## Abstract

This paper concerns the crest height statistics arising in sea states that are broad banded in both frequency and direction. A new set of laboratory observations are presented and the results compared with the commonly applied statistical distributions. Taken as a whole, the data confirm that the crest-height distributions are critically dependent upon the directionality of the sea state. Although nonlinear effects arising at third order and above are most pronounced in uni-directional seas, the present data show that they are also important in directionally spread seas, provided the seas are sufficiently steep and not too short crested. The data also highlight the limiting effects of wave breaking. With individual breaking events dependent upon the local wave steepness, the directionality of the sea state again plays a significant role. Indeed, the present observations confirm that the two competing processes of nonlinear amplification and wave breaking can have a profound influence on the crest-height distributions leading to significant departures from established theory. In such cases, the key parameters are the sea state steepness and directional spread; the latter acting to counter the former in terms of nonlinear changes in the crest-height distributions.

## 1. Introduction

The description of extreme waves and their associated exceedance probabilities represents a key input for the design of all marine structures. For example, if a structure lies in the slender-body regime, the magnitude of the drag loads will be proportional to the square of the linear component of the incident crest elevations. As a result, the load statistics, describing the exceedance probability of a particular load, must be based upon an accurate description of the crest-height statistics. Furthermore, for a fixed structure, a step change in the magnitude of the applied loads will arise if the incident crest elevation lies above the underside of the deck structure. In such cases, wave-in-deck loading occurs, with implications ranging from significant local damage to total structural collapse. Given the difficulty (and cost) of designing a structure to withstand such loads, the common approach is to avoid their occurrence by maintaining an effective air gap; the latter ensures that the largest expected crest elevation lies below the deck elevation. Once again, the success of such an approach is critically dependent on the crest-height statistics.

In recent years, there has been much discussion of *freak* or *rogue* waves; the term applied to individual wave events that are abnormally large given the characteristics of the sea state in which they arise. Broadly speaking, this work can be divided into: (i) the analysis of field data, (ii) the application of numerical models, and (iii) laboratory investigations. An important aspect of this work has been to identify whether the most extreme waves are associated with modified physical processes causing them to lie on a different crest-height distribution. The inherent difficulty in this task is that the data of primary interest lie in the extreme tail of the distribution, corresponding to small exceedance probabilities, and these data are subject to the largest sampling variability.

Unfortunately, this issue can only be resolved through the provision of more data, and this is seldom possible in the context of field observations; the largest most severe sea states being limited in terms of the duration over which the sea state parameters may be assumed constant. Likewise, although numerical simulations can be extremely informative (see §2), the number of calculations that adequately model both the frequency bandwidth and the directional spread is limited. Furthermore, the steepest sea states will also be subject to the effects of wave breaking, and this is seldom (if ever) adequately modelled in numerically generated data. The third option concerns laboratory generated sea states. Although these are not without their difficulties, provided appropriate checks are in place to ensure that the sea state is ergodic and its spectral properties spatially homogeneous, long random wave records involving many tens of thousands of wave cycles can be generated. This paper describes exactly this type of study, it presents the resulting crest-height distributions, indicates their dependence on the spectral properties of the sea state, and compares the data with the commonly applied linear and second-order distributions.

The paper continues in §2 with a brief review of earlier work, highlighting the crest-height distributions to which the laboratory observations will be compared. For further discussion of the key background material, the reader is directed to the reviews provided by Massel [1] and Ochi [2]. The experimental apparatus and instrumentation employed in the present study are described in §3. Section 4 outlines the method of wave generation, while §5 provides a number of preliminary observations necessary to ensure that the generated wave fields adequately describe the desired sea states. Although these preliminary checks may appear extensive, they are essential to establish the validity of both the generated data and the comparisons that follow. The main experimental results are presented in §6; the primary purpose of these data being to identify any systematic departures from the established crest-height distributions and provide a physical explanation for them. Finally, §7 offers some concluding remarks and highlights the practical implications of the generated data.

## 2. Background

If a random sea state is assumed to be linear and narrow banded, the crest heights, *η*_{c}, will be Rayleigh distributed; the probability of exceedance is given by
2.1where *H*_{s} is the significant wave height. However, real seas are broad banded and nonlinear; individual waves exhibit sharper and higher wave crests with flatter and shallower wave troughs. To capture these effects, Tayfun [3] derived a second-order correction, again based on the assumption that the frequency spectrum is narrow banded. Alternatively, Forristall [4] simulated a large number of realistic sea states using the second-order random wave model of Sharma & Dean [5]. Using these data, Forristall [4] fitted a two-parameter Weibull distribution to define what is commonly referred to as a second-order distribution of wave crest elevations,
2.2where *α* and *β* are constants defined in terms of a mean steepness parameter and the Ursell number, with different relationships being given for uni-directional and directionally spread seas. More recently, Fedele & Arena [6] and Arena & Ascanelli [7] proposed second-order crest-height distributions; the former appropriate to uni-directional seas, irrespective of the spectral shape, and the latter to directional seas specified in terms of a Joint North Sea Wave Programme (JONSWAP) spectrum and a directional spreading function (DSF) [8].

In addition to the Rayleigh and second-order models, several authors have sought to describe the crest-height distributions in terms of the Gram-Charlier series. This was first proposed by Longuet-Higgins [9], with more recent contributions from Tayfun [10] and Tayfun & Fedele [11]. The benefits of this approach are that the surface cumulants, involving certain combinations of spectral moments, can be used to provide guidance as to the importance of the wave nonlinearities. However, the evaluation of these terms is usually based upon an analysis of the recorded water surface elevation and, in this case, the model ceases to be entirely predictive. Nevertheless, important physical insights have been achieved using this approach, and these are considered further in the sections that follow.

Comparisons between the commonly applied crest-height distributions (equations (2.1) and (2.2)) and field observations have been sought by several authors. Unfortunately, many of these comparisons are limited by the steepness of the sea states involved, by uncertainty involving the measurement technique and by the finite length of the data records [12]. However, recent contributions [13,14] have analysed very extensive databases. In the latter case, more than 220 000 h of wave records were considered, and a rigorous quality assurance procedure applied to the measured data. Although this led to many of the largest individual wave records being discarded, analysis of the most severe sea states (*H*_{s}≥12 m) identified clear departures from the predicted second-order distribution; evidence of this is provided in figure 1.

