## Abstract

A series of laboratory experiments on the long-time evolution of nonlinear wave trains in deep water was carried out in a super wave flume (300×5.0×5.2 m) at Tainan Hydraulics Laboratory of National Cheng Kung University. Two typical wave trains, namely uniform wave and imposed sideband wave, were generated by a piston-type wavemaker. Detailed discussions on the evolution of modulated wave trains, such as transient wavefront, fastest growth mode and initial wave steepness effect, are given and the results are compared with existing experimental data and theoretical predictions.

Present results on the evolution of initial uniform wave trains cover a wide range of initial wave steepness () and thus, greatly extend earlier studies that are confined only to the larger initial wave steepness region (). The amplitudes of the fastest growth sidebands exhibit a symmetric exponential growth until the onset of wave breaking. At a further stage, the amplitude of lower sideband becomes larger than the carrier wave and upper sideband after wave breaking, which is known as the frequency downshift.

The investigations on the evolution of initial imposed sideband wave trains for fixed initial wave steepness but different sideband space indicate that the most unstable mode of initial wave train will manifest itself during evolution through a multiple downshift of wave spectrum for the wave train with the smaller sideband space. It reveals that the spectrum energy tends to shift to a lower frequency as the wave train propagates downstream due to the sideband instability.

Experiments on initial imposed sideband wave trains with varied initial wave steepness illustrate that the evolution of the wave train is a periodic modulation and demodulation at post-breaking stages, in which most of the energy of the wave train is transferred cyclically between the carrier wave and two imposed sidebands. Meanwhile, the wave spectra show both temporal and permanent frequency downshift for different initial wave steepness, suggesting that the permanent frequency downshift induced by wave breaking observed by earlier researchers is not permanent. Additionally, the local wave steepness and the ratio of horizontal particle velocity to linear phase velocity at wave breaking in modulated wave group are very different from those of Stokes theory.

## 1. Introduction

The evolution of nonlinear wave trains in deep water exhibits many subtle phenomena, such as chasing, integration and disintegration among waves. Pioneering research was done by Benjamin & Feir (1967), who showed that weakly nonlinear deep-water wave trains are theoretically unstable to modulational perturbations. Lake *et al.* (1977) further confirmed the validity of Benjamin and Feir's theory experimentally and discussions on how the energy interchanges between different modes are also given. Later, Melville (1982, 1983), Su (1982), Tulin & Waseda (1999) and Hwung & Chiang (2005*a*) all explored the wave modulation experimentally in detail.

Theoretical and numerical progresses to the evolution of nonlinear wave trains are further advanced by Zakharov (1968), Dysthe (1979), Lo & Mei (1985) and Krasitskii (1994) based on different assumptions and numerical schemes. Specifically, Zakharov showed that the leading order complex amplitude of the modulated surface wave satisfies the nonlinear Schrödinger (NLS) equation for weakly nonlinear and narrow-banded wave trains. On the basis of the NLS equation, the evolution of a wave train is expected to have a recurrence of initial state and the envelope of surface elevation is symmetric with respect to the peak of wave profile. On the other hand, previous studies on the numerical simulation of the NLS equation had revealed that a wave field with initial narrow bands could breakdown due to the energy leakage to high-frequency modes, resulting from the violation of the narrow bandwidth constraint of the NLS equation. Therefore, Dysthe (1979) derived the fourth-order NLS equation, which includes the small mean current induced by the radiation stresses of the modulated wave trains, whereby the asymmetric envelope of surface elevation with respect to the peak of the wave profile was found. Crawford *et al.* (1981) investigated the evolution of nonlinear wave trains using an improved approximation from Zakharov and showed that the predictions of Benjamin and Feir's theory regarding the most unstable wave mode and the maximum growth rate could differ as large as 30% from their numerical results for an initial wave steepness of 0.2. Stiassnie & Kroszynski (1982) analytically examined the long-time evolution of unstable wave trains, which combines with a carrier wave and two small sideband components; especially, the long waves induced by wave groups and the recurrence period of modulation–demodulation were discussed. Lo & Mei numerically solved the fourth-order NLS equation and demonstrated the asymmetric evolution of initial symmetric wave trains. Besides, the long-time evolution of wave trains reveals a periodic modulation and demodulation process, in which the recurrence of initial state of wave trains was found. The corresponding wave spectrum discloses a periodic frequency downshift and upshift. Using the same numerical scheme proposed by Lo & Mei, Trulsen & Dysthe (1990) included an artificial function of energy dissipation into the fourth-order NLS equation and produced permanent frequency downshift in their numerical simulation on the evolution of initial modulated wave trains. Later, they further proposed the band-modified nonlinear Schrödinger equation (BMNLS), in which the restriction of spectrum bandwidth was relaxed while retaining the same accuracy in nonlinearity. Numerical investigations with the BMNLS by Trulsen & Dysthe (1996, 1997) for the evolution of weakly nonlinear narrow band wave trains in deep water indicate that permanent frequency downshift is prohibited in two-dimension. It is worth noting that Trulsen & Dysthe's studies imply that energy dissipation is the dominant factor leading to permanent frequency downshift.

