## Abstract

This paper sets out an approximate analytical model describing the nonlinear evolution of a Gaussian wave group in deep water. The model is derived using the conserved quantities of the cubic nonlinear Schrödinger equation (NLSE). The key parameter for describing the evolution is the amplitude-to-wavenumber bandwidth ratio, a quantity analogous to the Benjamin–Feir index for random sea-states. For smaller values of this parameter, the group is wholly dispersive, whereas for more nonlinear cases, solitons are formed. Our model predicts the characteristics and the evolution of the groups in both regimes. These predictions are found to be in good agreement with numerical simulations using the NLSE and are in qualitative agreement with numerical results from a fully nonlinear potential flow solver and experimental results.

## 1. Introduction

The simplest nonlinear evolution equation to describe the dynamics of deep-water gravity waves is the nonlinear Schrödinger equation (NLSE). While this represents a gross simplification of the complex behaviour of real water waves, it does constitute a starting point for a theoretical investigation. Much work has been reported on the envelope instability of a regular wave train, dating back to Lighthill (1965) and the famous paper by Benjamin & Feir (1967), followed by Longuet-Higgins (1978) and many others.

The early work splits into two themes. First, the modulational approach was developed by Whitham and summarized in his monograph (Whitham 1974), and it leads eventually to the NLSE. This route can be followed through the work of Chu & Mei (1971). Their coupled pair of conservation laws was shown to be equivalent to the NLSE by Davey (1972). Second, the Fourier approach of Benjamin and Feir, based on Fourier decomposition, can be seen as the predecessor of the Zakharov equation described by Yuen & Lake (1980) and many others. Parallel to this was the work by Zakharov (1968) and Zakharov & Shabat (1972). Infeld & Rowlands (1990) also gave an interesting survey of the development of the theory of nonlinear waves on deep water. There is also a new review on the general notion of modulational instability by Zakharov & Ostrovsky (2009), co-authored by one of the main developers of the theory. Two important reviews of the general area of nonlinear water wave dynamics are given by Yuen & Lake (1982), and Dias & Kharif (1999).

In contrast to much work on the modulational instability of regular waves, there has been rather less work on the effects of the nonlinear dynamics of steep water waves on the dispersive properties of isolated wave groups, arguably a better idealization of an extreme wave event on the open sea, where linear dispersion is often assumed to dominate the global behaviour at least over time scales of 10–100 s of wave periods (Taylor & Haagsma 1994; Baldock *et al.* 1996; Johannessen & Swan 2003; Gibbs & Taylor 2005). The evolution of wave groups over 10 s of periods is also directly relevant to wave tank testing.

In this paper, we present some simple analytical results for the evolution of one-dimensional localized wave groups. While the NLSE is amenable to solution using the technique of inverse scattering (Zakharov & Shabat 1972; Ablowitz & Segur 1981; Ablowitz & Clarkson 1991), we have not attempted this. Instead, our aim is to present a simple description of the main features of the evolution by means of analytic approximations and then to compare these with numerical solutions. In future work, we aim to extend the simple approach to dispersive wave groups in two-dimensions for which the inverse scattering technique is not available.

The paper commences with the closed-form solution for the frequency dispersion of a water-wave packet with a Gaussian spectrum, presented in Kinsman’s classic book (Kinsman 1965). This is used as an approximate description for the evolution of a weakly nonlinear wave group, the amplitude and bandwidth being assumed to vary slowly with time. Conservation of the first two of the infinite number of conserved quantities for the NLSE is imposed to approximate the evolution of the group.

