Focusing of unidirectional wave groups on deep water: an...

Proc. R. Soc. A (2009) 465, 3083–3102doi:10.1098/rspa.2009.0224

Published online 29 July 2009

Focusing of unidirectional wave groups on deepwater: an approximate nonlinear Schrödinger

equation-based modelBY T. A. A. ADCOCK* AND P. H. TAYLOR

Department of Engineering Science, University of Oxford, Parks Road,Oxford OX1 3PJ, UK

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

Keywords: Gaussian; wave group; freak wave; nonlinear Schrödinger equation

1. Introduction

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

The early work splits into two themes. First, the modulational approach wasdeveloped by Whitham and summarized in his monograph (Whitham 1974),and it leads eventually to the NLSE. This route can be followed throughthe work of Chu & Mei (1971). Their coupled pair of conservation laws wasshown to be equivalent to the NLSE by Davey (1972). Second, the Fourierapproach of Benjamin and Feir, based on Fourier decomposition, can be seenas the predecessor of the Zakharov equation described by Yuen & Lake (1980)and many others. Parallel to this was the work by Zakharov (1968) andZakharov & Shabat (1972). Infeld & Rowlands (1990) also gave an interestingsurvey of the development of the theory of nonlinear waves on deep water. There is*Author for correspondence ([email protected]).

Received 27 April 2009Accepted 1 July 2009 This journal is © 2009 The Royal Society3083

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3084 T. A. A. Adcock and P. H. Taylor

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. Twoimportant reviews of the general area of nonlinear water wave dynamics are givenby 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 ofsteep water waves on the dispersive properties of isolated wave groups, arguablya better idealization of an extreme wave event on the open sea, where lineardispersion is often assumed to dominate the global behaviour at least overtime scales of 10–100 s of wave periods (Taylor & Haagsma 1994; Baldocket al. 1996; Johannessen & Swan 2003; Gibbs & Taylor 2005). The evolutionof wave groups over 10 s of periods is also directly relevant to wave tanktesting.

In this paper, we present some simple analytical results for the evolution ofone-dimensional localized wave groups. While the NLSE is amenable to solutionusing 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 evolutionby means of analytic approximations and then to compare these with numericalsolutions. In future work, we aim to extend the simple approach to dispersivewave groups in two-dimensions for which the inverse scattering technique is notavailable.

The paper commences with the closed-form solution for the frequencydispersion of a water-wave packet with a Gaussian spectrum, presented inKinsman’s classic book (Kinsman 1965). This is used as an approximatedescription for the evolution of a weakly nonlinear wave group, the amplitudeand bandwidth being assumed to vary slowly with time. Conservation of the firsttwo of the infinite number of conserved quantities for the NLSE is imposed toapproximate the evolution of the group.

We identify different regimes for the evolution of an initially perfectly focusedwave group, depending on the wave group’s nonlinearity. These are summarized infigure 1 (the numerical scheme is described in §3c). The dispersive regime, shownin figure 1b,c, is the most relevant to water waves in the open ocean. For thisregime, we derive an approximate analytical theory, which predicts that as thewave group focuses, its amplitude increases and it contracts along the mean wavedirection compared with the linear evolution. This is in qualitative agreementwith the results of Baldock et al. (1996). Thus a taller and more extreme wave isproduced, suggestive of a ‘freak’ or ‘rogue’ wave. However, we find that there isa limiting nonlinearity for which this theory is applicable: the nonlinearity mustbe 1/

√2 that of a solitary wave. Groups of Gaussian form more nonlinear than

those at focus are not accessible from an initially dispersed group. For initiallyfocused wave groups that are more nonlinear than this limit, we find the groupforms a soliton-like structure (figure 1d–f ), the characteristics of which can bepredicted using the conserved quantities and our Gaussian approximation. All ofthese analytical approximations agree closely with numerical results, except forwave groups that are slightly more nonlinear than the dispersive limit where theapproximation is not as good.