In contrast, there are only a few experiments available on the long-time evolution of nonlinear wave trains in deep water due to the limitation of appropriate experimental facilities. Lake *et al.* (1977) conducted experiments on the evolution of nonlinear wave trains and confirmed Benjamin & Feir's finding on the initial growth of sideband amplitudes. They reported that the evolution of wave trains occurs near recurrence of initial state and the corresponding wave spectrum reveals a frequency downshift. Later, Melville (1982) underwent a transformation analysis on initial regular waves in deep water with high nonlinearity, where the wave steepness ranges between 0.20 and 0.28. Melville found that the development of sideband energy mainly resulted from the repeated reflection of wavefronts in wave flumes. Following an initial exponential growth, two sidebands would grow asymmetrically and the location would mirror to the approximate point of wave breaking. The frequency downshift of wave spectrum was investigated experimentally after wave breaking and Melville concluded that the spectrum evolution of nonlinear wave trains due to the sideband instability is not only restricted to a few discrete frequencies, but also involves a growing continuous spectrum. On the other hand, Su *et al*. (1982) provided experimental findings on the evolution of a finite length wave packet. The frequency downshift during the evolution of a short wave packet is significant for initial wave steepness as low as 0.1. The evolution of longer wave packets is more complex than that of shorter wave packets. Thereafter, Su *et al*. conducted a series of experiments on the nonlinear instabilities of continuous wave trains in deep water, both in a wave tank and a wide basin. It was found that two-dimensional wave trains evolved into a series of three-dimensional spilling breakers for initial large wave steepness, followed by a series of two-dimensional wave groups. Melville (1983) further analysed the instantaneous amplitude, frequency, wavenumber and phase speed in deep-water wave trains using the Hilbert transform technique. The increasing modulation that evolves from sinusoidal perturbations finally results in a very rapid frequency jump or phase reversal near the location, where the local amplitude of wave trains approaches zero.

Recently, Waseda & Tulin (1999) demonstrated experimentally that the growth rate of sideband amplitudes decreases as wind speed increases, but the sideband amplitude is still larger for stronger wind conditions. Moreover, the near recurrence of the initial state of wave trains without frequency downshift was observed in a non-breaking case. For a wave-breaking case, the spectrum evolution at post-breaking stage indicates that the amplitude of the lower sideband almost coincides with that of the carrier wave in their experiment. Further evolution was not investigated due to the limitation on the length of the wave flume. More recently, some of the remarkable results on long-time evolution of nonlinear wave trains in deep water were reported by the research team of Tainan Hydraulics Laboratory. Particularly, Hwung & Chiang (2005*a*) studied experimentally the evolution of wave trains from the initial onset of instability on initial modulated wave trains to the long-fetch evolution of the wave trains. The modulation and demodulation of wave trains occur cyclically at post-breaking stages. The phenomenon was reconfirmed experimentally on the evolution of initial uniform wave trains by Hwung & Chiang (2005*b*).

Although successful results and interesting phenomena, such as disintegration of wave trains, recurrence of initial status of wave trains and frequency downshift of a wave spectrum, have been reported in earlier studies for different kinds of wave grouping, the observed evolutions of nonlinear wave trains in fetch are only a few (e.g. Stansberg 1998; Trulsen *et al*. 1999; Onorato *et al*. 2004) due to the limitation on the length of wave flume. To further extend the earlier studies, the present paper considers the evolution characteristics of nonlinear modulated wave trains for a wide range of initial wave steepness. In particular, the transient wavefront, the evolution of wave trains and related wave spectra, the fastest growth mode and time-frequency features of wave breaking are discussed in detail. A brief description of the experimental set-up, data acquisition system and the measurement technique is given in §2. The experimental data are analysed and discussed in §3. Section 4 outlines the conclusions obtained from this study.

## 2. Experimental apparatus and data analysis

The success of carrying out experiments on the modulation of nonlinear wave trains highly depends on the following terms (Hwung & Chiang 2005*b*): (i) a carefully constructed long wave tank, (ii) a high accuracy and programmable wavemaker, (iii) the experimental instrumentation, and (iv) a suitable analysis technique. The experimental facilities, instruments and the method of data analysis used in this study are described in the following sections.