We identify different regimes for the evolution of an initially perfectly focused wave group, depending on the wave group’s nonlinearity. These are summarized in figure 1 (the numerical scheme is described in §3*c*). The dispersive regime, shown in figure 1*b,c*, is the most relevant to water waves in the open ocean. For this regime, we derive an approximate analytical theory, which predicts that as the wave group focuses, its amplitude increases and it contracts along the mean wave direction compared with the linear evolution. This is in qualitative agreement with the results of Baldock *et al.* (1996). Thus a taller and more extreme wave is produced, suggestive of a ‘freak’ or ‘rogue’ wave. However, we find that there is a limiting nonlinearity for which this theory is applicable: the nonlinearity must be that of a solitary wave. Groups of Gaussian form more nonlinear than those at focus are not accessible from an initially dispersed group. For initially focused wave groups that are more nonlinear than this limit, we find the group forms a soliton-like structure (figure 1*d*–*f*), the characteristics of which can be predicted using the conserved quantities and our Gaussian approximation. All of these analytical approximations agree closely with numerical results, except for wave groups that are slightly more nonlinear than the dispersive limit where the approximation is not as good.

The key parameter in determining the behaviour of focused wave groups is the ratio of wave amplitude-to-wavenumber bandwidth, defined here as *A*/*S* for the non-dimensionalized NLSE. This is simply a version of the Benjamin–Feir index (BF index) (Janssen 2003), defined for a single group rather than an entire sea state.

The BF index is useful for characterizing the importance of modulation instability for wave trains. This dates back to the work of Alber (1978) who examined the competing effects of nonlinearity and randomness (bandwidth) on uni-directional wave trains. Recent work in this area has been carried out by Onorato *et al.* (2001, 2002*a*,*b*, 2003), Janssen (2003), Dysthe *et al.* (2003) and Socquet-Juglard *et al.* (2005). While mathematically sophisticated, all this work concentrates on the statistical properties of entire random fields influenced by the competing effects of nonlinearity and dispersion in an attempt to understand the occurrence of freak waves. Instead of large-scale computing, we seek a minimal theoretical description of the evolution of a single, isolated wave packet as it focuses, defined as a local concentration of wave energy. We return to this discussion at the end of the paper.

We also use the Gaussian approximation to explore the effects of the nonlinear dynamics on the local evolution time scale around a focused event. The nonlinearity slows down the evolution of a dispersive group by up to a half, compared with a linearly dispersing group. This is in agreement with accounts of freak wave events, where witnesses describe seeing a ‘wall of water’ which persists over several wave periods. Our analysis also predicts a solitary wave group, where time is frozen, which is very close in form to the exact soliton solution of the NLSE. Again, our approximate analytical predictions agree well with numerical results.

We conclude by comparing our predictions with real water waves simulated using a fully nonlinear numerical scheme and experimental results. We find that our results are in qualitative agreement, but that both the NLSE and our approximations overestimate the nonlinear changes to the group structure.

## 2. The nonlinear Schrödinger equation and its conserved quantities

The NLSE, given by equation (2.1), is the simplest equation for modelling the weakly nonlinear evolution of narrow banded wave groups in deep water. It was first derived by Zakharov (1968) and subsequently investigated by many researchers. An extensive discussion is given by Yuen & Lake (1980, 1982). NLSE solutions and those for the full water wave equations in one dimensions are compared in detail by Henderson *et al.* (1999):
2.1
where *u* is the complex wave envelope, ω_{0} and *k*_{0} are the carrier wave natural frequency and wavenumber, and the frame of reference moves with the group velocity of the carrier waves. The NLSE may be non-dimensionalized into standard mathematical form using the substitutions: *T*=−ω_{0}*t*, and to give
2.2

The NLSE (equation (2.2)) has an infinite number of conserved quantities (Zakharov & Shabat 1972), of which the first two are given by equations (2.3) and (2.4), and the first of these represents conservation of energy and the second is conservation of the Hamiltonian: 2.3 and 2.4

## 3. Dispersive wave groups

### (a) Approximate analytical model

The linear evolution of a localized group with a Gaussian wave envelope is given by Kinsman (1965):
3.1
where *A* is the amplitude of the group and *S* is the bandwidth, at *T*=0. This is an exact solution to the linear part of the NLSE. Curiously, the solution was derived before the NLSE.

