## Abstract

In deep water it is well known that the evolution of the largest waves in realistic, broad-banded frequency spectra is governed by dispersive focusing. However, as the water depth reduces this process weakens and the relative significance of wave modulation is shown to be increasingly important. This leads to very different extreme wave groups, the properties of which are critically dependent upon the local nonlinearity. To explore these effects, and to provide a physical explanation for their occurrence, two complementary wave models are employed. The combined numerical results show that the nature of large uni-directional waves varies depending on the relative water depth. As the water depth reduces, both the *bound* and *resonant* interactions become more significant. However, the third-order *resonant* terms have the most profound influence. By modifying both the amplitude and phase of the underlying linear wave components, the largest waves arise as a local instability within a truncated quasi-regular wave train; the latter appearing because of an initial narrowing of the underlying frequency spectrum. Furthermore, the numerical calculations show that, with large changes in both the spectral shape and the phasing of the wave components, both the maximum crest elevations and wave heights are less than those predicted by linear theory.

## 1. Introduction

This paper concerns the description of large non-breaking waves in intermediate and shallow-water depths. An accurate representation of these events provides a key input to the design of many coastal and offshore structures, particularly in terms of calculating the applied fluid loads, the possible occurrence of wave slamming, and the potential extent of any over-topping. The essence of the paper concerns the importance of the reduced water depth in uni-directional seas; it seeks to determine whether the consequent weakening of the dispersive properties of a wave field leads to a fundamental change in the nature of large wave events. In particular, it will establish whether recent advances in the description of large deep-water waves, not least the local and rapid energy transfers leading to large departures from established wave solutions (Gibson & Swan 2007), are equally appropriate to intermediate and shallow-water depths, and whether they lead to different and unexpected wave events.

In deep water the largest waves do not arise as part of a regular wave train, but occur as transient events in random and directionally spread wave fields owing to the focusing of the underlying or freely propagating wave components. In the present context the term freely propagating refers to wave components that satisfy the linear dispersion equation1.1where *ω* is the wave frequency or 2*π*/*T* where *T* is the wave period, *k* is the wavenumber or 2*π*/*λ* where *λ* is the wavelength, *d* is the water depth and *g* the gravitational acceleration. The focusing of these components produces a large transient wave crest that rapidly disperses in both space and time. Historically, these events have been observed in long time-domain simulations, with each wave component being allocated a random phase and amplitude (Forristall 1978) and calculations undertaken until a large wave event (or ideally several) arises.

Alternatively, if it is assumed that the phase velocity of the wave components is dependent upon linear dispersion, Lindgren (1970) and, more recently, Tromans *et al.* (1991) have described the expected or most probable shape of a large wave as being proportional to the auto-correlation function of the underlying wave spectrum. This approach, hereafter referred to as the *NewWave* model following Tromans *et al.* (1991), removes the need for long time-domain simulations and has been validated, subject to some nonlinear corrections, by comparison with field observations (Jonathan *et al.* 1994). The purpose of the present paper is, in part, to determine whether similar procedures can be adopted in intermediate and shallow-water depths.

To achieve the goals outlined, §2 provides a short review of the importance of nonlinearity in the evolution of large waves; particularly, recent evidence of local and rapid energy transfers leading to significant spectral change in the vicinity of an extreme wave event. Section 3 briefly outlines two nonlinear wave models appropriate to the evolution of realistic (broad-banded) sea states; their application in water of finite depth being described in §4. The main results are presented in §5 together with a physical explanation of why a reduced water depth has such a profound effect on the evolution of the largest waves. The paper concludes with §6, highlighting the practical implications of the present work and identifying areas for future study.

## 2. Background

Both Baldock *et al.* (1996) and Johannessen & Swan (2001) provide laboratory observations of large deep-water waves suggesting that there may be significant and rapid changes to the underlying or freely propagating wave spectrum, *S*_{ηη}(*ω*), in the vicinity of an extreme wave event. As a result, a linear solution based upon a constant underlying spectrum (similar to that adopted in the *NewWave* model and many other design-oriented solutions) will be inappropriate. In particular, *bound* interactions, arising at second order and above, and *resonant* interactions, arising at third order and above, must be taken into account. In the present context, *bound* waves are phase-locked to the underlying linear wave components and do not therefore satisfy the dispersion relation given in equation (1.1). In earlier work, notably Hasselmann (1962), these *bound* waves were referred to as steady-wave interactions, their amplitudes being entirely dependent upon the interacting free waves and therefore fixed. In contrast, *resonant* interactions involve the growth of new freely propagating wave components. These were referred to by Hasselmann (1962) as unsteady interactions: three (or more) freely propagating wave components interacting to produce a fourth which is also freely propagating and which continues to grow in size. As a result, the *resonant* interactions cause a redistribution of energy within the linear wave spectrum, altering both the amplitude and phase of the underlying linear wave components.

In respect of the *bound* wave interactions, Longuet-Higgins & Stewart (1960) first considered the interaction of two freely propagating wave components, identifying both the frequency-sum (high-frequency) and the frequency-difference (low-frequency) terms arising at second order. In subsequent work, Longuet-Higgins (1963) extended the analysis to include directionality. In applying these results to realistic wave spectra, Sharma & Dean (1981) summed up the interactions arising from every possible pair of free waves (assuming that the latter can be identified), producing a second-order directionally spread wave model appropriate to a range of water depths.

In contrast, Phillips (1960) provides the first investigation of *resonant* interactions; so named because they have the mathematical form of a linear resonator. In their simplest form these arise at a third order of wave steepness, O(*a*^{3}*k*^{3}); the interactions between three-wave components produce a fourth (hence the term four-wave interactions) that satisfies the linear dispersion equation. In this case, the wave component arising from the interaction is freely propagating and energy is exchanged between the various components. Further discussion of these important effects are given in the review articles of Yuen & Lake (1982), Hammack & Henderson (1993) and, in the context of wave breaking, Banner & Peregrine (1993). *Resonant* interactions account for both the instability of a regular wave train commonly observed in a laboratory flume (Benjamin & Feir 1967), although strictly speaking this is an example of *near resonance*, and the long-term evolution of a wave spectrum (Hasselmann 1962, 1963*a*,*b*). In this latter case the energy transfers occur in a random sea and are characterized as being extremely slow, a small percentage of the total energy being transferred over hundreds of wave cycles. In contrast, Johannessen & Swan (2003) and Gibson & Swan (2007) showed that, in a deep-water-focused wave group, with the phasing far from random but the surface profile believed to be representative of an extreme wave event, the local nonlinearity is such that the third-order *resonant* interactions can be responsible for highly localized and very rapid energy transfers. In this case, significant energy transfers can occur over 5–10 wave cycles immediately preceding the extreme wave event. The present paper will consider whether similar effects are equally appropriate to the evolution of extreme waves in intermediate and shallow-water depths.

