulation increases sharply. The local growth rate of instantaneous frequency f 1 within a ε 0  short distance of 0.6 m is above 40% and for Case 8 it reaches 66% in Fig. 14c. It also can be observed that the intense variation of the instantaneous frequency mainly exhibits within [-0.2 s, 0.2 s] for t-t m . After the extreme wave, f 1 decreases mildly within the rough distance of 0.6 m behind the extreme wave.
BFI is the parameter standing for instability, and CDN by Veltcheva and Soares (2016) means the combined degree of nonlinearity. From Fig. 15, it can be seen that the level of CDN is around -3.0 for all cases (including symmetrical and asymmetrical cases), where breaking cases have a little larger CDN but not obvious. However, in Veltcheva and Soares (2016), the parameter of CDN for the symmetrical and asymmetrical abnormal waves differs a lot. The possible explanation may be that the laboratory experimental data in this study are different from the field data of Veltcheva and Soares (2016). The IMF components are less, and the interaction between each other is weak, especially, the energy of the second IMF is very small and can be ignored here. Hence, the CDN here mainly stands for the contents of the first IMF and it suggests that CDN in this study has no direct relation with wave breaking of the single Gaussian wave group in finite water depth. Hwung et al. (2007) pointed out that the instability appears first at the wave front, and then extreme wave occurs, subsequently, breaking takes place. They only observed the evolution of the wave elevation and gave the whole change for the wave front, but by the Hilbert spectrum, local energy variation in the wave front can be reflected clearly, and the level of the frequency modulation also can be known. Zimmermann and Seymour (2002) pointed out that when wave initiates to break, the instantaneous frequency achieves the maximum, and then decreases as wave appears the peak of breaking. It is different from the cases with large energy loss after achieving the maximal surface elevation here. What is more, their study exhibits the twice occurrences for the maximal instantaneous frequency and wave crest elevation differ about 1 s around. Part of the breaking case here for f 1 has the similar results to theirs. Nevertheless, for some cases the difference here is just only 0.02-0.04 s. The possible reasons may be the different wave types, the initial conditions and the analyzing methods used. It needs to be emphasized that the Hilbert transform is employed directly to process original data in Zimmermann and Seymour (2002), but EMD analysis is used before the Hilbert transform in this study. The reason for adopting the HHT but not PTM (phase-time method) to analyze the wave packet is that when wave group evolves further more along  the direction of the wave channel, the asymmetry of wave surface appears obviously, applying the Hilbert transform directly maybe lead to the singular point in instantaneous frequency, thus, EMD is effective to avoid this.

Conclusions
The evolution of a Gaussian wave packet with various initial wave steepnesses and spectral widths in the water of finite constant depth was studied experimentally in a wave flume. The wave packet with an initial Gaussian shaped envelope underwent significant transformation and some extreme events with and without breaking as nonlinear wavewave interaction occurred during the propagation. The characteristics of transformation for the wave packet, including the surface elevation and geometric parameters for both the breaking and nonbreaking cases were investigated. The evolution process was analyzed with the Hilbert-Huang transform technique and detailed local energy variation around the extreme crest was presented. Some conclusions are drawn below.
First, the large wave in the wave packet runs from leading to the middle of the wave group due to the non-stop chasing of low-frequency waves. With the evolution, some envelope solitons emerge in some cases due to the interaction of the frequency dispersion and nonlinearity. From initiation to separation of the envelope solitons, the distance ΔL is much shorter for the breaking wave groups than that for non-breaking ones, and the envelope solitons appear late for wider non-breaking wave packet. In addition, the initial steepness has an important effect on the breaking, and the steeper waves tend to break earlier.
Second, the wave crest amplitude, trough depth, horizontal and vertical asymmetry parameters and skewness are calculated for the evolution of extreme waves. The trend of the depth of the front and rear troughs around the extreme crest suffers little sudden influence on spilling until wave refocuses again. The two vertical asymmetry parameters and skewness have the inverse tendency.
Third, with the time-frequency-energy distribution in the Hilbert spectrum, it reveals that high frequency waves play a key role in attributing to the nonlinearity and in the formation of extreme waves. For non-breaking waves, energy focuses around the carrier frequency contents at beginning, and with the evolution, the energy spreads to a wide frequency band, and some energy spreads to lower and higher frequency; a relatively weak intra-wave frequency modulation was observed. For breaking wave group, energy in wave front dissipates a lot suddenly after spilling, the back of crest shows little change, and strong intra-wave frequency modulation was observed. For the extreme wave, within one wave, the instantaneous frequency grows up to be 2.2 times the initial frequency. The local variation can support one to know the dynamic characteristics better and can also understand the energy distribution at any time and frequency.