Now it is assumed that this Gaussian group can be used to approximate a nonlinear group in the full NLSE, if the amplitude *A* and the bandwidth *S* are taken as slowly varying functions of time. Also we assume that the time scale, τ, is a time-like variable rather than physical time, *T*, with this new time scale being dependent on the nonlinear dynamics.

As pointed out by one of the referees of this paper, we note that it is strictly not necessary to introduce this new nonlinear time scale, τ. With slowly varying functions, *A*=*A*(*T*) and *S*=*S*(*T*), one can write
3.2
which corresponds to *S*^{2}=*R*_{T}. However, we choose to define
3.3
so that the square of the wave envelope can be written as
3.4
where *S* is the instantaneous bandwidth, changing as the group evolves, and τ is the associated nonlinear time scale, a direct measure of how close to focus the group is. While in this section we present results for groups focusing from infinity ( and τ=0), it is convenient to retain τ in the formulation. Physical experiments on focused wave groups in wave channels may be defined in terms of an initial wave group based on linear dispersion a fixed distance or number of periods before linear focus (Baldock *et al.* 1996; Gibbs & Taylor 2005). Thus, the initial starting condition gives (*A*,*S*,τ) directly.

We should stress the physical meaning of the amplitude term *A* as we use it here in the analytical work; this would be the height of the wave group when all components are perfectly focused if this focusing was perfectly linear. Obviously, this is also the sum of the amplitude coefficients of the Fourier wavenumber representation of the dispersed wave group, since focus simply implies the point at which the crests of all these Fourier components are in phase at a single point.

Equation (3.1) may be substituted into equations (2.3) and (2.4) to give equations (3.5) and (3.6), where *A* and *S* are functions of a nonlinear time scale τ:
3.5
and
3.6
Substituting τ=0 and gives two equations relating the parameters of a focused and a fully dispersed group. For the fully dispersed group, the second term in equation (3.6) goes to zero:
3.7
and
3.8
Eliminating gives
3.9
and similarly eliminating leads to
3.10
The equations can also be solved to give the properties of a focused group given those at infinity:
3.11
and
3.12
Equations (3.9) and (3.10) show that both the amplitude and bandwidth of the group increase at focus, compared with those on a purely linear basis. Thus we observe an increase in elevation owing to the nonlinear dynamics. However, the local BF index of the wave group, defined as the amplitude–bandwidth ratio, is reduced as shown in equation (3.13). These are shown in figure 2,
3.13
Equations (3.11) and (3.12) give a limit to the steepness of a wave group (at focus) which can be investigated using this approach, and thus a limiting steepness of a group which is accessible from an infinitely dispersed group, expressed as a BF index
3.14
Thus we have an analytical relationship between the parameters of a wave group, when perfectly focused and fully dispersed, and a limitation on the properties of a group at focus, if this arose from a fully dispersed and effectively locally linear group at .

### (b) Nonlinear time scale

The nonlinearity of the NLSE modifies the time scale over which any Gaussian group will evolve.

An approximate relationship may be derived between the nonlinear dispersive time scale (τ) and the true physical time scale (*T*). Taking equation (2.2) and its complex conjugate and combining the results give
3.15
As spatial integration of the first part of this equation over the whole *x*-axis produces the time derivative of the *I*2-conserved quantity and the second part evaluates by parts to zero, this equation is multiplied by *U**U*^{★} to give
3.16
Substituting in the Gaussian (equation (3.1)) and integrating over the whole *x*-axis gives
3.17
Thus we now have three equations (3.5), (3.6) and (3.17) that give a complete system for modelling the approximate evolution of a Gaussian-like wave group.