Alongside the analysis of field data, numerical calculations have also been used to quantify the crest-height distributions. For example, Gibson *et al.* [15] and Toffoli *et al.* [16] have recently undertaken numerical calculations and shown that at low exceedance probabilities, the second-order distribution under-estimates crest heights in uni-directional nonlinear sea states by as much as 20 per cent. These conclusions are further supported by the wave flume experiments reported by Onorato *et al.* [17]. However, in considering these results, it should also be noted that for nonlinear directional wave fields, Toffoli *et al.* [18] have shown that there is good agreement between the second-order distribution and numerical calculations based on nonlinear Schrödinger-type equations. However, these latter results do not relate to the steepest sea states, and this may account for the absence of nonlinear effects beyond second order. Other authors who have carried out similar work include Prevosto *et al.* [19] and Socquet-Juglard *et al.* [20]. In the latter case, the numerical calculations were based upon a JONSWAP spectrum with a fairly small directional spread and modest sea state steepness (, where *k*_{p} is the wavenumber corresponding to the spectral peak). Nevertheless, significant departures from the predicted crest-height distributions were observed.

As far as laboratory observations are concerned, numerous studies have been undertaken in uni-directional seas, and notable examples are given in earlier studies [17,21–23]. Provided the sea states are of sufficient steepness, these data highlight significant departures from the second-order predictions of Forristall [4], thereby confirming the importance of wave interactions arising at third order and above. Unfortunately, laboratory data describing the crest-height distributions arising in directionally spread seas are, by comparison, relatively rare. Nevertheless, in respect of long random wave records, recent contributions have been made by Onorato *et al.* [24] and Petrova *et al.* [25]. In the first example, two sea states with steepnesses of and 0.161 were investigated for a range of directional spreads. These data are important in that they provide clear evidence that the departures from second-order theory are strongly dependent upon the directional spread. In contrast, the second contribution identifies large departures from second order, but addresses the special case of crossing seas.

In addition to long random wave studies, several researchers have considered the occurrence of large deterministic wave events. Within a linear representation of a sea state, Lindgren [26], Boccotti [27] and Phillips *et al.* [28,29] define the most probable shape of a large linear wave as being proportional to the auto-covariance function of the wave spectrum. More recently, Arena *et al.* [30] describe the corresponding second-order solution in a directionally spread sea. In an experimental context, Baldock *et al.* [31] identified large departures from second-order theory in uni-directional focused wave groups. Likewise, Johannessen & Swan [32] identified similar effects in directionally spread seas, but noted that the magnitude of the departures reduces rapidly with increases in the directional spread. In addition, it was also noted that the onset of wave breaking, principally by spilling, was also dependent on the directional spread; larger non-breaking waves were generated in directionally spread seas. In a follow-up paper, Johannessen & Swan [33] combined experimental observations and numerical predictions to show that the dominant effect arising above second order relates to local changes in the wave spectrum, involving energy shifts to the higher frequencies. The influence of these changes on the directionality of a large wave event has been considered by Gibson & Swan [34] and Adcock *et al.* [35]; the former showing that such effects can be described in terms of the third-order resonant terms evaluated using the Zakharov [36] equation. Although these contributions provide considerable physical insight into the evolution of the largest waves, they do not immediately relate to the description of the crest-height distributions. The present paper will tackle this latter problem and, in so doing, will seek to build upon the physical understanding outlined above.

## 3. Experimental apparatus

### (a) Wave basin

The experimental study was undertaken in a directional wave basin located in the Hydrodynamics Laboratory at Imperial College London. This facility has a plan area of 20×10 m, operates with a maximum working depth of 1.5 m and is equipped with 56 individually controlled wave paddles mounted along its long axis. The wave paddles are dry-backed, flap-type machines, each 0.35 m wide and hinged 0.7 m below the still water level. The hydrostatic loads acting on the paddles are supported mechanically, and the drive system controlled numerically with active force-feedback absorption. The side walls of the basin are constructed from glass for maximum optical access, and the wave energy is dissipated on a parabolic beach extending 0.5 m below the still water level. Throughout the present tests, the wave basin was fitted with a rigid raised bed, maintaining a constant water depth of *d*=1.25 m over the entire plan area.

The design and primary purpose of this facility lies in the accurate generation of directionally spread waves. Earlier observations of Masterton & Swan [37] have shown that the paddles can accurately generate waves lying in the period range 0.3 s≤*T*≤3.3 s with propagation angles up to ±45^{°}; the latter limit imposed to avoid the generation of spurious error waves [38]. In addition, reflections from the downstream beach were typically shown to be less than 5 per cent. When combined with the success of the active paddle absorption, this ensures that both the spectral and the statistical properties of the generated wave fields are stable and uniform across the working area of the wave basin. Such conditions are essential to the success of the present study; evidence of the accuracy and stability of the generated sea states forming an important part of the preliminary data is presented in §5.

### (b) Instrumentation

Throughout this study, time histories of the water surface elevations, *η*(*t*), were recorded at a number of fixed spatial locations using resistance-type wave gauges. Each gauge consists of two 0.5 mm diameter tensioned wires, spaced 12 mm apart. These gauges cause no disturbance to the wave field, allowing the water surface elevation to be measured with an accuracy of ±0.5 mm. Data from each gauge were sampled at 128 Hz; the quality of the record being such that no post-processing (filtering) was necessary. A minimum of seven wave gauges were installed along the centre-line of the wave tank in each test case; the first gauge was located at a distance of *x*=3.75 m (or 3*d*) from the wave paddles. Established theory [39] confirms that this distance is sufficient to ensure that the recorded data will not be contaminated by evanescent wave modes generated at the wave paddles. The location of the seven wave gauges (numbered 1–7) are indicated on figure 2, with gauge 1 being located closest to the wave paddles. In a small sub-set of cases, the number of wave gauges was increased to 24, allowing the spatial evolution of the largest waves to be considered in detail.