### (a) Experimental facilities and set-up

The experiments were performed in a super long-wave flume at Tainan Hydraulics Laboratory. The flume is 300 m long, 5.0 m wide and 5.2 m deep. Figure 1 shows the schematic diagram of the experimental set-up. A programmable, high-resolution wavemaker is located at one end of the tank and an effective wave-absorbing structure at the opposite end. The wave-absorbent structure comprises two slopes covered by pebbles with a mean diameter of 10 cm. The first slope (1 : 7) of 14.9 m in length is located at 240 m and the second slope (1 : 10) of 25.8 m in length is located at 260.3 m from the position of the waveboard. Between the two slopes, there is a berm of 5.4 m in length in order to stabilize the armour pebbles. In general, the reflection coefficients are between 3 and 5% and the maximum reflection coefficient is about 7% in the experimental tests. The wavemaker is a piston-type paddle activated by a hydraulic cylinder. The motion of the hydraulic cylinder is prescribed by a programmable controller, which takes an external input signal for the motion. The input signal at 25 Hz can be either generated in real time or read from a data file stored on the hard disk. The wavemaker is designed to have optimal performance for wave periods between 1 and 3 s. The evolution of surface wave trains is recorded by using 66 capacitance-type wave gauges, which are located between 15 and 240 m downstream of the wavemaker. Each wave gauge has a dynamic range of 4.5 m at 12 bits digitization, with noise typically less than 4 counts (approx. 0.5 mm). The wave gauges are calibrated before and after the experiments. The linearity of wave-gauge response is indicated by a correlation coefficient of 0.999. The data acquisition is PC-based multi-node data-acquisition system, which is developed by Tainan Hydraulics Laboratory specially to cope with two problems. One is synchronization of signals from a large number of parallel inputs and the other is the signal decay due to long-distant transmission of data in the wave flume. The time delay of the measured data between first and last gauge in the data acquisition system is 0.01 s, which is relatively small compared to 0.04 s sampling interval and 1.6 s wave period. So, the time-series of water surface elevations are acquired almost simultaneously at 25 Hz and stored for further processing. For initial uniform wave trains, the experimental run is recorded for 40 min of real time in order to provide data samples sufficiently long for accurate calculation of the power spectra. Each experimental run is recorded for 10 min of real time for initial imposed sideband wave trains in order to provide data samples sufficiently long for accurate calculation of the power spectra. The time-interval between two successive measurements is 35 min for any long wave fluctuations to be damped out in the wave flume.

The accurate performance of the wave generator is obviously a key factor in successfully carrying out experiments on wave modulations and wave breaking. In order to ensure that the experimental data are of high quality and uncontaminated by background noise, the wave generator system was checked before carrying out the experiments. The detailed description can be found in Hwung & Chiang (2005*b*). The results suggest that the noise energy level generated by wave generator is very small compared to the given wave conditions both in time- and length-scale. Hence, the noise effect on the evolution of wave modulation can be ignored in the study.

### (b) Test conditions

The experiments were performed using computer-generated wave forms. Two different types of wave generation mechanisms are investigated, which includes uniform wave trains and imposed sideband wave trains, given by equations (2.1) and (2.2), respectively, as input to the wavemaker servo-system.

#### (i) Uniform wave trains

(2.1)where *η* is the surface displacement, *a*_{c} and *ω*_{c} are the given amplitudes and angular frequency of carrier wave, and *t* is the time.

#### (ii) Imposed sideband wave trains

(2.2)(2.3)(2.4)where *a*_{c} and *a*_{±} are the amplitudes of the carrier wave and imposed sidebands, *ω*_{c} and *ω*_{±} are the angular frequency of the carrier wave and imposed sidebands, Δ*ω* is the frequency difference between the carrier wave and imposed sidebands, is the initial phase difference between the carrier wave and imposed sidebands. The stroke of the wave paddle is related to the given wave amplitude by the linear wavemaker theory.

According to Benjamin & Feir's (1967) theory, the growth rates of the sidebands depend on the dimensionless frequency difference between the carrier wave and the sidebands, , where *k*_{c} is the wavenumber of the carrier wave. When is smaller than and the depth parameter is larger than 1.363, the Stokes wave is unstable to small sideband perturbations. The most unstable mode of sidebands corresponds to . Tulin & Waseda (1999) demonstrated that the initial growth rates of sidebands decrease as the initial wave steepness increases, indicating that initial wave steepness is also an important factor in the modulation of wave trains through sideband instability. Therefore, the effects of three important parameters, namely dimensionless frequency difference (), depth parameter and initial wave steepness parameter , are investigated in this study. For initial uniform wave trains, the wave period and amplitude are prescribed and the wave conditions are listed in table 1. In particular, the evolution of wave trains is examined for varied initial wave steepness parameter and water depth parameter.