The amplitude *A* may be eliminated using equation (3.5), giving
3.18
As the evolution has to be symmetric in time around focus (*T*=0, τ=0), assume
3.19
and use this with equation (3.6) and the value for *I*2 for the focused group. This gives
3.20
Substituting into equation (3.18) and taking the limit as , we get
3.21
and
3.22

This implies that a more nonlinear group produces a slower nonlinear time scale for evolution around focus. Physically, this means that steep events will persist for a longer period of time. The limiting case, where nonlinear time is frozen at focus, is given by
3.23
which again contains the BF index for the group. A Gaussian wave group with this steepness is very close in shape to the exact soliton solution shape for the one-dimensional NLSE, as shown in figure 3. For *A*_{f}/*S*_{f}>2^{3/4}, the local dispersion around the focus event runs backwards in time.

For the steepest group that can arise from focusing of an initially infinitely dispersed wave group, the rate at which wave group evolution (dispersion) occurs locally around focus is predicted to be exactly one-half of that for a linear group.

Note, it is possible to take these equations to a higher order by continuing the perturbation expansion in equation (3.19). Assuming 3.24 gives the non-dimensional time as 3.25 again containing the form of the solitary wave as a multiplier to freeze the evolution.

We note that equivalent results could be derived directly from *R*_{T} introduced in the previous section. However, we have been unable to solve the differential equation for (*R*,*S*) in the closed form and there seems little advantage in solving an approximate ODE numerically, rather than the original NLSE itself.

### (c) Comparison with numerical results

The predictions made by equations (3.9), (3.10) and (3.14) may be compared with the results of numerical simulations of the NLSE. The simulations use a fourth-order Runge–Kutta pseudo-spectral scheme to solve equation (2.1) with a time step of 2 s and a spatial discretization of 10 m. The spatial domain is periodic and 40 960 m long. In all simulations in this paper, the same carrier wave properties are used: 3.26

These equate to *t*_{p}=12 s, which is representative of a storm in the North Sea. We use a Gaussian spectrum, which is fitted to the peak of a JONSWAP spectrum with γ=3.3, as in Gibbs & Taylor (2005). Typically, for a simulation running for 20 periods, the total energy, *I*2, is conserved to one part in 10^{8} and the Hamiltonian, *I*4, to one part in 10^{4}.

The spatial evolution of a number of groups in this regime is shown in figure 1*b*,*c*. The maximum amplitude of the wave envelope is shown in figure 4 as a function of time. It can be seen that more nonlinear groups are slower to disperse as predicted.

We now look at the changes to the spectrum and the parts of the conserved quantities of an initially focused group as it disperses. Figure 5*a* shows the evolution of the terms in the *I*4-conserved quantity over time: as predicted, the first part goes to a constant limit and the second quadratic part decays to zero. Figure 5*b* shows the changes in bandwidth over time. The bandwidth is found by fitting a Gaussian to the wavenumber spectrum using a least-squares method.

This result is useful because it gives unambiguous information as to how long a simulation needs to continue for before the spectrum can be assumed to have stopped evolving.

A comparison between the evolution predicted by equation (3.10) and the numerical model is shown in figure 6. As the *A* value can be derived from the bandwidth using equation (3.5), only the bandwidth results are presented as the agreement for amplitude is identically good. The initially focused wave group is allowed to fully disperse, which is taken to be the time at which the spectrum stops changing and the spatial evolution becomes essentially linear.

The agreement between the analytical and numerical results shown in figure 6 is excellent. The numerical results for the two bandwidths are very close, as is expected from the scaling properties of the NLSE. The predicted bandwidth using the assumption that the group remains Gaussian is very close to that from the numerical solution. The biggest difference occurs near the limiting *A*_{f}/*S*_{f} ratio where the predicted bandwidth ratio is very large.

The predictions of nonlinear time scales (equation (3.22)) may be compared with numerical simulations. Once the *A* and *S* values have been found (as above) in the wavenumber domain, a Gaussian group (equation (3.1)) may be fitted to the envelope in the spatial domain, allowing τ to be estimated. This again illustrates the utility in introducing explicitly a nonlinear time scale as it facilitates comparison with both numerical simulations and physical experiments.