## 3. Wave modelling

The numerical calculations presented within this paper are based on two independent wave models. The first, due to Bateman *et al.* (2001, 2003), is hereafter referred to as BST. This is a fully nonlinear wave model, capable of describing the evolution of extreme waves in realistic frequency spectra, which has been extensively validated by comparisons with the laboratory observations of Baldock *et al.* (1996) and Johannessen & Swan (2001). The second is based on the Krasitskii (1994) Hamiltonian form of the wave evolution equation first proposed by Zakharov (1968); hereafter referred to as ZE. Although the implementations of these two wave models have some similarities, the information and understanding they provide is complementary. For example, although BST is a fully nonlinear wave model, essentially providing an exact solution to a specified wave problem, it provides no physical insight as to the relative importance of the wave interactions involved. Indeed, the nature of the interactions remains unclear; the model simply provides a description of the water surface elevation, *η*(*x*,*t*), and the corresponding velocity potential, *ϕ*(*x*,*η*,*t*). In contrast, ZE (or at least its Hamiltonian form proposed by Krasitskii) is of limited nonlinearity, having only been expanded to a fourth order of wave steepness, O(*a*^{4}*k*^{4}). However, its over-riding advantage lies in the fact that the wave interactions arising at progressively higher orders can be specifically identified. By using both wave models the full nonlinearity of the wave field can be addressed and, at the same time, important understanding as to the nature of the physical processes uncovered.

In choosing to apply these particular wave models, it is important to stress that the focus of the study concerns the nonlinear evolution of realistic sea states involving a broad range of frequency components. As such, solutions based upon the nonlinear Schrodinger equation, the Davey–Stewardson equation or the Kadomstev–Petviashvili equation are inappropriate given their assumed narrow-bandedness.

### (a) Wave model 1: BST

A full description of this wave model is given in Bateman *et al.* (2001, 2003), with further information concerning the application of the model to realistic wave spectra given in Bateman *et al.* (submitted). Although this is a fully three-dimensional solution, capable of describing directionally spread seas, the present application concerns uni-directional waves. As such, the applied wave model is closely related to the original contribution of Craig & Sulem (1993). With the fluid assumed to be inviscid and incompressible, and the motion two-dimensional and irrotational, the velocity vector , where *ϕ*(*x*,*z*,*t*) is the velocity potential and (*x*,*z*) are the usual cartesian co-ordinates in which *x* is measured horizontally in the direction of wave propagation and *z* is measured vertically upwards from the mean water level. Within this description the governing field equation, representing mass continuity, is given by Laplace’s equation,3.1This is valid throughout the fluid domain, bounded by a horizontal bed at *z*=−*d* and the water surface at *z*=*η*(*x*,*t*), and can be solved subject to the usual boundary conditions.

If the bed is assumed to be impermeable, the normal velocity must be zero,3.2After some rearrangement, the nonlinear free-surface boundary conditions can be expressed as3.3and3.4Equation (3.3) describes the required kinematic condition and equation (3.4) describes the dynamic condition; the former ensuring that the water surface is a streamline and the latter that the pressure on the water surface is a constant. In considering these equations, it is important to note that the right-hand side of each equation is entirely dependent upon spatial descriptions.

The methodology underpinning the BST model, and many other related solutions, is thus one in which series solutions for *η* and *ϕ* are adopted, the latter satisfying both equations (3.1) and (3.2). Using these descriptions, a (known) spatial representation for both *η* and *ϕ* is defined at some initial time (*t*=*t*_{0}) and the free-surface boundary conditions time-marched to determine *η* and *ϕ* at all subsequent times.

Assuming that the wave motion is periodic over some large fundamental wavelength, *λ*_{x}, then *η*(*x* + *λ*_{x})=*η*(*x*) and *ϕ*(*x* + *λ*_{x})=*ϕ*(*x*). Within this periodic domain, *η* and *ϕ* can be defined by3.5and3.6where *A*_{k} and *a*_{k} are functions of time only and, within a Fourier system, *k* is an integer multiple of the fundamental wavenumber so that *k*=*nk*_{0}, where *k*_{0}=2*π*/*λ*_{x} and *n* is an integer. Although equations (3.5) and (3.6) could be applied directly in equations (3.3) and (3.4), a dimensional reduction is sought to maximize the efficiency of the numerical procedure. This requires the entire solution to be based on surface values alone: the water surface elevation, *η*(*x*,*t*), and the velocity potential on that surface, *ϕ*(*x*,*η*,*t*)=*φ*(*x*,*t*).

However, in adopting this approach, attention needs to be paid to the calculation of ∂*ϕ*/∂*z* on *z*=*η* in equations (3.3) and (3.4). Craig & Sulem (1993) overcame this difficulty by adopting a Taylor series expansion of a Dirichlet–Neumann operator. BST adopts a very similar approach in which the expansion is referred to as a ‘G-operator’ and is defined by3.7Using this definition and noting that the time derivative of the velocity potential on the surface is given by3.8where the second term on the right reflects the motion of the water surface, the free-surface boundary conditions can be expressed as3.9and3.10In adopting this approach the required dimensional reduction is achieved, both *η* and *φ* can be expressed as Fourier series and, most importantly, the evaluation of the surface derivatives in equations (3.9) and (3.10) can be rapidly achieved using fast Fourier transforms. This latter point is fundamentally important since it ensures that the computational effort increases as , where is the number of surface points or twice the number of wave components. This is in marked contrast to the relationship that arises when a model is based upon a global potential, *ϕ*, the solution of which requires large matrix inversion (e.g. Fenton & Rienecker 1982; Johannessen & Swan 1997). In truth, the efficiency of the BST model is of primary importance when it is employed in fully three-dimensional computations, since this is essential to ensure adequate resolution in both the frequency and the directional domains. However, given the length of the calculations undertaken in the present study (§4), the efficiency of the BST model proved very convenient. Nevertheless, it is important to stress that the present calculations could have been undertaken with other efficient uni-directional wave models; notable examples include Dold & Peregrine (1984) and Dommermuth *et al.* (1988).