The initial imposed sideband wave trains are composed of a carrier wave of prescribed angular frequency and a pair of sideband components of prescribed frequency difference between the carrier wave and imposed sidebands. The initial magnitudes of the sideband amplitudes relative to that of the carrier waves are also prescribed. A wide range of initial wave steepness and the frequency difference between the carrier wave and sideband components are generated and details are listed in table 2.

We noted that the existence of background noise might mislead our understanding of the wave evolution. In this study, the measured noise data in the wave flume are composed of the electronic noise and wind ripple, which are obtained by the analysis of the Hilbert-Huang transform (Huang *et al.* 1998). Generally, the mean amplitude of noise is less than 0.2 cm depending on the environmental condition, which is not substantial in comparison with the given wave amplitudes in the experiments. Additionally, the peak frequency is about 4.0 Hz according to the results of the Fast-Fourier transform analysis, which is far beyond the carrier frequency used in the experiments and the corresponding instability region. The results demonstrate that the electronic noise of wave gauges and the ripple due to light wind blowing over the water surface in the flume are much smaller than the wave generated by wavemaker both in length- and time-scale. More detailed analysis can be referred to Chiang (2005).

### (c) Data analysis techniques

#### (i) Analysis of the local maximum wave

In order to quantitatively describe the evolution of a wavefront, the wave data before wave reflection are extracted from the experiments. According to the measured surface elevation at the station near the end of the tank, the duration before the wave approaching the wave absorber is identified. The measured wave data before wave reflection from the wave absorber at all gauges are used in the following analysis.

The normalized local wave parameters provide a key to understanding the transient large wave. Following Grue *et al.* (2003), the local wavenumber is estimated from Stokes third-order wave theory in conjunction with the measured surface elevation and the dispersion relation. A set of local wave parameters in equations (2.5)–(2.7) are defined according to the Stokes third-order wave theory and the measured surface elevation data. By solving the coupled equations (2.6) and (2.7), the local wave steepness *ϵ*_{m} and wavenumber *k*_{m} that corresponds to the measured maximum crest elevation are calculated. Then, the local wave steepness based on crest elevation, , and maximum wave height, , can be obtained.(2.5)(2.6)(2.7)(2.8)(2.9)where is the wave period of measured surface elevation defined by zero up crossing method, is the measured maximum crest elevation, *H*_{m} is the wave height corresponding to the crest elevation and *u*_{m} is the water particle velocity at crest elevation.

#### (ii) Analysis of wave spectrum

Spectrum analysis is used to analyse the phenomena of wave evolution, such as the sideband energy level, the initial growth of sidebands and the evolution of the wave spectrum. The spectrum is obtained by using a discrete Fourier transformation with the Hanning window. In the present study, the spectrum is calculated within the frequency range of 0–12.5 Hz and a resolution bandwidth of 0.006 Hz, which is smaller than the frequency difference between the carrier wave and the naturally evolved and initial imposed sidebands in all the experiments.

#### (iii) Analysis of initial growth rate

From the analysed wave spectrum, the amplitudes of wave modes can be expressed as the square root of the total energy of the corresponding wave spectral peaks. The normalized amplitudes of sidebands, , are defined in equation (2.10). The first-order effect of dissipation is removed by using the normalized amplitudes due to the small frequency difference between the carrier wave and sidebands. The dissipation rates are almost identical for the sidebands and carrier wave owing to the small frequency difference between them. Next, the normalized amplitudes are plotted as a function of normalized fetch, , which is defined in equation (2.11). Then, the exponential curve, equation (2.12), is fitted by the least square method to the plotted data for normalized sideband amplitudes between 0.05 and 0.2, whereby the initial growth rate, *β*, of the sideband amplitudes is determined.(2.10)

(2.11)

(2.12)

Note that *x*_{0} is the initial position of the sideband growth downstream of the wavemaker, *k*_{c} is the wavenumber of the carrier wave and, *x* is the distance measured downstream of the initial position, *x*_{0}.

## 3. Experimental results

### (a) Modulation of wavefront and quasi-steady state

The evolution of wave trains from initial transient wave generation up to quasi-steady state is investigated in this section. The modulation of initial uniform wave trains is initiated from transient wavefront accompanied by the increase of sideband energy. The sideband energy further enhances from the effect of wave reflection until a quasi-steady state is achieved. Details are given as follows.