Figure 7 shows the averaged nonlinear time scale over the first four periods after the focus. While the agreement is close for the less-steep cases, there is some discrepancy for the more nonlinear runs, but the results are still in reasonable agreement.

## 4. Non-dispersive groups

### (a) Approximate theory for 2^{1/4}<*A*_{f}/*S*_{f}<2^{3/4}

Numerical solutions run from a focused Gaussian group in this regime do not disperse, but after some initial transient behaviour, they oscillate around a steady structure (figure 1*d*–*f*). The initial Gaussian group is close in form to the *sech*-shaped group used in Mei (1989)—our results are in good agreement with the results presented in §12.6.1 of Mei. We can estimate this limit by again considering conservation of *I*2 and *I*4. *I*2 corresponds to conservation of energy, and *I*4, the Hamiltonian integral, is a measure of wave-group nonlinearity. First, we note that if *A*_{f}/*S*_{f}>2^{1/4}, there is a second focused Gaussian group that has identical *I*2 and *I*4 values. These are given by
4.1
and
4.2

During the evolution, the wave-group parameters appear to remain bounded between the focused value and this second value during its evolution.

Assuming that the initial wave packet of amplitude *A*_{f} (and hereafter we set the bandwidth as *S*_{f}=1) evolves into a single soliton and a very long and low tail, it is reasonable to assume that all the initial *I*4 is carried by the soliton. Then, we simply match the initial value of *I*4 to that of a single soliton, writing the soliton amplitude as *A*_{s} and taking its bandwidth as given by (*A*_{s}/*S*_{s})=2^{3/4}, which is our Gaussian representation of a soliton. Then the height of the soliton can be estimated as
4.3

We can also estimate the energy leakage away from the soliton, which has the form of a dispersive tail 4.4

When *A*_{f}=2^{1/4}, the energy leakage goes to 1 and the system is fully dispersive and the soliton disappears. When *A*_{f}=2^{3/4}, no energy is leaked and *A*_{s}=*A*_{f}, which is identical to the value of *A* that is found to freeze dispersion in §3*b* and is very close in form to a soliton.

This approximate analysis is similar in spirit to the discussion by Miles (1980), in his review of solitary waves on shallow water, of Green’s law for shoaling as originally discussed by Boussinesq. There, energy conservation rather than mass conservation is used to model the evolution of the wave as it gradually moves into shallower water leaving a long low ‘hump’ of water behind, which contains a negligible proportion of the incoming energy.

### (b) Approximate theory for *A*/*S*>2^{3/4}

In this regime, the initial focused group is taller than a simple Gaussian approximation for a single soliton, so it reasonable to assume that two solitary waves will emerge together with a very weak tail. Thus, we assume that the strength of the two solitary waves can be estimated from *I*2 and *I*4 conservation, neglecting the weak contributions to each conserved quantity carried by the dispersive tail. Assuming that each of the emerging solitons has a height/bandwidth *A*/*S* of 2^{3/4}, and that bandwidth of the initial group at focus is *S*_{f}=1, then the strength of each soliton can be written as
4.5
Although these are closed form approximations for the strength of the two solitary waves that emerge, the physical basis of the behaviour is clearer if we perform a series expansion around *A*_{f}=2^{3/4}. Then the solitons are
4.6
and
4.7

Thus, perturbations upwards away from the single soliton limit *A*_{f}=2^{3/4} produce linear variation in the size of the larger soliton and a quadratic birth of a very weak second soliton, in a manner comparable to hump solitons emerging from a sech^{2}-hump in the Korteweg–de Vries equation for waves on shallow water. Although this analysis is only appropriate for *A*_{f}>2^{3/4}, the series expansion for *A*_{s1} given here matches that for *A*_{s} given in the previous section expanded about the same soliton value to *O*[*A*_{f}−2^{3/4}]^{5}, giving confidence in both forms. It is expected that as nonlinearity increases a third soliton would emerge.