## 4. Experimental method

The purpose of this study lies in the generation of long random wave records from which the crest-height statistics can be deduced and, specifically, any systematic departures from the established second-order theory identified. To achieve these goals and to ensure that the data are representative of conditions observed in the open ocean, the data will inevitably focus on the steeper sea states and must be generated such that the underlying linear wave components are well defined, even in those cases where the nonlinear evolution of the largest waves leads to significant change. The methodology outlined in this section details how this is achieved, while the preliminary data presented in §5 establishes the accuracy of the adopted procedures, both of which are fundamental to the success of this study.

### (a) Sea state specification

To ensure that the sea states generated in the wave basin are representative of realistic storm conditions, JONSWAP spectra [40] were applied throughout. Accordingly, the spectral density function, *S*_{ηη}(*ω*), is defined by
4.1where *ω* is the circular (wave) frequency given by *ω*=2*π*/*T*, *T* is the wave period, *ω*_{p} is the wave frequency corresponding to the spectral peak period *T*=*T*_{p}, *γ* is the peak enhancement factor, *α* is the Phillips parameter, *β*=1.25 and *σ*=0.07 for *ω*≤*ω*_{p} and 0.09 for *ω*>*ω*_{p}. For the chosen values of the shape parameters *σ*, *β* and *γ*, a clearly defined relationship exists between *T*_{p} and *H*_{s} [41]. Using this relationship, *α* can be adjusted to obtain the desired *H*_{s} for a given *T*_{p}. With realistic sea states typically being broad banded in respect of their frequency distribution, a value of *γ*=2.5 is commonly adopted in design practice, and has been widely applied in the present simulations.

Real seas are also directionally spread, with individual wave components propagating at an angle to the mean wave direction. This accounts for the short crestedness of a sea state, or the finite length of any individual wave crest, and is commonly represented by a normal distribution of the form
4.2where *D*(*θ*,*ω*) is the DSF, *A* is a normalizing coefficient and *σ*_{θ} is the standard deviation. Adopting this distribution, *σ*_{θ}=0^{°} corresponds to a uni-directional sea state, and severe storms are typically characterized by 15^{°}≤*σ*_{θ}≤30^{°} if *σ*_{θ} is assumed to be frequency independent [42]. Alternatively, if the directional spread is assumed to be frequency dependent, *D*(*θ*,*ω*) with *σ*_{θ}(*ω*), Ewans [43] suggests *σ*_{θ}≈22^{°} in the vicinity of the spectral peak, increasing to *σ*_{θ}≈45^{°} in the tail of the distribution. Within this study, both frequency-independent (*σ*_{θ}=0^{°}, 15^{°} and 30^{°}) and frequency-dependent [43] directional spreads have been adopted; comparisons between these cases allow the influence of directionality to be assessed.

### (b) Test cases, scaling and practical relevance

The full range of sea states considered in this study is outlined in table 1. The data presented in this table, and throughout the remainder of the paper, are given at laboratory scale. The generated sea states were all based on JONSWAP spectra (equation (4.1)), covering a broad range of *H*_{s}, *T*_{p} and *σ*_{θ}. Taken together, the sea states outlined in table 1 allow the separate effects of sea state steepness () and directionality to be assessed, with all other parameters held constant. Most importantly, the test conditions include sea states ranging from linear () to highly nonlinear (), the latter being heavily influenced by wave breaking, with directional spreads varying from uni-directional (*σ*_{θ}=0^{°}) to very short crested (*σ*_{θ}=30^{°}). Based upon the *k*_{p}*d* values, the sea states may be categorized as being at the deep end of the intermediate range (*k*_{p}*d*≈2.0), through to genuinely deep water (*k*_{p}*d*>3.0).

In considering the practical relevance of the sea states outlined in table 1, it is essential that they can be related to field conditions. Although the scales linking the laboratory and field conditions must be based upon Froude number similarity, the magnitude of the scaling is entirely arbitrary. However, if a length scale of *l*_{s}=1 : 100 is adopted, the corresponding time scale will be *t*_{s}=1 : 10. In this case, the full-scale equivalent peak periods lie in the range 12 s≤*T*_{p}≤16 s, with significant wave heights in the range 3 *m*≤*H*_{s}≤20 m. These conditions are closely related to commonly applied design conditions. For example, *T*_{p}=12–14 s and *H*_{s}=10 m would be representative of the 100 year storm conditions in the Southern North Sea, while *T*_{p}=16 s and *H*_{s}=15 m would correspond to the 100-year conditions in the northern North Sea or the Gulf of Mexico. Furthermore, *T*_{p}=16 s, *H*_{s}=20 m corresponds to a very severe storm, and would be representative of the 10 000 year storm associated with a tropical cyclone. On the basis of these calculations, it is clear that the sea states outlined in table 1 not only cover an appropriate parameter range, but are also relevant to commonly applied design conditions.

### (c) Calibration and wave generation

Having specified a number of target sea states (table 1), their accurate generation is dependent upon the paddle calibration and the method of wave generation. Taking each of these points in turn, the paddle calibration can either be empirical [37] or theoretical [44,45]; the latter approach has been adopted herein. In considering these approaches, it is important to stress that both represent an effective paddle calibration. In other words, they seek to ensure that the wave paddles generate the desired wave components. This is in marked contrast to a basin or facility calibration that seeks to achieve a given target spectrum at a specified location. If the sea state is linear and there are no unwanted (or spurious) wave reflections, the results of these two approaches will be identical. However, if the sea state is highly nonlinear, as is the case for several of the examples noted in table 1, the underlying linear wave components generated at the wave paddles will correspond to the desired JONSWAP spectrum, but the nonlinear interactions within the evolving sea state may lead to important spectral changes; the latter effects were discussed by Johannessen & Swan [32,33] and Gibson & Swan [34].

To provide the best possible representation of the random nature of a real sea state, the desired wave components were generated with both random phase and random amplitudes. To achieve the former, each wave component generated at the wave paddles was assigned a starting phase (*ϕ*) chosen randomly from a uniform distribution lying in the range 0≤*ϕ*≤2*π*. To achieve the latter, the amplitudes of the generated wave components were calculated based upon a weighted Rayleigh distribution following the discussion outlined by Tucker *et al.* [46]. In the case of directionally spread seas, it is also important that the sea state remains ergodic [47]. This is achieved by ensuring that each frequency component is generated in a single direction. Within the present tests, the direction of propagation of each wave component was assigned randomly based upon a weighted normal distribution reflecting the desired directional spread (equation (4.2)). This approach is very similar to that adopted by Waseda [48]; the preliminary data presented in §5 confirm the successful generation of the desired sea states.