The key parameter in determining the behaviour of focused wave groups isthe ratio of wave amplitude-to-wavenumber bandwidth, defined here as A/S for

Proc. R. Soc. A (2009)



Gaussian wave groups on deep water 3085

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(a) (b)

(c) (d )

(e) ( f )

Figure 1. The evolution of Gaussian groups of various heights from focus. Contours are |U |/Af .(a) Linear evolution, (b) Af/Sf = 0.4(21/4), (c) Af/Sf = 0.8(21/4), (d) Af/Sf = 1.3(21/4), (e) Af/Sf =(21/4)2, and (f ) Af/Sf = 1.55(21/4).

the non-dimensionalized NLSE. This is simply a version of the Benjamin–Feirindex (BF index) (Janssen 2003), defined for a single group rather than an entiresea state.

The BF index is useful for characterizing the importance of modulationinstability for wave trains. This dates back to the work of Alber (1978) whoexamined the competing effects of nonlinearity and randomness (bandwidth) onuni-directional wave trains. Recent work in this area has been carried out byOnorato et al. (2001, 2002a,b, 2003), Janssen (2003), Dysthe et al. (2003) andSocquet-Juglard et al. (2005). While mathematically sophisticated, all this workconcentrates on the statistical properties of entire random fields influenced by thecompeting effects of nonlinearity and dispersion in an attempt to understand theoccurrence of freak waves. Instead of large-scale computing, we seek a minimaltheoretical description of the evolution of a single, isolated wave packet as itfocuses, defined as a local concentration of wave energy. We return to thisdiscussion at the end of the paper.

We also use the Gaussian approximation to explore the effects of the nonlineardynamics on the local evolution time scale around a focused event. Thenonlinearity slows down the evolution of a dispersive group by up to a half,compared with a linearly dispersing group. This is in agreement with accountsof freak wave events, where witnesses describe seeing a ‘wall of water’ whichpersists 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 solitonsolution of the NLSE. Again, our approximate analytical predictions agree wellwith numerical results.

We conclude by comparing our predictions with real water waves simulatedusing a fully nonlinear numerical scheme and experimental results. We findthat our results are in qualitative agreement, but that both the NLSE and ourapproximations 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 theweakly nonlinear evolution of narrow banded wave groups in deep water. Itwas first derived by Zakharov (1968) and subsequently investigated by manyresearchers. An extensive discussion is given by Yuen & Lake (1980, 1982). NLSEsolutions and those for the full water wave equations in one dimensions arecompared in detail by Henderson et al. (1999):

i(

∂u∂t

)=

(ω0

8k20

)∂2u∂x2

+ 12ω0k2

0 |u|2 u, (2.1)

where u is the complex wave envelope, ω0 and k 0 are the carrier wave naturalfrequency and wavenumber, and the frame of reference moves with the groupvelocity of the carrier waves. The NLSE may be non-dimensionalized intostandard mathematical form using the substitutions: T = −ω0t, X = 2

√2k 0x and

U = (ko/√

2)u to give

i∂U∂T

+ ∂2U∂X 2

+ |U |2U = 0. (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 secondis conservation of the Hamiltonian:

I2 =∫∞

−∞|U |2 dX (2.3)

and

I4 =∫∞

−∞

∣∣∣∣dUdX

∣∣∣∣2

− 12|U |4 dX . (2.4)

3. Dispersive wave groups

(a) Approximate analytical model

The linear evolution of a localized group with a Gaussian wave envelope is givenby Kinsman (1965):

U (X , T ) = A(1 + 2iS2T )1/2 exp

[−(1/2)S2(X − (1/2)T )2

(1 + 4S4T 2)+ i

2S4T

1 + 4S4T 2

(X − 1

2T

)2],

(3.1)





where A is the amplitude of the group and S is the bandwidth, at T = 0. Thisis an exact solution to the linear part of the NLSE. Curiously, the solution wasderived before the NLSE.

Now it is assumed that this Gaussian group can be used to approximate anonlinear group in the full NLSE, if the amplitude A and the bandwidth S aretaken 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 scalebeing dependent on the nonlinear dynamics.