#### (i) Transient wave transformation

The unsteady wavefront, where the local wave amplitude increases gradually from zero to a steady state, is observed on the surface elevation due to the variation of initial transient motion of the wavemaker. Transformation of the unsteady wavefront develops with the propagation downstream of the wave flume. The combining effect of nonlinearity and dispersion results in the interesting and complicated evolution of the unsteady wavefront. Owing to the nonlinearity of wave trains, the wave crest with large amplitude travels fast. So, the evolution of wave trains following the initiation of modulation shows that the wave energy focuses into a short group of steeper waves. As the wave trains evolve further, the modulation becomes stronger, which either initiates wave breaking or reaches a maximum modulation. Roughly, at this stage, the modulated wave trains form a pulse train, where the minimum level of individual wave amplitude approaches zero and a crest pairing is observed. However, the wave trains demodulate at a later stage and crest splitting is exhibited during the demodulation process. A typical evolution of initial uniform wave trains without breaking is shown in figure 2. The initial wave steepness of the wave trains is 0.110 and the relative water depth is 5.5. The top time-series of surface elevations measured at 15 m fetch is about four wavelengths away from the wavemaker. It shows that the wave trains just off the paddle are almost uniform except the initial transient wavefront. The following time-series illustrates the developing modulation and the formation of leading group in front. Owing to small initial wave steepness, the evolution of the wave trains is relatively slow.

To understand the effect of water depth, the evolution of wave trains with and is shown in figure 3. Under the same wave steepness, it is evident that the evolution of wave trains evolves faster as the relative water depth increases. Moreover, two wave groups form at the head of the following uniform wave trains shown at fetch . No breaking is observed in this case according to the video recording of the experiment. For sufficiently large wave steepness, the initial transient wavefront evolves due to the Benjamin–Feir instability leading to wave breaking. Then, the wave front goes through a region of breaking and becomes a region of unbroken gentle wave as shown in figure 4 with and .

The above-mentioned experimental data suggest that the modulation of wave trains induces a large transient wave group, characterized by an extreme wave height in front of the wave trains, which is significantly higher than all the proceeding and the following waves. Besides, these experimental results demonstrate that the source of modulational disturbances lies in a disturbance travelling with the wavefront, and the nonlinear instability of wavefronts in initial uniform wave trains does not lead to permanent disintegration of wave trains. It is worth noting that Yue (1980) solved the NLS equation numerically for the evolution of a transient wavefront. His calculated results indicated that the undulation developed behind the wavefront is qualitatively consistent with the present experimental results.

On the other hand, the evolution of transient wavefronts of initial imposed sideband wave trains is also examined as shown in figures 5 and 6 for and 0.173, respectively. Generally, the same qualitative features as the evolution of transient wavefront of initial uniform wave trains are found. However, the modulation of wave trains occurs simultaneously on the transient wavefront and the following wave trains due to the nonlinear interaction between the carrier wave and initial imposed small perturbations through the mechanism of sideband instability. In figure 6, wave breaking was observed during the propagation of the wavefront. First breaking region locates between and second breaking region locates between *k*_{c}*x*=303 and 360. It is observed that these breaking regions of transient wavefront may not correspond to those of the quasi-steady state.

Furthermore, based on the visual investigation of experiments, the observed wave breaking during the evolution of the wavefront shows both two- and three-dimensional features, which depend not only on the initial wave steepness, but also on the frequency difference between the carrier wave and sideband components. Figure 7 displays two typical images of two- and three-dimensional wave breaking. The breakers were observed intermittently at the centre and the sidewall of the wave flume with the propagation of the wavefront.

In order to reveal the evolution of amplitude spectra of wave trains in space domain, the amplitude contours of wave trains are plotted versus frequency and space at different time instants for initial uniform wave trains as shown in figure 8. It indicates the energy distribution of wave trains in the frequency and fetch at the initial stage of wave generation, *t*=10 s, in which the wave energy focuses in the fundamental and its first higher harmonic frequencies. At *t*=50 s, the wave trains propagates a little further and the corresponding energy spreads to the sidebands of fundamental frequency, which are enhanced after the reflection of wavefront from the absorbing structure as shown at *t*=120 s. However, the peak frequencies of naturally evolved sidebands are obscure at this stage. At *t*=2000 s, the contours of wave amplitudes reveal evident energy peaks at the carrier wave and naturally evolved frequencies of sidebands, especially a pair of fastest growth sidebands. These results demonstrate that the sideband energies initiate from the transient wavefront and increase as the wavefront propagates downstream of the wave flume. Ideally, the wave trains will evolve further in an infinite domain. However, the wave trains reflect at the end wall of wave flume, which further magnifies the strength of the sidebands through nonlinear interaction between the carrier wave and the sidebands. Eventually, a quasi-steady state is reached after 40 min of wave generation for initial uniform wave trains. For initial imposed sideband wave trains, the quasi-steady state of wave modulation is achieved earlier than the initial uniform wave trains. Since the characteristics of evolution are similar to that of initial uniform wave trains, the detailed patterns are omitted here.