### (d) Experimental procedure

When generating long random wave records, the software controlling the wave paddles is based upon a pre-determined repeat period. In the present study, this was set to 1024 s. On the basis of this value, individual wave components are generated at integer multiples of the fundamental frequency, *f*_{n}=*n*/1024 Hz, such that the resolution in the frequency domain is defined by Δ*f*=1/1024 Hz. Given the working range of the wave paddles, the spectra described in table 1 were generated using frequency components lying in the range 0.40625 Hz≤*f*≤1.875 Hz; the latter correspond to three times the spectral peak period of the longest wave components.

Having established the frequencies of the generated wave components, the amplitude, phase and direction of propagation of each component were defined as described earlier. Given the adoption of random amplitudes, together with the nonlinearity of many of the generated sea states, it was not possible to predetermine the exact value of the significant wave height, *H*_{s}, in any one random simulation. This arises because the paddle calibration defines the amplitudes of the freely propagating linear wave components generated at the wave paddles, and the sum of all of these components defines (on average) the underlying JONSWAP spectrum. In contrast, the actual recorded wave heights are dependent upon any nonlinear wave evolution/interactions arising in the wave basin; these latter effects represent an important part of the sea state we seek to investigate. Nevertheless, the measured crest statistics must be related to the target *H*_{s} values. To overcome this difficulty, an iterative procedure was adopted in which the *H*_{s} value was calculated at each measuring location and the input wave amplitudes linearly scaled such that the average *H*_{s} was in good agreement with the target value; the same scaling was applied to each random simulation (see below).

Having defined the experimental procedure, it is important to note that with the assumed scaling of *l*_{s}=1 : 100, a laboratory sample of 1024 s duration represents an equivalent full-scale sample of approximately 3 h. To ensure that the crest statistics are adequately defined, a total of 20 random simulations (or seeds) were undertaken for each test case; each seed was based upon a different set of random amplitudes, phases and directions of propagation. This corresponds to the equivalent of 60 h at full scale, and involves a minimum of some 15 000 individual wave events for each sea state, more for cases involving the smaller spectral peak periods. This was sufficient to define the statistical distributions arising at low exceedance probabilities (10^{−4}) without the need for extrapolation.

## 5. Preliminary results

Before discussing the core laboratory data, it is important to recognize that there are a number of key issues that have the potential to adversely affect the data analysis and any conclusions that can be drawn. Since the nature of the tests involves the repeated generation of long random records, the stability (and repeatability) of the generated wave conditions and the occurrence of spurious wave reflections are fundamentally important. In addition, it is clear from earlier work [24,32] that the directionality of a generated sea state will be important in determining both the extent of any nonlinear amplification and the limiting effects of wave breaking. It therefore follows that if the present data are to be relevant to the characterization of real seas, the directional spread must be representative and accurately generated. Preliminary data addressing each of these three key points (stability, reflections and directionality) are briefly discussed below. These data correspond to both deterministic focused wave events and long random wave simulations. The former are produced by pre-determining the phasing of the wave components such that constructive interference (or the superposition of wave crests) occurs at one point in space and time, producing a large isolated wave event. Such waves are representative of the largest waves arising in a given sea state [26,27], and are useful in the present context in that they allow the accurate identification of the generated wave components and the presence of any reflected wave modes. In contrast, the long random wave records are more difficult to interpret, but directly relevant to this study.

### (a) Focused wave groups

Figure 3 provides data relating to a focused wave event based upon a JONSWAP spectrum with *T*_{p}=1.2 s, a linear amplitude sum of mm and a directional spread of *σ*_{θ}=30^{°}. Figure 3*a* shows the time history of the water surface elevation, *η*(*t*), at the focal location; figure 3*b* shows the transverse variation in the water surface at the instant of wave focusing, *η*(*y*), and figure 3*c* shows the amplitude spectrum, *a*_{n}(*ω*). In figure 3*a*,*b*, comparisons are made between two generations of the same wave event taken several days apart and the predictions of the second-order random wave theory of Sharma & Dean [5]. In figure 3*c*, the amplitude spectra are defined using data recorded in a single simulation involving three repeat periods. In this case, three successive focused wave events were generated; the amplitude spectrum associated with each event was compared with both linear and second-order predictions. Taken as a whole, these data confirm that the generated wave conditions are entirely repeatable and in good agreement with expected theoretical results; the amplitude, phasing and direction of propagation of the generated wave components are very close to their target values. In particular, there is no evidence of error waves associated with the second-order sum and difference terms and, most importantly, no build-up of low-frequency error waves because of repeated reflections within the wave basin.

### (b) Random wave records

Although the data presented in §5*a* are important, they do not fully represent the long random wave records on which this study relies. To address this point, figures 4–6 provide some initial analysis of exactly these data. Figure 4 concerns the wave spectra, *S*_{ηη}(*ω*), recorded in the wave basin; figure 4*a* relates to a linear sea state defined by *H*_{s}=0.10 m, *T*_{p}=1.6 s and *σ*_{θ}=15^{°}, and figure 4*b* relates to a nonlinear sea state defined by *H*_{s}=0.20 m, *T*_{p}=1.6 s and *σ*_{θ}=15^{°}. The data presented on figure 4 define the average wave spectra recorded at three locations corresponding to gauges 1, 4, 7 and provide comparisons with the (linear) JONSWAP spectrum. In considering these data, it is important to note that the average is taken across the full 20 (3 h) simulations and is required because of the adoption of random amplitudes. In both cases, there is good agreement between the spectra recorded at the three gauge positions. This confirms the homogeneity of the sea states and provides further evidence that wave reflections do not represent a significant issue. Comparisons with the linear JONSWAP spectrum also show very good agreement. In particular, it is clear from the semi-log plots that while the mechanically generated frequency spectra is truncated at 3*ω*_{p}, incorporating more than 99 per cent of the total wave energy, the high-frequency tail (*ω*>3*ω*_{p}) is consistent with expectation; the energy within this region decays according to *ω*^{−5}, as discussed by Tucker & Pitt [49].