Clearly, as the height of the starting Gaussian profile increases, there is scope for more complex soliton solutions to result. However, these are probably of little interest in terms of focused water wave groups with physically realistic steepness and bandwidth. The result that two solitons form if the initial condition is slightly taller than a single isolated soliton is consistent with the results of Satsuma & Yajima (1974) and confirmed by Yuen & Lake (1982). These authors present an explicit formula for the number of solitons emerging from an initial wave group of the soliton shape but the ‘wrong’ height. However, as Mei (1989) discusses in §12.6 of his book, the main features of this work are that ‘an arbitary shaped envelope will eventually evolve into a finite number of soliton, plus minor oscillations which decay as *t*^{−1/2}’. Other than this section, this paper is devoted to approximately modelling the evolution of these ‘minor oscillations’, which are the wave groups that are most comparable with those on the open ocean.

### (c) Comparison with numerical simulations

The spatial evolution of the wave envelope for *A*_{f}/*S*_{f}>2^{1/4} is shown in figure 1*d*–*f*.

We can also track the evolution of the spectrum over time by fitting a Gaussian as was done for the dispersive groups. This is shown in figure 8 along with values of *S*_{2} and *S*_{s}. During the initial transient behaviour, the group shape approaches the shape of a Gaussian with amplitude *A*_{2} and bandwidth *S*_{2}, but does not reach this.

Comparison of the amplitude of the soliton produced in the numerical solution agrees well with that predicted by the approximate analytical solution (equations (4.4) and (4.6)), as shown in figure 9. As expected, the greatest discrepancy is for values of *A*_{f}/*S*_{f} close to 2^{1/4}, where most of the energy leaks away from the soliton.

In figure 8, it is clear that there is a very slow decay of the numerical solution for *A*/*S*>2^{3/4}. We believe that this may be real, rather than an artefact of the numerical scheme, and is probably linked to the prediction of two solitons. Unlike hump solitons of the Korteweg–de Vries equation, in which the propagation speed increases with the height of the soliton, solitons in the NLSE do not show any nonlinear modifications to the group velocity. Therefore, the separation of large solitons from small will not occur in the NLSE, unlike the behaviour of solitary wave envelopes in the full water wave equations that does exhibit such amplitude-dependent modifications (Bryant 1983). The oscillation at long times represents the continued interaction of two wave groups, and this interaction slowly leaks energy, unless the combined structure happens to correspond with a breather.

## 5. Comparison with fully nonlinear simulations and experiments

### (a) Fully nonlinear simulations

Gibbs (2004) carried out fully nonlinear simulations of focused wave groups using the numerical scheme developed by Bateman *et al.* (2001). The evolution of initial Gaussian wave group at −80 periods before focus was investigated.

Gibbs (2004) found that the effect of nonlinearity was to increase both the amplitude at focus and the local bandwidth of the wave group as it focused from an initially dispersed initial condition. The NLSE produces similar changes in the shape of the wave group, although not all nonlinear effects are captured. A typical comparison is shown in figure 10. It can be seen that in this case the Gaussian is not an accurate model for the wave group, which has pronounced sidebands (Yuen & Lake 1980), which are symmetrical in the NLSE and non-symmetrical in the fully nonlinear model. Henderson *et al.* (1999) report similar differences betweeen the solutions of the NLSE and the full water wave equations in one dimension.

The Gaussian approximation may be used to predict the bandwidth changes observed in the fully nonlinear simulations. The predictions are based on comparing the values of the conserved quantities at −80 periods with those at focus.

Owing to the nonlinear dynamics, both the NLSE and fully nonlinear numerical models reach a maximum amplitude and bandwidth sometime before focus. The point in time when the group reaches maximum amplitude is used in the comparison with the predicted values. The bandwidths were found by fitting a Gaussian to wavenumber spectrum, as before, which is different from the method used in Gibbs (2004).