As pointed out by one of the referees of this paper, we note that it is strictlynot necessary to introduce this new nonlinear time scale, τ . With slowly varyingfunctions, A = A(T ) and S = S(T ), one can write

R =∫T

0S2 (

T ′) dT ′, (3.2)

which corresponds to S2 = RT . However, we choose to define

S2(T ) · τ(T ) =∫T

0S2(T ′) dT ′, (3.3)

so that the square of the wave envelope can be written as

A2

(1 + 4R2)= A2

(1 + 4S4τ 2), (3.4)

where S is the instantaneous bandwidth, changing as the group evolves, and τis the associated nonlinear time scale, a direct measure of how close to focusthe group is. While in this section we present results for groups focusing frominfinity (T = ∞, τ = ∞ �⇒ T = 0 and τ = 0), it is convenient to retain τ in theformulation. Physical experiments on focused wave groups in wave channels maybe defined in terms of an initial wave group based on linear dispersion a fixeddistance 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 ithere in the analytical work; this would be the height of the wave group when allcomponents are perfectly focused if this focusing was perfectly linear. Obviously,this is also the sum of the amplitude coefficients of the Fourier wavenumberrepresentation of the dispersed wave group, since focus simply implies the pointat 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 giveequations (3.5) and (3.6), where A and S are functions of a nonlinear time scale τ :

A2

S= I2 (3.5)

and

A2S − 1√2

A4

S(1 + 4S4τ 2)1/2= I4. (3.6)

Substituting τ = 0 and τ = ∞ gives two equations relating the parameters of afocused and a fully dispersed group. For the fully dispersed group, the second





term in equation (3.6) goes to zero:

A2∞S∞

= A2f

Sf(3.7)

and

A2∞S∞ = A2

f Sf − 1√2

A4f

Sf. (3.8)

Eliminating S∞ gives (A∞Af

)4

= 1 − 1√2

(Af

Sf

)2

, (3.9)

and similarly eliminating A∞ leads to(S∞Sf

)2

= 1 − 1√2

(Af

Sf

)2

. (3.10)

The equations can also be solved to give the properties of a focused group giventhose at infinity:

Af

A∞=

√√√√ 1

2√

2

(A∞S∞

)2

+√

1 + 18

(A∞S∞

)4

(3.11)

and

Sf

S∞= 1

2√

2

(A∞S∞

)2

+√

1 + 18

(A∞S∞

)4

. (3.12)

Equations (3.9) and (3.10) show that both the amplitude and bandwidth of thegroup increase at focus, compared with those on a purely linear basis. Thus weobserve an increase in elevation owing to the nonlinear dynamics. However, thelocal BF index of the wave group, defined as the amplitude–bandwidth ratio, isreduced as shown in equation (3.13). These are shown in figure 2,

Af/Sf

A∞/S∞= 1√

1/(2√

2)(A∞/S∞)2 + √1 + (1/8)(A∞/S∞)4

. (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 steepnessof a group which is accessible from an infinitely dispersed group, expressed asa BF index

21/4 <Af

Sf. (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 ofa group at focus, if this arose from a fully dispersed and effectively locally lineargroup at T = ∞.





0 0.2 0.4 0.6 0.8 1.00.6

0.8

1.0

1.2

1.4

1.6

A∞/S∞

Figure 2. The predicted change in amplitude, bandwidth and BF index from a fully dispersedwave group to one at focus. Thick line, Af/A∞; thin line, Sf/S∞; dotted line, (Af/Sf )/(A∞/S∞).

(b) Nonlinear time scale

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

An approximate relationship may be derived between the nonlinear dispersivetime scale (τ ) and the true physical time scale (T ). Taking equation (2.2) and itscomplex conjugate and combining the results give

i(U �UT + UU �T ) + U �UXX − UU �

XX = 0. (3.15)

As spatial integration of the first part of this equation over the whole x-axisproduces the time derivative of the I2-conserved quantity and the second partevaluates by parts to zero, this equation is multiplied by UU � to give

i(UU �)2T + UU �(U �UXX − UU �

XX ) = 0. (3.16)

Substituting in the Gaussian (equation (3.1)) and integrating over the wholex-axis gives

∂

∂TA2

(1 + 4S4τ 2)1/2= −4

A2S4τ

(1 + 4S4τ 2)3/2. (3.17)

Thus we now have three equations (3.5), (3.6) and (3.17) that give a completesystem for modelling the approximate evolution of a Gaussian-like wave group.