Given the potential importance of the directional spread, additional preliminary observations were undertaken to address the spread of the generated wave components. This analysis was undertaken in two parts. First, the surface elevation data corresponding to a near-linear sea state () was recorded at a large number of spatial locations and the maximum-likelihood method (MLM) presented by Young [50] used to determine the direction of propagation of individual frequency components within a single seed. Comparisons between these calculated directions and the corresponding target values gave a root-mean-square-error of ±1.5^{°} for frequency components close to the spectral peak. Although these results confirm the accuracy of the generated wave components in a linear sea state, a similar approach cannot easily be applied to a nonlinear wave field. In this latter case, a second approach involving a more conventional application of the MLM to define the directional distribution averaged across the generated frequency range and incorporating data recorded in all 20 seeds was applied. The results of this analysis (including an iterative improvement to remove inherent smearing) are presented in figure 5; figure 5*a* corresponds to a target spread of *σ*_{θ}=15^{°} and figure 5*b* corresponds to *σ*_{θ}=30^{°}. In both cases, the measured spread is shown to be in very good agreement with the target value.

Finally, figure 6 provides a sample set of crest elevation data arising from 20 random simulations. These data correspond to a sea state defined by *T*_{p}=1.6 s, *H*_{s}=0.1 m and *σ*_{θ}=15^{°}, but are representative of the wide range of data that will be presented in §6. In this case (and all subsequent cases), data were recorded at the seven gauge positions noted in figure 2*a*, and the individual crest heights identified using an up-crossing analysis. Figure 6*a* concerns the data recorded at a single gauge (gauge 2); the 20 crest-height distributions (one from each sample realization) are presented as a non-dimensional crest height, *η*_{c}/*H*_{s}, plotted against its exceedance probability, *Q*. As expected, the distributions show some variability; the spread of the data becomes larger for smaller values of *Q*. However, since all of these data relate to a single measuring location and the same underlying sea state, the crest height data arising from the 20 individual realizations (each involving different random amplitudes, phasing and directions of propagation) can be treated as independent parts of a single distribution. Merging and re-ordering the data allow the crest-height distribution to be presented to a very small exceedance probability, *Q*<10^{−4}; the present data give rise to the single black line in figure 6*a*.

Figure 6*b* also shows the crest-height distributions in 20 sample realizations and contrasts the data recorded at each of the seven gauge locations. The agreement between these distributions confirms that the generated sea state is homogeneous, thus providing further evidence that wave reflections do not substantially influence the working area of the wave basin. Indeed, taken altogether, the data presented in figures 3–6 confirm that the calibration of the wave paddles, the methodology underpinning the generation of the various sea states, and the performance of the wave basin generally, are such that wave conditions representing the target (underlying) linear sea states can be successfully generated.

## 6. Experimental data

With the purpose of this study being to highlight any systematic departures from the commonly predicted crest-height distributions, figures 7–9 present the data recorded at gauge 2 relating to a wide range of sea states. All these data relate to sea states with *T*_{p}=1.6 s giving an effective water depth of *k*_{p}*d*=2.0 and hence correspond to the deep end of the intermediate range. Figure 7 addresses a number of uni-directional (*σ*_{θ}=0^{°}) seas, whereas figures 8 and 9 concern the additional effects of directional spreading; the former corresponding to *σ*_{θ}=15^{°} and the latter *σ*_{θ}=30^{°}. Each figure provides a number of sub-plots relating to different sea state steepnesses, and the range extends from linear () to highly nonlinear () sea states; substantial wave breaking is observed in the latter case. In each sub-plot, the non-dimensional crest heights, *η*_{c}/*H*_{s}, based upon 20 (3 h) simulations recorded at a single location (gauge 2), are plotted against their probability of exceedance, *Q*, and compared with the Rayleigh and the Forristall distributions (equations (2.1) and (2.2), respectively).

Comparisons between the measured data and the model predictions highlight a number of important trends; variations within a single figure highlight the effect of the sea state steepness, and comparisons across the three figures 7–9 highlight the role of directional spreading. Figure 8*a* concerns a linear sea state () and is representative of equivalent cases generated with other directional spreads. In this case, there is little difference between the Rayleigh and Forristall distributions, confirming the linearity of the sea state, and the measured data are in very good agreement with these predictions. Indeed, the only departures from the model predictions occur in the extreme tail of the distribution, involving the three or four largest wave crests. This simply reflects the statistical uncertainty associated with these points, or the fact that they may represent events with an even smaller probability of exceedance. The only way this latter issue can be resolved is through the generation of substantially more data, which from an experimental perspective, this ceases to be a viable option.

Figure 8*b* concerns a weakly nonlinear sea state with a steepness of . In this case, the second-order increase in the crest elevation becomes more substantial and, as a consequence, the measured data are in close agreement with the Forristall model, both showing marked departures from the Rayleigh distribution. In the tail of this distribution, the measured data lie above the Forristall predictions. However, given the uncertainty that exists in this region of the distribution, no firm conclusions can be drawn without considering the sampling variability attributed to the measured data. This has been calculated using the method proposed by Tayfun & Fedele [11] and is indicated on figure 8*b* using the grey lines. Based on these comparisons, it is clear that the data presented on figure 8*b* are in very good agreement with the second-order theory, except in the extreme tail of the distribution where small (additional) increases in the crest elevation are observed; the likely explanation for these is wave nonlinearity arising at third order and above.

Figure 7*a* provides a similar set of comparisons relating to a uni-directional sea state (*σ*_{θ}=0^{°}), in which the sea state steepness is again defined by . At large exceedance probabilities, the measured data are in good agreement with second-order theory. However, with increases in the crest elevation, the departures from the second-order distribution become larger in magnitude, and are sustained over a wider range of exceedance probabilities when compared with the directionally spread case (*σ*_{θ}=15^{°} in figure 8*b*). By contrast, figure 9*a* also relates to a sea state in which , but the directionality is increased to *σ*_{θ}=30^{°}. In this case, the departures from second-order theory are notably smaller. Indeed, they are only apparent in the extreme tail of the distribution and, given the size of the sampling variability within this region, it is difficult to draw any conclusions. Nevertheless, comparisons between figures 7*a*, 8*b* and 9*a* clearly indicate that the directionality of the sea state has an important role in any amplification of the crest heights above the second-order distribution.