In applying the BST model, all the present calculations were undertaken with a total of 256 wavenumber components, the first seven terms evaluated in the Taylor series expansion of the G-operator, and a time step of Δ*t*=0.05 s; the latter corresponding to *T*_{p}/200. Previous calculations (Bateman *et al.* submitted) have shown that these modelling parameters are consistent with accurate predictions of a highly nonlinear (near-breaking) uni-directional wave field; the maximum variation in the total wave energy being less than 0.01 per cent after more than 40 000 time steps. In addition, separate calculations undertaken with 512 wavenumber components produced near-identical results; the variation in the maximum recorded crest elevation being 0.05 per cent.

### (b) Wave model 2: ZE

The Zakharov equation (ZE) is an integro-differential equation that can describe the evolution of a broad-banded, directionality spread, wave field. A derivation of this equation, valid to a fourth order of wave steepness and preserving the Hamiltonian structure of the formulation, is provided by Krasitskii (1994). In addition, Annenkov & Shrira (2001) describe its numerical implementation, Rasmussen & Stiassnie (1999) consider the important effects of discretization, and Shemer *et al.* (2001) demonstrate that the equation is in good qualitative agreement with laboratory data concerning the evolution of both a bimodal and a Gaussian wave spectrum. A brief explanation of ZE, following the derivation of Krasitskii (1994), is given below. To avoid any confusion regarding the nature of the kernels describing the various interactions, the full directionality of Krasitskii’s solution has been retained at this stage. The main purpose of this explanation is to contrast the formulation with that adopted in BST, to highlight the physical significance of the various terms, and to clarify the advantages of using this model.

With the fluid again assumed to be inviscid and the fluid motion irrotational, Zakharov (1968) defines an evolution equation for gravity waves propagating in water of constant depth, *d*. Expressed in Hamiltonian form3.11where *δ* are functional derivatives, *η*(**x**,*t*) and *φ*(**x**,*t*) are as defined previously, and *H* is the Hamiltonian or total energy defined as the sum of the kinetic and potential energies divided by the fluid density. Within this solution, the velocity potential must satisfy Laplace’s equation (3.1) and the bottom boundary condition (equation (3.2)), but the kinematic and dynamic boundary conditions (equations (3.3) and (3.4)) are not required since they are taken into account in the Hamiltonian formulation. Introducing a Fourier representation for *η*(**x**,*t*) and *φ*(**x**,*t*)3.12and3.13where **k** is the wavenumber vector, integration with respect to **k** is extended over the entire k-plan, an asterisk denotes a complex conjugate, and the explicit dependence of *η* and *φ* on *t* is suppressed for simplicity of notation. Since a Fourier transformation is canonical, equation (3.11) reduces to3.14Adopting a further canonical transformation, a new pair of canonically conjugate variables, and , can be defined as follows:3.15
3.16where and ; the latter representing the linear dispersion relation given in equation (1.1). Using these new variables, equation (3.14) reduces to a single equation3.17where *H* is a functional of and . Expanding *H* into an integral power series in powers of and , Krasitskii (1994) expresses the evolution equation (3.17) to fourth order. Up to and including terms at third order, this is reproduced as follows:3.18In presenting equation (3.18) Krasitskii introduces a compact notation in which the arguments **k**_{j} in , *ω*, *U*^{(n)} and *V*^{(n)} and the *δ*-functions are replaced by subscripts *j*; the subscript zero is assigned to **k**. For example, , *ω*_{j}=*ω*(**k**_{j}), and *δ*_{0−1−2}=*δ*(**k**−**k**_{1}−**k**_{2}); the value of the *δ*-function is equal to 1 if the argument is zero (in this case **k**−**k**_{1}−**k**_{2}=0) and zero otherwise. For differentials d**k**_{0}=d**k**, d**k**_{012}=d**k** d**k**_{1} d**k**_{2}, etc. and the integral signs denote corresponding multiple integrals with limits from to . Within equation (3.18), the kernels *U*^{(n)} and *V*^{(n)} are known functions of wavenumber and water depth. This integral power series consists of both *bound* interactions that do not alter the underlying linear spectrum and *resonant* interactions that do.

Equation (3.18) could be time marched to model the evolution of the spectrum. This would include both the *bound* and the *resonant* interactions. However, as it is only the *resonant* interactions that alter the underlying spectrum, it is possible to reduce this equation by transforming the variables *a*(**k**) and *ia**(**k**) to the canonically conjugate variables *b*(**k**) and *ib**(**k**),3.19The various *kernels* *A* and *B* are now exclusively the *bound* interactions, and, hence, the evolution of the underlying linear wave components can be expressed entirely in terms of the *resonant* interactions (expressed below to third order)3.20where *B*(**k**,*t*)=*b*(**k**,*t*)e^{iω(k)t}.

Equation (3.20), expressed in the form given by Annenkov & Shrira (2001), describes the evolution of an underlying linear wave component in terms of third-order *resonant* interactions, . It can be time marched using a standard Runge–Kutta algorithm and then the surface profile recreated at each time step to the desired order by re-incorporating the *bound* interactions of equation (3.19). However, the essential benefit of this approach is that it is possible to choose which interactions are included and which are neglected, and, hence, to isolate the physical processes (or wave interactions) that control the evolution of the wave field.

In an earlier study, Gibson & Swan (2007) applied the ZE model to describe the evolution of a number of deep-water wave cases. In outlining the theory they also demonstrated the validity of comparisons between the BST and ZE models. These results provide important background material to the present study and, in a small number of cases, direct comparisons between the earlier deep-water results and the present intermediate depth studies will be used to demonstrate the importance of a reduced water depth.

## 4. Applications in water of finite depth

All of the cases presented within this paper concern the evolution of large uni-directional waves within a realistic JONSWAP spectrum. In this case, the spectral density function *S*_{ηη}(*ω*), defining the distribution of wave energy across the frequency domain, is given by4.1where *ω* is the wave frequency or 2*π*/*T* where *T* is the wave period, *ω*_{p} is the frequency of the spectral peak corresponding to *T*_{p} the peak period, *α*=0.0081, *β*=1.25, *σ*=0.07 for and 0.09 for *ω*>*ω*_{p}, and *γ* is the peak enhancement factor. In the present calculations the intermediate wave case arises in a water depth of *d*=15 m, with *ω*_{p}=0.628 rad s^{−1} (corresponding to *T*_{p}=10 s) and *γ*=2.5. In the comparative deep-water cases, but all other parameters remain constant.