#### (ii) Quasi-steady modulation of wave trains

Figure 9 shows the records of water surface elevations and the related amplitude spectra, when the quasi-steady state is approached at several locations downstream of the wave flume for the initial uniform wave train with and . Initially, the wave profile is almost uniform with the wave spectrum comprising a carrier wave and its related weak higher harmonics. Modulational instability causes an almost uniform wave train to gradually develop a well-defined wave group at the fetch accompanied with the growth of a pair of sideband amplitudes, which can be seen in the evolution of wave spectra. Further evolution shows that the large wave amplitude occurs in front of the wave group and then initiates wave breaking. Around the initiation of the breaking stage, the wave spectrum indicates an asymmetrical development of amplitudes of the lower and upper sideband components. Meanwhile, the energy of the wave train spreads over frequencies that are higher than the upper sideband. During the breaking process, the energy of the lower sideband is selectively amplified compared to those of the carrier wave and upper sideband. Therefore, the frequency downshift of wave spectrum is observed after wave breaking. The wave train then undergoes a demodulation process until the end of the wave flume, in which the carrier wave gradually recovers part of its initial energy through nonlinear wave interaction due to wave breaking, an earlier observed irreversible process. Finally, the amplitudes of the carrier wave and lower sideband are almost identical in this case.

#### (iii) Spatial evolution of normalized wave amplitudes

To describe the modulation of wave trains quantitatively, the spatial evolution of wave mode amplitudes is discussed in this section. The spectrum discussed in §3*a*(ii) showed that most of the energy of the wave trains resides in the carrier wave and some discrete sidebands. The amplitudes of wave mode components are calculated from the spectrum of each corresponding wave mode. The spatial evolution of the normalized amplitudes of the carrier wave and a pair of fastest growth sidebands with four different wave steepness, , 0.165, 0.220 and 0.252, are discussed and the results are shown in figure 10. For the sake of comparisons, the theoretical predictions based on Benjamin & Feir (1967) and Tulin & Waseda (1999) are also given. As a result, the initial evolution of sideband amplitudes displays an exponential growth tendency that is consistent with the prediction of Benjamin & Feir. However, the growth rate of Benjamin & Feir is overestimated, especially for initial larger wave steepness.

Detailed inspection further shows that the initial growth rate of the sidebands matches well with Tulin & Waseda's calculation up to the breaking area. Moreover, the symmetric growth of the lower and upper sidebands develops until it reaches a level, where the energy is defined by the square of the normalized sideband amplitudes, approximately . Subsequently, the amplitudes of the lower and upper sidebands diverge, which roughly corresponds with the onset of wave breaking. While the lower sideband retains tendency of exponential growth during the breaking stage, the upper sideband declines slowly after approaching a certain smaller maximum value. When the wave breaking ceases, the energy of the lower sideband reaches its local maximum. Meanwhile, the peak of the wave spectrum shifts to the lower sideband. At post-breaking stages, the evolution of wave trains discloses a periodic demodulation and modulation process. The amplitude of the lower sideband increases while that of the carrier wave decreases with the increase of the modulation. However, a contrary trend of the evolution of the carrier and lower sideband is found during the demodulation process. For initial larger wave steepness, the energy transfer back to the carrier wave during the demodulation process is lower than that for smaller wave steepness after wave breaking. More specifically, a permanent frequency downshift is observed for initial larger wave steepness, while the temporary frequency downshift is found for initial smaller wave steepness. The present experimental results provide much longer spatial evolution of wave trains, compared to previous researchers' results, and reveal that the frequency downshift induced by wave breaking is not permanent but partial recurrence of the initial state of wave trains due to the wave breaking is an irreversible process.

#### (iv) Effect of sideband space on evolution of normalized wave amplitudes

For the wave trains with constant wave steepness but different imposed sideband space, the evolution of normalized amplitudes of one carrier wave and two pairs of sidebands versus dimensionless fetch is shown in figure 11 for . Note that the hatched area indicates the breaking region identified according to the visual inspection during the experiments. Consistently, the sidebands exhibit a symmetrical growth at the initial stage until the dimensionless energy of the imposed sidebands approaches , where the onset of asymmetric evolution of the sidebands is observed.

As the wave trains evolve further, the modulation is strong enough to initiate wave breaking in front of the wave trains. Several different types of evolution are shown at the post-breaking stages. It is observed that the multiple downshift occurs for smaller imposed sideband space as shown in figure 11*a*, suggesting that the most unstable mode eventually manifests itself instead of the imposed sideband components during the evolution of the wave trains. When the imposed sideband components coincide with the most unstable mode, wave breaking occurs earlier, as shown in figure 11*b*, due to the larger initial growth rate of the imposed sidebands. Furthermore, the breaking event is seen to cover a larger area and last for a longer time. While the lower sideband continues to grow during this period, the upper sideband almost remains the same and the carrier wave is further dissipated through the breaking process. The permanent downshift of peak of wave spectrum is observed at the post-breaking stage in this case. In figure 11*c*, the growth rate of the imposed sideband components is smaller than that shown in figure 11*b*, as the imposed sideband space increases. The amplitudes of the carrier wave and lower sideband almost coincide at post-breaking stage in this case. For the wave trains with imposed sideband space, , no frequency downshift is observed after wave breaking due to the decrease of the growth rate of imposed sidebands as shown in figure 11*d*. Interestingly, these wide sideband spaces lead to less modulation and downshift shown in figure 11*c*,*d* confirms earlier findings from irregular wave spectra (Stansberg 1995).