In figure 8*c*,*d*, the sea state steepnesses are increased to and 0.122, respectively. As a result, the second-order increase in the crest elevation becomes larger and, most importantly, the measured data describe a clearly identifiable trend lying above the second-order predictions. In considering figures 8*c* and particularly 8*d*, it is important to note that these increases are not restricted to a small number of individual wave events lying in the tail of the distribution. As a result, the departures from second-order theory cannot be discounted on the basis of the expected sampling variability. Indeed, in figure 8*d*, the additional nonlinear amplification in the crest elevation is comparable in size to the second-order increase above the linearly predicted Rayleigh distribution. While amplifications of this magnitude are not unexpected, the fact they occur in a directionally spread sea is surprising.

Figure 8*d* also incorporates a second dataset (denoted by triangles) relating to a sea state with very similar non-dimensional parameters (*k*_{p}*d*>*π*, and *σ*_{θ}=19^{°}). These data were reported by Buchner *et al.* [51] and based upon observations in the offshore basin at the Marine Research Institute Netherlands (MARIN), which is one of the largest facilities of its kind worldwide. The agreement between these data and the present observations are important in two respects. First, it confirms that the present observations are unaffected by the horizontal dimensions of the Imperial College wave basin. Second, the method of wave generation employed in the MARIN basin is based upon the single summation method outlined by Miles & Funke [52] and is thus very different to that described herein. The fact that two different methods of wave generation, adopted in two different facilities using very different wave makers and operating at different scales, produces near-identical results represents an important validation of the present data.

Although the effects of directional spreading will be further considered below, it is important at this stage to draw an initial comparison between figures 7*b*, 8*d* and 9*b*. These cases relate to sea states with *T*_{p}=1.6 s, and directional spreads of *σ*_{θ}=0^{°}, *σ*_{θ}=15^{°} and *σ*_{θ}=30^{°}, respectively. As expected, the largest nonlinear amplification beyond second order occurs in the uni-directional sea (figure 7*b*). In this case, the departures from second order begin at a relatively large exceedance probability (*Q*≤0.1), growing in size as the crest elevation, and hence the steepness of the individual waves increases. Interestingly, in the tail of the distribution, there is some evidence that the size of the amplification begins to reduce; the data tending back towards the second-order distribution. This is an important effect that will become clear in subsequent cases. With the introduction of directionality, *σ*_{θ}=15^{°}, figure 8*d* suggests that the nonlinear amplifications beyond second order reduce, particularly for *Q*≥0.01. Nevertheless, significant amplifications remain. In contrast, further increases in the directional spread (*σ*_{θ}=30^{°} in figure 9*b*) lead to a rapid reduction in the additional nonlinear contribution; the measured data lye very close to the predicted second-order crest elevations.

Given the data presented in figure 8*a*–*d*, further increases in the sea state steepness might be expected to produce progressively larger departures from the second-order distribution. Interestingly, this does not appear to be the case. Indeed, the data suggest that there is a second (competing) mechanism that limits the maximum nonlinear increase in the crest elevation. For example, figure 8*e* presents data relating to and confirms that although the nonlinear amplification beyond second order is clearly defined, its relative contribution reduces for *Q*<0.01. Further evidence of this second effect and its influence on the crest-height distributions is provided in figure 8*f*. This corresponds to a sea state steepness of and provides clear evidence of an initial (small) amplification of the crest heights above second order. However, this rapidly reduces such that the crest heights corresponding to *Q*≤0.01 are consistently smaller than the second-order predictions. Indeed, in this case, the largest crest heights are reduced to values approaching the linear Rayleigh distribution; the latter are some 10–15% smaller than the corresponding second-order predictions.

Having considered both the characteristics of the sea state in which these reduced crest heights arise, and the nature of numerous individual wave events, this second mechanism is believed to be associated with the occurrence of wave breaking, both spilling and over-turning. Further evidence of the importance of this effect and its dependence on directionality is provided in figures 7*c*,*d* and 9*c*. It has already been noted that the uni-directional cases exhibit the largest nonlinear amplification. Figure 7*c*,*d* also suggests that these sea states are strongly influenced by wave breaking. Given that both the nonlinear amplification (including effects beyond second order) and the occurrence of wave breaking are dependent upon the local (wavefront) steepness; this result is to be expected. Indeed, based on the steepness arguments alone, the nonlinear amplification and the onset of wave breaking will be largest in the uni-directional waves, reducing with increasing directional spread. The data presented in figures 7*c*,*d*, 8*e*,*f* and 9*c* confirm this trend; figure 8 establishes the potential importance of nonlinear effects beyond second order in some directionally spread seas, and figure 9 suggests that such effects become much less significant in very short crested seas (*σ*_{θ}=30^{°}).

Figures 10–12 provide an alternative, more compact, presentation of the crest height data, facilitating comparisons between the various sea states. In these plots, the *x*-axis defines the exceedance probability, *Q*, and the *z*-axis defines the crest elevation normalized with respect to the linear or Rayleigh predicted value, *η*_{c}/*η*_{L}. Adopting this approach, values of *η*_{c}/*η*_{L}>1 indicate a nonlinear amplification, while comparisons between the measured data and the second-order predictions identify effects arising at third order and above. Figure 10 reconsiders the data recorded in sea states with a spectral peak period of *T*_{p}=1.6 s; the three sub-plots address directional spreads of (*a*) *σ*_{θ}=0^{°}, (*b*) *σ*_{θ}=15^{°} and (*c*) *σ*_{θ}=30^{°}. Comparisons between these cases highlight the relative importance of the two competing mechanisms; the first represents a nonlinear amplification of the crest heights due to wave interactions arising at third order and above, while the second concerns the limiting effects of wave breaking. In figure 10*a*,*b*, both mechanisms are immediately apparent and are shown to be critically dependent upon the sea state steepness. In considering these cases, it is also clear that both effects are more pronounced in the uni-directional sea states (figure 10*a*); evidence of this is provided by both the magnitude of the amplifications arising at low probabilities of exceedance and the point at which the ratio *η*_{c}/*η*_{L} achieves its maximum value. In contrast, the directional data presented in figure 10*c*, corresponding to *σ*_{θ}=30^{°}, show little evidence of appreciable amplification beyond second order. However, the limiting effects of wave breaking continue to be relevant in the steepest sea state.