With the fundamentals of the models explained in §3, the only difficulty that remains is the specification of the initial conditions. In essence, this process sets out to establish some initial time, (*t*=*t*_{o}), well in advance of the focal event, at which the wave energy is fully dispersed across the computational domain. In the absence of any large wave events, the wave field at this initial time can be specified using either a linear or a second-order (Sharma & Dean 1981) wave model. With the initial phasing of the wave components specified such that, according to linear dispersion, a focused wave event arises at a specific location in both space and time, the initial conditions are time marched using the models outlined in §3.

In deep water this process is relatively straightforward: frequency dispersion remains strong and, as a result, the calculations do not have to go too far back in time to identify appropriate initial conditions. However, as the water depth reduces, the dispersive processes are weakened, equivalent to a reduction in the spectral bandwidth, and the initial conditions need to be more remote from the focal time. However, there is now a compromise to be made in the sense that, if calculations go back too far in time, the slow *resonant* interactions discussed by Hasselmann (1962, 1963*a*,*b*; see §2), will become apparent and the observed nonlinearity will no longer reflect the desired underlying spectrum. In contrast, if calculations do not go back sufficiently far, the wave field will not be fully dispersed, leading to larger waves in the initial conditions. These may be inadequately represented by either a linear or a second-order solution causing the evolution of spurious error waves that may be incorrectly interpreted as nonlinear wave–wave interactions. To ensure that the present calculations both relate to the desired underlying wave spectrum, specified in equation (4.1), and avoid the occurrence of error waves, the choice of the initial start time must be carefully considered. Data relating to this choice are presented in figures 1 and 2. These results confirm that, provided the numerical calculations are initiated using a second-order model (Sharma & Dean 1981) with *t*_{0} lying in the range −120*T*_{p}≤*t*_{0}≤−70*T*_{p}, the evolution of the largest wave event is independent of the chosen start time, *t*_{0}.

Figure 1*a* concerns the maximum crest elevation, , arising anywhere within the computational domain for a sequence of times preceding the focal event, where the latter occurs at *t*=0. The data are based on simple linear calculations and directly compare the intermediate depth case (*d*=15 m) with the corresponding deep-water case. In both examples reduces with increasing (negative) times, but the differences in the dispersive properties are clearly noted by the rate of reduction: the deep-water case is more dispersive and hence describes smaller values of for all times *t*<0. Figure 1*b*,*c* presents a related sequence of data; the former corresponding to deep water and the latter intermediate. In each case the time variation in the second-order correction to the maximum crest elevation, expressed as a percentage of the linearly predicted , is described on the right-hand axis. Earlier deep-water work (Johannessen & Swan 2003) suggests that, provided this percentage is less than 2 per cent, initial conditions defined by a simple linear solution were adequate. Unfortunately, this approach was based upon simplified laboratory spectra that were both relatively narrow-banded and truncated at the high-frequency end. As a result, this criterion appears not to work in either deep or intermediate water depths. More recently, Bateman *et al.* (submitted) suggested that, provided the second-order increase is less than 3 per cent, initial conditions based on a second-order model are appropriate. In the present example, this occurs in deep water (figure 1*b*) for *t*_{0}<−550 s. Unfortunately, the intermediate depth data (figure 1*c*) suggest that the second-order correction remains surprisingly high (approx. 7%), even though the absolute value of eventually becomes asymptotic to the deep-water value (figure 1*a*).

Clearly, these results suggest that a different criterion for the specification of the initial conditions is required in intermediate and shallow water. To overcome this difficulty, figure 1*b*,*c* also describes the time variation in the maximum crest elevation, , on the left-hand axis. However, in these cases the results are based on fully nonlinear calculations, as opposed to the linear computations compared in figure 1*a*. In figure 1*c* it is important to note that the nonlinear remains almost constant over the range −2000 s ≤ *t*_{0}≤−700 s. Given these results several numerical runs were initiated with different start times, lying in the range −2000 s ≤ *t*_{0} ≤ −550 s, and with the initial conditions specified using a second-order model. In each case the calculations were continued up to and beyond the point at which the largest wave occurs. Although these results were conspicuously different from the linearly predicted events, they were remarkably similar; the variation in both the maximum crest elevation and the maximum wave heights was less than 1 per cent at the location of the extreme event. These results suggest that, for this case, provided *t*_{0} ≤ −700 s, giving at least 70 peak periods of evolution, and the initial conditions specified in terms of a second-order model, the extreme wave event is effectively independent of the initial conditions. The only exception to this arises if the evolution becomes very long (*t*_{0}<−120*T*_{p}), in which case the slow interchange of energy associated with the Hasselmann-type (unsteady) interactions begins to become apparent. This, however, does not imply that the solution is incorrect, but merely that the extreme event is associated with a slightly different underlying wave spectrum. Figure 2 also considers the importance of the initial start time, *t*_{0}, contrasting the extreme wave profiles associated with different start times lying within the range −100*T*_{p} ≤ *t*_{0} ≤ −55*T*_{p}. The agreement between these results confirms that the calculated results are indeed independent of the chosen *t*_{0}. In the results that follow, all of the numerical calculations were initiated at *t*_{0}=−850 s with a second-order wave input. In each case the phasing of the wave components was adjusted to create a linearly focused event at *x*=*t*=0.

## 5. Discussion of results

### (a) Wave evolution

Figure 3 concerns the intermediate (*d*=15 m) wave case and contrasts the evolution of the wave field predicted by the fully nonlinear BST wave model with the equivalent linear predictions. In both cases the evolution of the water surface elevation is described by a sequence of spatial histories, *η*(*x*), at various times up to and including the occurrence of the largest wave event. The nonlinear results are presented in figure 3*b*; the equivalent linear predictions, at identical times, are presented in figure 3*a*. The only exception to this is the final pair (or bottom row), which defines the spatial profile with the largest crest elevation, , and therefore occurs at different times. In both calculations the input amplitude sum corresponds to m, where *a*_{n} is the amplitude of the *n*th wave component and *N* is the total number of wave components; the latter corresponds to *N*=256 wavenumber components calculated at 2*N* spatial locations.