### (b) Wavelet analysis of wave breaking

In this section, the wavelet analysis (Torrence & Compo 1998) is employed to investigate the energy transport mechanism of modulated wave trains. Figures 12 and 13 show the free surface variation and related wavelet spectra before and after wave breaking, respectively. Specifically, on the leading wavefront, both the frequency and the amplitude modulate to high frequency before wave breaking as shown in figure 12. This phenomenon is very similar to the findings from wave group experiments presented by Stansberg (1998), where the Hilbert envelope technique was used. As wave trains propagate further downstream, the wave highly deforms due to nonlinearity and eventually a breaking wave occurs in front. Meanwhile, the peak frequency shifts to lower frequency after wave breaking as depicted in figure 13. The above-mentioned phenomena can also be verified by the growth rate of evolving sidebands as shown in figure 10. For the cases of seeded wave trains, similar results are obtained with faster growth of wave amplitude under the same initial wave steepness.

### (c) Extreme wave

Wave trains propagating with the focus of energy on the strongly modulated wavefront due to sideband instability are summarized in earlier sections, whereby the transient large wave in front of the wave trains is observed during their evolution. The maximum wave crest in the transient wavefront is investigated up to the onset of incipient wave breaking and the quantitative features of transient large waves are presented in this section.

#### (i) Large transient wave on the wavefront

Based on the background analysis described in §2, the wave period of typical background noise existing in the experiment is less than 0.25 s. Herein, the measured data are low-pass filtered at a cut-off frequency of 4 Hz. Then, the local parameters of large transient wave on the unsteady wavefront are defined by the method proposed by Grue *et al*. (2003).

For initial uniform wave trains with and , the spatial evolution of local maximum wave parameters is shown in figure 14. In this case, the wave modulation increases with fetch until it is strong enough to initiate wave breaking at , according to the visual experiment. The wave trains then demodulate before they increase modulation again. At the fetch, , second breaking region is identified in the experiment. Although we did not show it here, it is found that the evolution of wave trains in shallow water with about the same initial wave steepness is slower than that in deep water.

The measured maximum transient waves for all the experimental runs are exhibited in figures 15 and 16 for local maximum wave steepness based on the wave amplitude and crest elevation, respectively. Note that the experimental data include initial uniform and imposed sidebands wave trains, in which the data cover a wide range of initial wave steepness. Clearly, the maximum local wave steepness increases rapidly with the increase of the initial wave steepness, and levels off at an initial wave steepness roughly equal to 0.16, despite the fact that the data exhibits little scattering. Practically, it is difficult to exactly identify the initiation of wave breaking in a physical experiment, especially, in a large wave flume. On the other hand, the wave gauge stations in the wave flume are not close enough to identify the breaking point accurately. Theoretically, even for the same initial wave steepness, each breaking wave could have different maximum wave steepness due to the different sideband frequencies, which either naturally evolves from initial uniform wave trains or initially imposed in the wave trains. The above-mentioned factors introduce scattering in the local maximum wave steepness. However, the tendency of maximum wave steepness distribution versus initial wave steepness is consistent, which provides useful information for the determination of the onset of wave breaking in deep water. It is interesting to point out that the wave steepness at wave breaking is lower than the corresponding criterion of Stokes wave, suggesting that the breaking criteria of modulated wave trains in deep water may be quite different from that of the uniform wave trains.

#### (ii) Large transient wave at quasi-steady state

Su & Green (1985) found that the ratio of the maximum wave amplitude, , to the initial wave amplitude, *a*_{c}, is not a monotonic increasing function of initial wave steepness, but has a maximum value around . Their experiments were conducted with initial uniform wave trains. Tulin & Waseda (1999) performed a series of experiments on the evolution of initial imposed sidebands wave trains and analysed both the local wave steepness and wave height at wave breaking. Generally, Tulin & Waseda's experimental data on the amplification factor of wave height are slightly larger than Su & Green's results, especially for the lower wave steepness.

Normalized horizontal particle velocity at the crest of wave breaking is plotted versus initial wave steepness in figure 17. Note that this particle velocity is calculated based on equation (2.9). In general, the ratio of horizontal particle velocity to phase velocity for a wave breaking in strongly modulated wave trains is approximately equal to 0.5, which is much less than the kinematic criterion of wave breaking () based on Stokes wave theory.