For fully nonlinear outputs, data have to be ‘linearized’ to remove bound harmonics. This is performed by running the numerical model for two cases 180^{°} out of phase to produce ‘crest’ and ‘trough’ focused cases. The even harmonics are removed using
5.1
Higher order harmonics are removed by filtering.

Figure 11 shows a comparison between the predicted bandwidth changes and those observed in the fully nonlinear numerical model and the NLSE numerical model. All three are in good agreement for the cases with input steepness less than *a**k*=0.1. For the more nonlinear cases, the nonlinear group is less well modelled by a Gaussian shape, largely owing to the asymmetry in the groups, and is rather different from the numerical NLSE and the Gaussian prediction. The Gaussian prediction over-estimates the contraction of the group compared with the NLSE. This is partly owing to the deviation from the Gaussian shape in the numerical model, but also owing to the group not focusing perfectly because of the nonlinear dynamics.

The question of focal quality is important and is discussed in Johannessen & Swan (2003). If the initial condition for a nonlinear focused group is based on the relative phases required for linear focusing, then there is no guarantee that, at the instant when the highest peak envelope occurs, all the wave components will be exactly in phase. In fact, this will not occur. However, it is possible to start from an exactly focused initial condition and then allow the group to de-focus. In the first case, the evolution from −80 periods, through focus (which may not be at the same time as linear focus), to +80 periods will not be precisely symmetric in time. In contrast, starting at perfect focus and allowing the simulations to run both backwards to −80 periods and forwards to +80 periods will give exactly symmetric evolution. This latter case is the problem that our simple Gaussian model is intended to address.

### (b) Comparison with experimental results

Baldock *et al.* (1996) investigated the focusing of uni-directional wave groups experimentally. The wave-group parameters used were somewhat different from those used in this paper. The wave groups were generated by a paddle 8 m away from linear focus in a 20 m long tank. This equates to around 10 periods before linear focus for wave components around the peak of the spectrum. The wavetank was 0.7 m deep. A typical *k**d* based on the mean wavenumber of the components was around 3, although the lowest wavenumber components generated had *k**d*∼1.4.

A number of spectral widths were used, but all spectra had the form *S*(ω)∼ω^{−4}, which is very different from the Gaussian upon which our analysis is based. Because of this difference, and because of the effect of finite depth on the low wavenumber components, we do not present cases A and B (Baldock *et al.* 1996), which are the most broadbanded cases. Case D has a very narrowbanded spectrum, and Vijfvinkel (1996), who used a fully nonlinear scheme based on Craig & Sulem (1993), found poor agreement for this case when starting the simulations with a spatial profile at a fixed point in time. Our approach also requires an initial spatial profile at a point in time so we do not present results for this case. Instead, we look at case C that has an initial spectrum over a frequency range of 0.77–1.42 Hz.

We calculate initial values for *I*2 and *I*4 using equations (3.5) and (3.6). The value of *A* is found directly from the summation of the amplitude components of the initial spectrum. The spectral width, *S*, is found from the variance of the spectrum about the mean wavenumber. For the spectrum in case C, this gives a mean wavenumber of 4.42 m^{−1} and *s*_{x}=1.54 m^{−1}. The time τ is set as 10 periods before linear focus. The equations may then be solved for *A*_{f} and *S*_{f} at τ=0. We can then compare the extra amplitude predicted with that found experimentally. The experimental data have been linearized by combining crest and trough focused groups (Baldock 1994), as for the fully nonlinear simulations described earlier.

A comparison is shown in figure 12. It can be seen that, despite the very different spectra used, there is good agreement, except for the most nonlinear case in which it is likely that the focal quality in the experimental results was relatively poor.