If this were a deep-water wave case, the laboratory observations of Baldock *et al.* (1996) and the numerical calculations of Johannessen & Swan (1997, 2003) show that, although the nonlinear effects may be very important, they lead to a modification of the underlying linear process (frequency dispersion) rather than its wholesale replacement. Evidence of this is provided by the fact that although the focal position (*x*_{f},*t*_{f}) may occur earlier in both space and time, *x*_{f} and *t*_{f}<0, and the maximum crest elevation may be larger than the linearly predicted maximum, , the process of dispersive focusing remains dominant; the overall wave group structure becomes progressively more compact until such a point that a focal event, or a near-focal event, arises corresponding to a global maximum in the water surface elevation.

In contrast, figure 3 shows that with a reduction in the water depth the evolution of the wave field may be fundamentally different from that driven by linear dispersion. Moreover, the extreme wave event corresponding to the largest crest elevation no longer resembles a focused wave event and, as such, is at odds with the predictions of the *NewWave* model. Indeed, figure 3 suggests that wave modulation rather than dispersive focusing becomes the driving mechanism. From a very early stage (*t*=−450 s) *resonant* or *near-resonant* interactions, since these and these alone are able to modify the freely propagating wave components, create a quasi-regular wave pattern not previously present within the wave field. This propagates, almost without change, until the modulation between closely spaced frequency components, or side-band instabilities, leads to the rapid generation of a large wave event appearing at the front of an elongated wave group. This view is supported by the fact that over the final 450 s (or 45 peak periods) the wave group shows almost no evolution and, certainly, no evidence that it is becoming progressively more compact.

The difference between the deep and intermediate wave cases is further investigated in figure 4. This concerns the evolution of a large wave event and describes the change in the amplitude spectrum, *a*(*k*), where *a* is again the amplitude of each wave component and *k* its corresponding wavenumber. Figure 4*a* concerns a deep-water case (*A*=5 m) and shows that, as the wave field evolves from its initially dispersed condition at *t*=−850 s to the extreme event at *t*=−18.2 s, the amplitude spectrum progressively broadens particularly in the later stages. Evidence of this is provided by a reduction in the peak spectral value, together with a spread of wave energy to the higher wavenumber components. In contrast, figure 4*b* concerns the intermediate water case (*d*=15 m) and shows that the evolution of the amplitude spectrum involves a two-stage process. First, there is an initial narrowing of the wave spectrum with a corresponding increase in the peak spectral value. This is followed by a local broadening involving the development of notable side-bands either side of the spectral peak. This two-stage development is further clarified in figure 4*c*,*d*; the former contrasts the input amplitude spectrum (*t*=−850 s) with that arising some 40 wave periods later at *t*=−450 s. In this case, the narrowing of the amplitude spectrum is clearly apparent and accounts for the development of the quasi-regular wave train identified in figure 3. This latter spectral shape effectively remains constant until eight wave periods immediately preceding the evolution of the largest wave event. During the second stage of development the amplitude spectrum broadens with energy transferred to both the second-order sum and difference terms, as well as the growth of significant side-band components; the latter appears to be closely related to the evolution of the largest wave which arises as a modular instability of the quasi-regular wave train. Details of this second stage are highlighted in figure 4*d*, which contrasts the amplitude spectra at *t*=−100 s and *t*=−19.2 s, where the latter corresponds to the occurrence of the largest wave event.

In considering these results, particularly the evolution of a large individual wave event owing to a modular instability at the front of an elongated wave group, it is relevant to note that recent work by Janssen & Onorato (2007) has shown that in uni-directional waves the classical Benjamin–Feir instability ceases to exist for values of *kd*<1.36, where *k* is the wavenumber and *d* is the water depth. In the present calculations the effective water depth may be characterized by *k*_{p}*d*=0.86, where *k*_{p} is the wavenumber corresponding to the spectral peak. Although this is smaller than the limit noted above, the present calculations relate to a broad-banded (JONSWAP) wave spectrum. As a result, appreciable wave energy exists at *ω*>1.5*ω*_{p}, for which the corresponding *kd* value will be substantially larger than the limit noted above. The fact that a broad-banded wave spectrum evolves differently from established theory based upon a narrow-banded assumption is (perhaps) not surprising. Indeed, it emphasizes the importance of adopting the wave models described in §3, both being appropriate to broad-banded problems.

Before attempting to explain the differences in the evolution of the largest waves, particularly its dependence on the water depth, it is relevant to consider the role of changes in the spectral bandwidth. In a focused wave event, the linearly predicted height of the highest wave crest is defined by the amplitude sum , where again *a*_{n} is the amplitude of the *n*th component and *N* is the total number of wave components. Within the computational domain the total wave energy *E*, which must remain constant, is proportional to the sum of the squares of the component wave amplitudes, . If, for simplicity, the amplitude sum, *A*, is spread uniformly over the *N* wave components, *a*_{n}=*A*/*N*, then it follows that . In this case, the amplitude sum of the spectrum is proportional to the square root of the number of components over which the amplitude is spread. It, therefore, follows that, as a spectrum becomes more broad-banded (*N* increases), its amplitude sum increases giving the potential for larger maximum crest elevations. Conversely, if the spectral bandwidth reduces, so will its amplitude sum and hence the potential .

### (b) Bound-wave interactions

To explain the evolution of the largest waves (figure 3), and the corresponding spectral changes (figure 4), it is instructive to isolate the effects of the *bound* and *resonant* wave interactions. This can be achieved using the ZE model outlined in §3. With reference to equations (3.19) and (3.20), the inclusion of the kernels (with *n*=1,2,3) in equation (3.19) reproduces the second-order solution of Sharma & Dean (1981); the *A*^{(1)} and *A*^{(3)} kernels represent the ‘frequency-sum’ or (*k*_{1} + *k*_{2}) terms, while the *A*^{(2)} kernel describes the ‘frequency-difference’ or (*k*_{1}−*k*_{2}) terms. In contrast, the inclusion of the kernels in equation (3.19) defines the third-order *bound* waves. Comparisons between these solutions and the fully nonlinear model of BST are presented in figure 5. Figure 5*a*–*c* concerns the deep-water case: the first describes the amplitude spectrum *a*(*k*); the second describes the water surface elevation, *η*(*x*) at the instant of wave focusing, or when the largest crest elevation arises; and the third describes the contribution to the water surface elevation provided by the second- and third-order terms, the former being separated into the frequency-sum and frequency-difference terms. In contrast, figure 5*d*–*f* provides a similar sequence of figures relating to the intermediate depth case (*d*=15 m). In both cases the fully nonlinear predictions of *η*(*x*), denoted by BST in figure 5*b*,*e*, have been shifted so that the maximum crest elevation arises at *x*=0. This merely serves to facilitate comparisons.