### (d) Initial growth rate and fastest growth mode

The amplitude spectra have been used as a measure of the growth of amplitude modulation on nonlinear wave trains. The initial growth rates of the naturally evolved sidebands from initial uniform wave trains are analysed by the method described in §2. A typical comparison of the initial growth rate of the sideband amplitudes between theoretical predictions and experimental data for initial wave trains, , is shown in figure 18. This result indicates that the sideband components, which meet the instability criteria, develop during the wave propagation. The initial growth rate of sideband components increases with normalized frequency difference between the carrier wave and sidebands until a maximum value is reached. The initial growth rate then decreases; the initial growth rates of the sideband amplitudes in the experiment consist of the prediction of Crawford *et al.* (1981) before the peak growth rates are attained. However, the theory over predicts the growth rate for an even larger normalized frequency difference.

From the investigation on the evolution of initial uniform wave trains, it is evident that a pair of specific sideband frequencies grows faster due to the nonlinear wave interaction through sideband instability. The naturally evolved fastest growth modes of the naturally evolved sidebands from our experiments together with previous experimental results are compared with those by several theoretical predictions as shown in figure 19. We note that the existing experimental results are confined in initial larger wave steepness due to the limitation of their experimental facility. Extended results on smaller wave steepness are provided by the present experiments. From figure 19, it is clear that Benjamin and Feir's prediction consistently overestimated the frequency difference between the carrier wave and the fastest growth mode for almost an entire range of wave steepness due to the assumption of infinitesimal wave steepness made in their theory. Besides, Dysthe's result, which is derived based on a narrow band assumption, is valid only for a wave steepness smaller than 0.15 and significantly underestimates the growth rate for larger wave steepness. Overall, the experimental results agree reasonably well with the prediction of Krasistkii's theory derived from the Zakharov equation and that of Longuet-Higgins solving the fully nonlinear equations.

## 4. Conclusions

Without a stable, sophisticated and programmable wave generator or a long enough flume, it is difficult to investigate the long-time evolution of nonlinear wave trains. Owing to these restrictions, only limited experimental studies are available since Benjamin & Feir (1967) and most of them did not show the complete story. To further picture the long-time evolution of nonlinear wave trains, a series of systematic experiments were conducted in a long-wave flume (300×5×5.2 m) at Tainan Hydraulics Laboratory. Several types of wave trains were initially generated including uniform, imposed small sidebands and detailed analyses of their evolution are given in the present study. The following main findings can be drawn from this study.

The evolution of a transient wavefront exhibits complicated and highly nonlinear wave trains, in particular, a strongly modulated wave group is observed as propagation of the wavefront, in which larger waves appear in front. Later, a series of intermittent wave breaking occurs at the large crest elevation. The wave group demodulates at post-breaking stage. Meanwhile, the second modulated wave group forms at the following wave trains. The mentioned modulation and demodulation processes appear periodically as the wave trains propagate downstream along the flume.

Our results also indicate that the sideband energies are initiated by the initial transient wave generation and enhanced with time and fetch until the critical value initiates wave breaking or reaches a maximum modulation.

The present experimental results provide much longer spatial evolution of wave trains compared to earlier researchers' results and reveal that the frequency downshift induced by wave breaking is not permanent, but partial recurrence of initial state of wave trains due to the wave breaking is an irreversible process.

The wavelet spectrum of a modulated wave train shows that both the frequency and amplitude modulate to high frequency on leading wavefront before wave breaking. However, the peak frequency shifts to lower frequency after wave breaking.

According to the experimental results, the breaking wave steepness in a modulated deep water wave is very different from that derived by Stokes, . Besides, the ratio of horizontal particle velocity to phase velocity at wave breaking in strongly modulated wave trains is approximately 0.5, which is much less than the kinematic criterion of wave breaking () based on Stokes wave theory.

From the investigation on the evolution of initial uniform wave trains, it is evident that a pair of specific sideband frequencies grows faster due to nonlinear wave interactions through sideband instability. In addition, our results show that the fastest growth modes of the naturally evolved sideband components and related initial growth rates of initial uniform wave trains match reasonably well with the predictions of Tulin & Waseda's calculation based on Krasiskii's theory.

## Acknowledgments

The authors gratefully acknowledge the financial support provided by the Ministry of Education, Taiwan for the Programme for Promoting University Academic Excellence under grant no. A-91-E-FA09-7-3. They also thank the research staff of Tainan Hydraulics Laboratory for their assistance to conduct elaborate experiments.

## Footnotes

- Received January 13, 2006.
- Accepted June 29, 2006.

- © 2006 The Royal Society