The amplitude A may be eliminated using equation (3.5), giving

∂

∂TS

(1 + 4S4τ 2)1/2= −4

S5τ

(1 + 4S4τ 2)3/2. (3.18)

As the evolution has to be symmetric in time around focus (T = 0, τ = 0), assume

S = Sf(1 − εT 2) + O(T 4), (3.19)

and use this with equation (3.6) and the value for I2 for the focused group.This gives

ε = 2√

2A2f S

4f

4S2f − √

2A2f

. (3.20)





–5 –4 –3 –2 –1 0 1 2 3 4 50

0.5

1.0

1.5

2.0

X

|U|

Figure 3. Comparison of Gaussian with steepness set to freeze nonlinear time and a soliton ofequivalent amplitude. Dotted line, Gaussian, A = 23/4, S = 1; solid line, soliton.

Substituting into equation (3.18) and taking the limit as T → 0, we get

1 = dτ

dT

(4S2

f

4S2f − √

2A2f

)+ O(T 2) (3.21)

and

τ = T

[1 − A2

f

2√

2S2f

]+ O(T 3). (3.22)

This implies that a more nonlinear group produces a slower nonlinear timescale for evolution around focus. Physically, this means that steep events willpersist for a longer period of time. The limiting case, where nonlinear time isfrozen at focus, is given by

Af

Sf= 23/4, (3.23)

which again contains the BF index for the group. A Gaussian wave group withthis steepness is very close in shape to the exact soliton solution shape for theone-dimensional NLSE, as shown in figure 3. For Af/Sf > 23/4, the local dispersionaround the focus event runs backwards in time.

For the steepest group that can arise from focusing of an initially infinitelydispersed wave group, the rate at which wave group evolution (dispersion) occurslocally 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 theperturbation expansion in equation (3.19). Assuming

S = T(1 − ε1T 2 − ε2T 4) + O(T 7) (3.24)

gives the non-dimensional time as

τ = T

[1 − A2

f

2√

2S2f

] [1 +

(5

3√

2

(Af

Sf

)2(

1 − 3

5√

2

(Af

Sf

)2))

T 2

]+ O

(T 5) ,

(3.25)

again containing the form of the solitary wave as a multiplier to freeze theevolution.





0 10 20 30 40 50 60 70 800.5

0.6

0.7

0.8

0.9

1.0

time (periods)

|Um

ax|/A

f

Figure 4. The maximum of the wave envelope of dispersing wave group over time. Solid line,Af/Sf = 0.67; dotted line, Af/Sf = 0.335.

We note that equivalent results could be derived directly from RT introducedin the previous section. However, we have been unable to solve the differentialequation for (R, S) in the closed form and there seems little advantage in solvingan 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 comparedwith the results of numerical simulations of the NLSE. The simulations use afourth-order Runge–Kutta pseudo-spectral scheme to solve equation (2.1) witha time step of 2 s and a spatial discretization of 10 m. The spatial domain isperiodic and 40 960 m long. In all simulations in this paper, the same carrierwave properties are used:

ω0 = 0.523 s−1 and k 0 = 0.0279 m−1. (3.26)

These equate to tp = 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 spectrumwith γ = 3.3, as in Gibbs & Taylor (2005). Typically, for a simulation runningfor 20 periods, the total energy, I2, is conserved to one part in 108 and theHamiltonian, I4, to one part in 104.

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

We now look at the changes to the spectrum and the parts of the conservedquantities of an initially focused group as it disperses. Figure 5a shows theevolution of the terms in the I4-conserved quantity over time: as predicted, thefirst part goes to a constant limit and the second quadratic part decays to zero.Figure 5b shows the changes in bandwidth over time. The bandwidth is found byfitting a Gaussian to the wavenumber spectrum using a least-squares method.