In deep water, the amplitude spectrum, *a*(*k*) in figure 5*a*, shows that, while the second-order *bound* wave components account for some of the energy transfers to the higher wavenumbers (and hence higher frequencies), comparisons with the fully nonlinear BST model indicate significant additional transfers for *k*>0.05 rad m^{−1}. Furthermore, comparisons between the second- and third-order *bound* wave solutions confirm that the latter contribution is very small. Further evidence of this is provided in figure 5*b*,*c*. In terms of the resulting deep-water wave profile, figure 5*b* shows that, while the general shape of the wave group is similar to that predicted by linear dispersion, the fully nonlinear wave profile (BST) is higher and steeper than that predicted by linear, second- or even third-order *bound* wave solutions; the difference in terms of the crest elevation (see insert) is 17, 8 and 7 per cent, respectively. Furthermore, the fact that the maximum crest elevation calculated to third order is only marginally (1.5%) greater than that predicted at second order is consistent with the perturbation expansion on which the solution is based and suggests that the inclusion of yet higher order *bound* wave components will not explain either the increase in the maximum crest elevation or the steepening of the extreme wave profile. The relative magnitude of the *bound* wave components is further considered in figure 5*c*. This confirms that in deep water the *bound* waves are dominated by the second-order frequency-sum terms; their phasing is such that they add to the height and steepness of the linearly predicted wave form. The second-order frequency-difference terms act counter to these effects, but remain relatively small, accounting for only 30 per cent of the frequency-sum terms. Figure 5*c* also shows that, in deep water, the third-order *bound* waves are unimportant.

Figure 5*d*–*f* provides the equivalent intermediate water depth results. These indicate that, with a reduction in water depth (*d*=15 m), the relative magnitude of the *bound* wave components and their contribution to the resulting water surface elevation is fundamentally different. First, the amplitude spectrum, *a*(*k*) in figure 5*d*, confirms that the *bound* wave components are substantially larger. At second order this is equally true of the frequency-sum terms and, perhaps more importantly, the frequency-difference terms; the former lead to the growth of high wavenumber components and the latter low wavenumbers. The changes in the spectral shape owing to the third-order *bound* waves are also significant, accounting for both a small down-shifting of the spectral peak and the growth of high wavenumber components in the tail of the spectrum; the latter are largely accounted for by a reduction in the amplitude of the components lying between the spectral peak and its second harmonic (0.06 rad m^{−1} ≤ *k* ≤ 0.12 rad m^{−1}).

Although the *bound* wave components lead to a substantial modification of the amplitude spectrum (figure 5*d*), it remains markedly different from the fully nonlinear results based on BST. In particular, these latter results are characterized by smaller amplitudes in the tail of the spectrum, a narrowing of the spectral form in the vicinity of the peak, and the existence of marked side-bands at *k*=*k*_{p}±0.02, where *k*_{p} is the wavenumber of the spectral peak. Irrespective of the order of the terms included, none of these features of the amplitude spectra are reproduced by the *bound* wave models. Furthermore, their importance in respect of the water surface elevation, *η*(*x*), is clearly demonstrated in figure 5*e*. In this case, it is clear that the surface elevation corresponding to the largest crest elevation bears no resemblance to the focal event that would arise if the evolution of the wave field were governed by linear dispersion. Indeed, given that the initial phasing was specifically chosen so that such an event would occur, it is evident that other mechanisms are controlling the evolution of the wave field in shallower water. It is also important to note that, with the absence of wave focusing, the wave energy remains dispersed over a larger spatial area; evidence of this is provided by the significant wave amplitudes arising at *x*<−200 m corresponding to the quasi-regular wave train discussed in relation to figure 3. As a result, both the maximum crest elevation and, in particular, the maximum wave height are significantly less than those predicted by linear theory. Furthermore, the inclusion of second- and third-order *bound* waves leads to increases in the predicted crest elevation and hence further deviations from the fully nonlinear behaviour predicted by BST.

Although the *bound* wave solutions are clearly inappropriate in this intermediate depth (figure 5*d*,*e*), it is nonetheless instructive to consider the relative magnitude and phasing of the *bound* wave components and to contrast these with the equivalent deep-water results. Figure 5*f* concerns the intermediate depth case and provides these data in an identical format to the deep-water results presented in figure 5*c*. Comparisons between these results highlight two important effects. First, although the second-order increase in crest elevations is limited, both the second-order sum and difference terms are substantially larger than those arising in deep water, the former being three times larger and the latter almost eight times larger. However, the phasing of these components is such that, while the sum terms add to the crest elevation, the difference terms subtract. Accordingly, the second-order increase in the crest elevation should not be used as an indicative measure of the importance of the second-order terms in general. Second, the third-order *bound* waves are also substantially larger. Indeed, far from being negligible, as was the case in deep water, they are of a similar magnitude to the second-order sum and difference terms. In intermediate and shallow-water depths this raises important questions concerning the convergence of a series expansion truncated at low order, such as the second-order model of Sharma & Dean (1981). Furthermore, it also limits our ability to isolate the freely propagating wave components on the basis of the difference between focused and inverse-focused (*η*_{crest} and *η*_{trough}) wave groups (see below).