Further evidence of the importance of wave steepness is given in figure 11*a*. This superimposes the normalized crest-height distributions, *η*_{c}/*η*_{L}, for three sea states corresponding to *H*_{s}=0.15 m, *σ*_{θ}=15^{°} and spectral peak periods of *T*_{p}=1.6, 1.4 and 1.2 s. These examples correspond to , 0.156 and 0.210, respectively; the latter being the steepest sea state observed in this study. These distributions confirm that the nonlinear amplifications, both in total and the component arising beyond second order, increase with the sea state steepness. However, the limiting effect of wave breaking exhibits a similar dependence. As a result, the largest nonlinear amplification across the full range of exceedance probabilities does not occur in the steepest sea state. Figure 11*b* also considers three sea states with *σ*_{θ}=15^{°} and *T*_{p}=1.6, 1.4 and 1.2 s, but in this case, the *H*_{s} values have been adjusted to maintain a constant sea state steepness, . In this example, the normalized crest-height distributions, *η*_{c}/*η*_{L}, are in close agreement; amplification effects beyond second order are clearly defined, but there is no evidence of the limiting effects of wave breaking. The agreement between these cases also suggests that as far as the crest-height distributions are concerned, all the present wave cases are effectively propagating in deep water.

To further investigate the effects of directional spreading, figure 12 superimposes the non-dimensional crest-height distributions, *η*_{c}/*η*_{L}, for *σ*_{θ}=0^{°}, *σ*_{θ}=15^{°} and *σ*_{θ}=30^{°}. All of these data relate to a spectral peak period of *T*_{p}=1.6 s; the three sub-plots address steepnesses of (figure 12*a*), 0.122 (figure 12*b*) and 0.163 (figure 12*c*). In figure 12*a*, the uni-directional data exhibit the largest nonlinear amplification, including effects beyond second order. In contrast, the two directionally spread sea states exhibit little or no departures from second-order theory and, when plotted in this non-dimensional form, describe very similar distributions. Indeed, given the associated sampling variability, there are no practical differences between these measured distributions. At this stage, it is important to note that the data presented on figure 12*a* are very similar to the crest-height distributions proposed by Toffoli *et al.* [18], with the latter being based upon numerical simulations using the Euler equations. On the basis of these calculations, Toffoli *et al.* [18] concluded that while wave nonlinearities beyond second order could be significant in uni-directional seas, they were unlikely to be important in directionally spread seas.

The data presented in figure 12*b*, corresponding to , add to this important practical discussion. In this case, the uni-directional data again exhibit the largest amplification. However, appreciable amplification also arises in the *σ*_{θ}=15^{°} case. In contrast, the *σ*_{θ}=30^{°} distribution remains very close to the second-order predictions. These results confirm that directionally spread distributions can exhibit significant amplifications beyond second order, but that the extent of any amplification is critically dependent upon the directional spread.

One likely explanation for this lies in the steepness of the largest waves. Adopting the description of a large linear wave outlined by Phillips *et al.* [28], figure 13 contrasts three representative wave profiles, *η*(*x*), arising in the target JONSWAP spectra (*T*_{p}=1.6 s, *σ*_{θ}=0^{°}, 15^{°} and 30^{°}) used to generate the random sea states addressed in figure 12. In considering these profiles, the crest elevations have been normalized to 1.0, and an increase in the directional spread is observed to produce both an increase in the wavelength and a corresponding reduction in the wave height (*H*); the latter arises because the adjacent wave troughs are less deep. If the wave steepness is defined in terms of *η*_{c}*k* or , figure 13*b* indicates how the steepness of a large linear wave (normalized with respect to the uni-directional wave steepness) varies with directional spread. Based upon these calculations, the introduction of a *σ*_{θ}=15^{°} spread leads to a 3.8 per cent reduction in *η*_{c}*k* and a 4.1 per cent reduction in . In contrast, the introduction of a *σ*_{θ}=30^{°} spread produces a 15 per cent reduction in *η*_{c}*k* and a 17 per cent reduction in . Although these results are at best indicative, it is clear that the magnitude of the directional spread has a significant effect on the wave steepness. With the nonlinear wave interactions arising beyond second order dominated by third-order effects and hence proportional to , it is to be expected that nonlinear changes in the crest-height distributions will be strongly influenced by the directional spread.

A second reason for the importance of directionality lies in the random nature of the sea states. With the target directionality applied to the sea state as a whole, individual waves will exhibit a varying directional spread; evidence of this is provided by the varying crest lengths. It therefore follows that in a sea state with a small directional spread, there will be a higher probability of observing a large wave that is unusually long crested when compared with a sea state with a large directional spread. Since such waves experience a larger nonlinear amplification, it is to be expected that the distribution of crests heights will be heavily influenced by the directional spread.

When considering figure 12*b*, it is also interesting to note that the uni-directional data exhibit a clearly defined maxima in *η*_{c}/*η*_{L} at *Q*≈10^{−3}; further reductions in *Q* lead to reduced *η*_{c}/*η*_{L} ratios. In contrast, the *σ*_{θ}=15^{°} distribution exhibits no clearly defined maxima, suggesting that the limiting effects of wave breaking are less significant in this directionally spread sea. Again, this is consistent with the earlier discussion of wave steepness.

In figure 12*c*, the crest-height distributions for each of the three sea states are markedly different. For high probabilities of exceedance (*Q*>0.05), the *σ*_{θ}=0^{°} and 15^{°} sea states include a substantial amplification, whereas the *σ*_{θ}=30^{°} case is closer to the second-order distribution. However, for smaller exceedance probabilities, the trend of *η*_{c}/*η*_{L} reverses, indicating a progressive reduction in the total nonlinear contribution. This occurs in each of the three sea states, although a reversal in the *σ*_{θ}=30^{°} case is less well defined and occurs at smaller exceedance probabilities, indicating the reduced influence of wave breaking.