## 6. Discussion: group and field properties

This paper has concentrated on the evolution of an isolated focused wave group on still water. Using the conserved quantities of the NLSE, we have shown how simple but apparently new analytical approximations can be obtained for the nonlinear evolution of group amplitude and bandwidth as focusing of the group occurs. Our approach appears incapable of simple generalization to include the effect of a random background sea state, at least without using rather more sophisticated mathematical techniques. In contrast, much recent work on the properties of wave fields in both one and two spatial dimensions has concentrated on the statistics of random fields.

While the recent work of Onorato *et al.* (2001, 2002*a*,*b*, 2003), Dysthe *et al.* (2003) and Janssen (2003) is clearly all of great importance for the exploration of freak waves, published results from random simulations concentrate rightly on the statistics of the field, particularly looking for divergence from the normal distribution for the surface, or the Rayleigh distribution for wave height. There is seldom any detailed discussion of the local temporal or spatial structure of extreme events that may arise. Our work is perhaps a small contribution in this direction.

The paper by Onorato *et al.* (2003) is particularly interesting in the context of this paper. They look at evolution from random initial conditions (based on bandwidth and steepness) for the one-dimensional NLSE, showing that robust coherent structures emerge from random initial conditions and propagate throughout the entire simulation if the BF index is high enough. What is the average shape of such structures, do they resemble solitons or perhaps our dispersing wave groups that are modified by the competition of nonlinearity and bandwidth? Although this work is directly relevant only to waves in one-dimension, the issue of coherent structures in two-dimensions is directly relevant to offshore engineering. Are freak waves as observed on the open sea linked to these coherent structures? If an average shape can be extracted from the sophisticated nonlinear dynamical modelling, what are the wave kinematics associated with such events? Could knowledge of the surface shape and internal wave kinematics be combined into a ‘design wave’ for structural design or re-assessment of oil and gas production facilities?

The work of Janssen (2003) on the statistical properties of random fields is also important in terms of both importance of four-wave interactions and in the context of warning shipping of the increased likelihood of unusually severe extreme waves. Can a useful BF index be validated for directional spread or perhaps crossing seas? If so, it could then be a reliable indicator of the danger posed by freak waves, when coupled with the standard predictions of significant wave height, wave period, wave directionality, and so on. In a subsequent paper extending the approach presented here to directional spread waves, we intend to address some of these issues.

## 7. Conclusions

In this paper, we show that it is possible to model the nonlinear changes to the shape of a Gaussian wave group using the conserved quantities of the NLSE. We use this approach to predict the evolution for a number of regimes and to predict the shape of the most nonlinear group accessible from an initially dispersed group. Our results are in good agreement with numerical solutions to the NLSE. However, the agreement is not as good, although still qualitatively correct, when compared with solutions to the full water wave equations.

We also derive a nonlinear time-scale and find that the greater the nonlinearity of an initially focused group, the slower the group disperses, and that for a group in the dispersive regime, this can be a maximum of half as fast as the group would evolve on a linear basis. We also predict a soliton solution that is very close in form to the exact soliton solution of the NLSE.

Overall, our approach of using the conserved quantities to derive approximate evolution equations for Gaussian wave groups captures much of the physics of the problem and allows robust predictions to be made. Thus in a subsequent paper, we propose to extend this approach to directionally spread wave groups using the two-dimensional NLSE. The two-dimensional NLSE cannot be solved by inverse scattering, and the additional lateral dimension means wave-group behaviour cannot be simply categorized using the BF index, via a simple ratio of wave steepness to bandwidth. The directionality of the wave field also needs to be accounted for. However, the two-dimensional NLSE is a much better model for real ocean waves.

## Acknowledgements

The authors are grateful for the data provided by Dr Richard Gibbs. T.A.A.A. is supported by an EPSRC studentship and a PhD plus fellowship.

## Footnotes

- Received April 27, 2009.
- Accepted July 1, 2009.

- © 2009 The Royal Society