### (c) Resonant-wave interactions

The purpose of this section is to consider whether the third-order *resonant* wave interactions, described by the kernel in equation (3.20), may account for the hitherto unexplained evolution of the intermediate depth wave field. In previous work, notably Johannessen & Swan (2003) and Gibson & Swan (2007), third-order modifications to the free-wave regime corresponding to *resonant* or near-*resonant* interactions were identified by considering the difference between focused and inverse-focused wave groups, where the latter correspond to the focusing of wave troughs rather than wave crests and are simply achieved by inverting the input signal. If *η*_{c}(*x*,*t*) corresponds to the focusing of wave crests at *x*=*t*=0 and *η*_{t}(*x*,*t*) the focusing of wave troughs, it follows that if5.1then5.2where *f*, *g*, *h* and *l* are arbitrary functions. The difference between these records therefore allows the identification of the *odd*-order wave harmonics5.3while the sum of these records identifies the *even*-order harmonics5.4In deep water the third-order *bound* waves are shown to be negligible (figure 5) and consequently in equation (5.3) provides a good first approximation to the third-order *resonant* terms accounting for the modification of the free-wave regime. Alternatively, in equation (5.4) isolates the *even* harmonics which will be dominated by the second-order *bound* waves, both sum and difference terms.

In shallower water it has already been shown that the third-order *bound* waves may be appreciable (figure 5). As a result, it follows that the *odd*-order terms will not precisely equate to the free-wave regime. Nevertheless, given the fundamental changes that arise in the freely propagating wave components, not least the evolution of a quasi-regular wave train, it is instructive to compare the time evolution of the water surface elevation corresponding to the *odd*-order terms calculated from the fully nonlinear BST model with the sum of the linear and third-order *resonant* terms predicted by ZE. The purpose of these comparisons is to establish whether the growth of the third-order *resonant* terms may account for the significant departures from a linear model based on dispersive focusing.

Figure 6 concerns a spatial description of the water surface elevation, *η*(*x*), at a number of times covering the evolution of the wave field from its initial input at *t*=−850 s to the occurrence of the highest crest elevation at *t*=−19.2 s. Several of these times correspond to those previously investigated in figures 3 and 4. In each case the sum of the linear and third-order *resonant* terms, the latter identified by the kernel in equation (3.20), is shown to be in very good agreement with the *odd*-order terms of BST. Most significantly, the development of a truncated quasi-regular wave train, around *t*=−450 s, and its subsequent propagation without change of form until the occurrence of a large wave event owing to a modular instability at the front of the wave train are clearly noted and well described by the evolution of the new freely propagating wave components described in ZE. Indeed, these comparisons confirm that the very significant departures from linear theory, giving rise to a fundamentally different extreme wave event, are controlled by the third-order (four-wave) *resonant* interactions allowing the growth of new freely propagating wave components and hence permanent changes to the underlying wave spectrum.

In related work, Gibson & Swan (2007) compared the predictions of ZE with a number of deep-water wave cases, highlighting both the importance of local changes in the wave spectrum and the dominant role played by the third-order *resonant* terms. However, comparisons between this deep-water study and the present intermediate depth case identifies a number of significant differences. Several of these have already been addressed, but it is important to consider the extent to which the spectral changes (both narrowing and broadening) can be described by third-order changes to the free-wave regime and to contrast the results with equivalent deep-water cases. Figure 7*a* concerns a deep-water sea state and contrasts *a*(*k*) for the input JONSWAP spectrum (*A*=5.0 m) with the *odd*-order terms arising from BST and the freely propagating terms arising at first and third order in ZE; the latter description arises at the instant of wave focusing or the occurrence of the highest wave crest. In this deep-water case it is clear that the changes in the free-wave regime involve a movement of energy to the higher wavenumber components, accounting for both the increased crest elevation and wave steepness, and that these changes are well described by the *resonant* third-order wave interactions.

In shallower water (*d*=15 m) the changes in the free-wave regime are more complex: the amplitude spectra *a*(*k*) at the time of the largest wave crest (figure 7*b*) are dependent upon an initial narrowing of the spectral shape, corresponding to the evolution of a quasi-regular wave train, followed by a broadening which includes the development of distinct side-band instabilities. These separate stages are considered in figure 7*c*,*d*, respectively, in which third-order *resonant* interactions described by the kernel in equation (3.20) are shown to reproduce, at least in qualitative terms, both the initial narrowing and the subsequent broadening of the amplitude spectrum.

Although it is clear that the third-order *resonant* terms play a dominant role in the evolution of the intermediate depth wave field, the resulting amplitude spectrum (figure 7*b*) is less well defined than the corresponding deep-water case (figure 7*a*). This is undoubtedly because in shallower water the third-order *bound* waves may be appreciable. Unfortunately, these will be present in the *odd*-order terms isolated from the fully nonlinear BST calculations and therefore inhibit our ability to fully separate the freely propagating wave components.

### (d) Laboratory and field observations

With two very different sets of numerical calculations suggesting that the nonlinear evolution of large surface water waves in intermediate and shallow-water depths may be very different from that which occurs in deep water, it is interesting to consider whether such effects can be identified in either laboratory or field data. In a recent industrially funded research project concerning the limits on waves in shallow water, a large number of laboratory observations were made of uni-directional random or irregular waves propagating over gradual bed slopes . These observations were undertaken in a long purpose-built wave flume allowing the measurements to extend from relatively deep water to very shallow water. Figure 8*a* shows a typical wave shape associated with the highest crest elevations recorded in deep water (*kd*>2.0). In contrast, figure 8*b* shows a typical large wave event arising in shallower water.

More recently, a second joint industry project concerning extreme seas and their impact on offshore structures has considered field data from a wide variety of sources. In analysing these data each record was cut into 20 min intervals, the corresponding significant wave height (*H*_{s}) calculated, and the largest or most extreme wave events identified according to the criteria or . These criteria are commonly adopted as the definition of *freak* or *rogue* waves, but in the present study are merely used as a means to identify extreme waves. Having applied a rigorous quality control algorithm to remove those records involving spurious or physically unreliable data (typically spikes, lock-in or drop-down), more than 4000 individual wave records have been considered covering water depths from *d*=1000 to 7 m. Considering these records it is clear that the deep-water cases are typified by the data given in figure 8*c*, while the intermediate and shallow-water cases are more commonly represented by the profile given in figure 8*d*.

The laboratory and field data presented in figure 8 are in good qualitative agreement with the earlier numerical calculations: the deep-water records (figure 8*a*,*c*) are representative of focused wave events, their evolution governed by linear dispersion, while the intermediate and shallow-water wave events (figure 8*b*,*d*) are markedly different, suggesting the importance of modular instabilities. Indeed, the similarity between these records and the earlier numerical calculations is perhaps surprising given that the laboratory data involve waves propagating up a sloping bed (albeit gradual) and the field data inevitably involve directionally spread sea states; the latter are the subject of an ongoing study and a possible part II paper.