In considering the steepest sea states, in which the occurrence of wave breaking is significant, the definition of the sea state adopted in the laboratory study is such that the two processes influencing the crest-height distributions, nonlinear amplification and wave breaking, cease to be independent. The explanation for this, and its relevance to field data, is described as follows. In defining the target sea states, *H*_{s} represents a key parameter; evidence of its importance was provided in figure 10. However, if a generated sea state involves a large proportion of breaking waves, with the majority involving localized white capping or spilling, the dissipation of wave energy may be such that *H*_{s} is less than the target value. In such circumstances, the input amplitude of all the generated wave components is increased until the target *H*_{s} is achieved. As a result, the non-breaking waves arising within the sea state are associated with larger generated waves components, effectively belonging to a more severe sea state that would have been characterized by a larger *H*_{s} had it not been for the dissipative effects of wave breaking. It therefore follows that the occurrence of large non-breaking waves, in excess of those predicted by second-order theory, will be due to the combined effects of higher-order nonlinear amplification (arising at third order and above) and the distortion of the distribution due to the occurrence of wide spread wave breaking. In many sea states, the occurrence of occasional wave breaking will have no impact on the crest-height distributions. However, in the most severe sea states, this effect should not be ignored and may contribute to larger than expected ‘higher-order’ amplifications arising at relatively modest exceedance probabilities.

The arguments outlined above have been explained in the context of the present laboratory study. However, they are equally appropriate to measured field data. In this latter case, the concept of a target *H*_{s} has no relevance. However, there is a measured *H*_{s} that is used to characterize the sea state. If wide spread wave breaking is present, this will be smaller than might otherwise have been the case. Evidence of the importance of this effect is readily observed in shallow water crest-height statistics. In this case, the composite Weibull distribution proposed by Battjes & Groenendijk [53] is based upon a fit to measured data and describes substantially larger wave heights at high exceedance probabilities when compared with other theoretical distributions. This increase is, in part, due to the occurrence of wave breaking. In the context of this study, the potential importance of this effect should not be discounted. However, with an on-going study of deep water wave breaking, the quantification of this effect will be addressed in a subsequent paper.

Having established the importance of the directional spread, it should be acknowledged that the analysis of field data [8,43,54] suggests that the directional spread is typically found to be frequency dependent. With this study seeking to provide a fundamental understanding of the role of directionality, a variety of directional spreads have been examined. However, these have been applied uniformly across the frequency range (§2). To ensure that the present conclusions concerning the amplification of crest heights above the second-order predictions are equally applicable to field data, figure 14 contrasts the non-dimensional crest-height distributions, *η*_{c}/*η*_{L}, corresponding to three directional spreads: *σ*_{θ}=15^{°}, *σ*_{θ}=30^{°} and Ewan's (frequency-dependent) spreading. All of the data presented in figure 14 relate to a sea state steepness of , so that the only change concerns the applied directional spreading. In adopting Ewan's spreading, the recommended fit to the field data has been applied; the frequency components in the vicinity of the spectral peak, *ω*_{p}, are spread by *σ*_{θ}≈22^{°}, and those in the tail of the distribution are spread more widely, *σ*_{θ}≈45^{°}. Comparisons between the recorded crest-height distributions (figure 14) show that the data relating to Ewan's spreading lies approximately mid-way between the *σ*_{θ}=15^{°} and *σ*_{θ}=30^{°} cases. The conclusions that can be drawn from these data are twofold. First, nonlinear amplifications beyond second order can occur in sea states based upon the best possible representation of the (frequency-dependent) directional spreading. Second, the characteristics of the directional spread in the vicinity of the spectral peak appear to exert a controlling or dominant influence. This latter result is hardly surprising given that the majority of the wave energy resides in this region.

Finally, it is important to stress that the emphasis of the present study lies in the crest-height distributions. The fact that nonlinear effects beyond second order do not produce significant increases in the crest elevation for large directional spreads does not necessarily imply that effects arising at third order and above will not influence other local wave properties. In particular, having established the importance of wave breaking, the horizontal fluid velocity arising high in the wave crest is clearly a key parameter; earlier work by Johannessen & Swan [33] and Adcock *et al.* [35] discusses the importance of high-order nonlinearities in this regard.

## 7. Conclusion

A new experimental study involving the generation of long random wave records has allowed the investigation of crest height statistics in a number of realistic deep water sea states. The generated conditions cover a wide range of spectral peak periods, significant wave heights and directional spreads. In particular, the sea state steepness varies from linear to highly nonlinear, the latter involving significant wave breaking, with the directionality ranging from uni-directional to very short crested. In each case, sufficient data have been gathered to quantify the crest heights arising at small (10^{−4}) exceedance probabilities and the results compared with established models.

While much of the recorded data are in broad agreement with the second-order crest-height distribution proposed by Forristall [4], systematic departures lying well outside the normal sampling variability are observed in the steeper sea states. Although nonlinear effect occurring beyond second order are expected to be significant in uni-directional seas, the present data also confirm their potential importance in directionally spread seas. This is of considerable practical relevance, not least because it suggests that commonly adopted design procedures may be non-conservative in some sea states.

Most importantly, the present data highlight the significance of the directional spread. If a sea state is very short crested, nonlinear increases in the crest elevations above second-order predictions are unlikely to be large. However, as directionality reduces, significant amplifications can arise; the departures from second-order theory are comparable in size to the difference between linear and second-order theory. In the most severe sea states, the limiting effects of wave breaking also become important. As a result, the largest departures from existing crest-height distributions do not necessarily occur in the most severe sea states or at the smallest exceedance probabilities. Indeed, in some sea states, the influence of wave breaking is such that the largest crest heights arising in the tail of the distributions are smaller than the predicted second-order values. With wave breaking, both spilling and over-turning, which is dependent upon the local wave steepness, the directionality of the sea state is again a key criterion; the maximum non-breaking crest height increases with directional spread. The data obtained in this study have highlighted the significant changes to the crest-height statistics that can arise due to the competing influences of nonlinear amplification and wave breaking; the nature of any changes is strongly dependent upon the directional spread.

- Received November 26, 2012.
- Accepted January 11, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.