### (e) Spectral bandwidth: changes in the amplitude sum

Having provided insights into the nature of a large wave event, and its dependence on water depth, it is also relevant to question whether the evolution process has implications for the maximum crest elevation. It has already been noted that nonlinear wave interactions, giving rise to changes in the spectral bandwidth, can account for changes in the linear amplitude sum, *A*=*Σa*_{n}. In a ‘linear’ sense this defines the maximum crest elevation. Figure 9 contrasts a number of wave cases in which the amplitude sum is normalized by the corresponding input value, *Σa*_{n}/*A*_{input}. In a linear solution there can be no change in the spectral bandwidth and hence this non-dimensional value remains constant at 1.0. Any departures from this value reflect nonlinear changes in the spectral bandwidth, as described earlier. Figure 9*a* concerns three deep-water wave fields, each based on the earlier JONSWAP spectrum (*T*_{p}=10.0s, *γ*=2.5), with input amplitude sums of *A*=3.5, 4.0 and 5.0 m; the last case being the subject of our previous investigation. In each of these cases the nonlinear evolution of the wave field leads to a local broadening of the wave spectrum, owing to third-order *resonant* interactions, and a corresponding increase in the amplitude sum. The larger the input amplitude sum (*A*), the greater the nonlinearity, the more effective the spectral broadening, and hence the larger the increase in the amplitude sum. These results, together with the fact that large deep-water waves arise as focused or near-focused wave events, explains why the large crest elevations in deep water may be higher than expected, particularly where predictions are based upon a constant regime of freely propagating wave components coupled with their associated *bound* waves.

In contrast, figure 9*b* provides a similar sequence of results relating to the same JONSWAP spectrum evolving in a water depth of *d*=15 m. Identical input amplitude sums are considered (*A*=3.5, 4.0 and 5.0 m) and, once again, the largest wave event relates to the wave case considered previously. In these examples the implications of the two-stage evolution, involving an initial narrowing and subsequent broadening of the spectral form, are clearly noted. This is particularly apparent in the *A*=5 m case. In each of these examples the narrowing of the wave spectrum initially leads to a reduction in the amplitude sum. This effect is the opposite to that which occurs in deep water; is of an equivalent or larger magnitude, accounting for up to 20 per cent of the input amplitude; and occurs at an earlier stage in the evolution owing to the reduced effect of frequency dispersion. It is only at a relatively late stage that the local broadening of the wave spectrum leads to an increase in the amplitude sum. However, the magnitude of the initial reduction is larger than the subsequent increase (at least for the cases studied) and consequently there is a net reduction in *A* for each case considered. This result, coupled with the nature of the extreme wave event involving the spread of wave energy over a large spatial domain rather than its focusing at one location in space and time, explains why in this and many other simulations of extreme waves in intermediate and shallow-water depths the maximum crest elevation is less than the linear amplitude sum, .

Finally, given the importance of changes in the spectral bandwidth and hence the amplitude sum, it is instructive to consider whether the ZE model based upon the sum of the linear terms and the third-order *resonant* interactions is able to reproduce the predicted changes in *A*. Comparisons provided in figure 9*c*,*d*, respectively, concern the deep and intermediate depth cases with *A*_{input}=5 m. In both examples, the changes in the amplitude sum are well described by the inclusion of the third-order *resonant* interactions. These results highlight the importance of these terms in defining the nature of an extreme wave event, with comparisons between the deep and intermediate wave cases emphasizing the critical importance of water depth.

The results presented in figure 9 are consistent with the recent numerical calculations reported by Toffoli *et al.* (2009). While the present paper has considered (in detail) the properties of a single large wave event, Toffoli *et al.* (2009) considered the statistical properties of both uni-directional and directionally spread waves in random seas using numerical calculations based upon the truncated potential Euler equations. In the case of uni-directional waves, it is shown that as the water depth reduces the number of extreme waves also reduces. The explanation for this result lies in both the nonlinear change in the spectral bandwidth (and hence the change in the linear amplitude sum, *A*) and the extent to which the largest waves correspond to focused wave events, both aspects having been shown to be strongly dependent upon the effective water depth (*kd*).

## 6. Conclusions

This paper has considered the effect of water depth on the evolution of large surface water waves. By using two very different but complementary wave models, and by providing direct comparisons between the evolution of highly nonlinear waves in deep and intermediate (*d*=15 m) water depths, it has been shown that the water depth has a profound effect on the evolution of large uni-directional waves in realistic frequency spectra. The explanation for this lies in the nature of the third-order *resonant* or *near-resonant* interactions since these, and these alone, are able to alter both the amplitude and the phasing of the freely propagating wave components.

In deep water these interactions are important, but not so strong that they dominate the process of linear dispersion. As a result, the evolution of a deep-water wave group is governed by frequency dispersion, but modified by nonlinear effects, the latter accounting for local increases in the spectral bandwidth and hence larger maximum crest elevations. In contrast, with a reduction in water depth, frequency dispersion is weakened and the relative importance of the nonlinear interactions increased. This is true of both the *bound* and the *resonant* interactions, but the latter have the potential to create more significant changes. Indeed, the present calculations have shown that, with reduced water depths, the third-order or four-wave *resonant* interactions led to a narrowing of the spectral bandwidth and hence the formation of a quasi-regular wave train. At this stage the wave field ceases to evolve until the rapid growth of a modular instability leads to a large wave event at the front of the wave train. This wave is fundamentally different from that which would be produced by dispersive focusing and cannot therefore be predicted by typical design solutions such as the *NewWave* model. Most importantly, with the wave energy more widely dispersed across the spatial domain, and with the wave components no longer possessing a strong phase correlation, the maximum crest elevation and wave height are much reduced relative to the linearly predicted solutions.

Although the results presented in this paper have important practical implications, it must be stressed that they take no account of directionality, particularly the additional contribution that this makes to the focusing process. Indeed, recent work by (among others) Gibson & Swan (2007) and Toffoli *et al.* (2009) has shown that directionality can be very significant. Such effects are the subject of an ongoing study and will be reported in a subsequent (part II) paper.

- Received June 1, 2010.
- Accepted August 3, 2010.

- This journal is © 2010 The Royal Society