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





0 200 400 600 800

0.2

0.4

0.6

0.8

1.0(a) (b)

time from focus (periods)0 200 400 600 800

0.70

0.75

0.80

0.85

0.90

0.95

1.00

time from focus (periods)

Figure 5. The evolution of parameters for a typical dispersive case (Af/Sf = 0.713). (a) Evolutionof the two parts of the I4-conserved quantity, normalized by their values at focus. Thin line,∫∞

−∞ |Ux |2; thick line,∫∞

−∞ |U |4. (b) Change in bandwidth and amplitude parameter. Thin line,S/ST=0; thick line, A/AT=0. All values are normalized by the value at focus.

0.2 0.4 0.6 0.8 1.0 1.2 1.40

0.2

0.4

0.6

0.8

1.0

(Af /Sf )2

(S∞

/Sf)

2

Figure 6. Comparison of the analytically predicted change in bandwidth with numerical simula-tions. Circle, numerical (Sf = 0.58); plus, numerical (Sf = 1.16); solid line, analytical prediction.

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

The agreement between the analytical and numerical results shown in figure 6is excellent. The numerical results for the two bandwidths are very close, as isexpected from the scaling properties of the NLSE. The predicted bandwidth usingthe assumption that the group remains Gaussian is very close to that from thenumerical solution. The biggest difference occurs near the limiting Af/Sf ratiowhere the predicted bandwidth ratio is very large.

The predictions of nonlinear time scales (equation (3.22)) may be comparedwith numerical simulations. Once the A and S values have been found (asabove) in the wavenumber domain, a Gaussian group (equation (3.1)) maybe fitted to the envelope in the spatial domain, allowing τ to be estimated.





0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.5

0.6

0.7

0.8

0.9

1.0

(Af /Sf )2

τ/T

Figure 7. Comparison of predicted nonlinear time scale close to focus with numerical results. Circle,numerical (Sf = 0.58); plus, numerical (Sf = 1.16); solid line, analytical prediction.

This again illustrates the utility in introducing explicitly a nonlinear timescale as it facilitates comparison with both numerical simulations and physicalexperiments.

Figure 7 shows the averaged nonlinear time scale over the first four periodsafter the focus. While the agreement is close for the less-steep cases, thereis some discrepancy for the more nonlinear runs, but the results are still inreasonable agreement.

4. Non-dispersive groups

(a) Approximate theory for 21/4 < Af/Sf < 23/4

Numerical solutions run from a focused Gaussian group in this regime donot disperse, but after some initial transient behaviour, they oscillate arounda steady structure (figure 1d–f ). The initial Gaussian group is close in form tothe sech-shaped group used in Mei (1989)—our results are in good agreementwith the results presented in §12.6.1 of Mei. We can estimate this limit by againconsidering conservation of I2 and I4. I2 corresponds to conservation of energy,and I4, the Hamiltonian integral, is a measure of wave-group nonlinearity. First,we note that if Af/Sf > 21/4, there is a second focused Gaussian group that hasidentical I2 and I4 values. These are given by

A2 = A1

√A2

1√2S2

1

− 1 (4.1)

and

S2 = S1

(A2

1√2S2

1

− 1

). (4.2)

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

Assuming that the initial wave packet of amplitude Af (and hereafter we setthe bandwidth as Sf = 1) evolves into a single soliton and a very long and lowtail, it is reasonable to assume that all the initial I4 is carried by the soliton.





Then, we simply match the initial value of I4 to that of a single soliton, writingthe soliton amplitude as As and taking its bandwidth as given by (As/Ss) = 23/4,which is our Gaussian representation of a soliton. Then the height of the solitoncan be estimated as

A3s = 23/4A2

f

(A2

f

21/2− 1

). (4.3)

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

�E = 1 − 2A−4/3f

(A2

f

21/2− 1

)1/3

. (4.4)

When Af = 21/4, the energy leakage goes to 1 and the system is fully dispersiveand the soliton disappears. When Af = 23/4, no energy is leaked and As = Af ,which is identical to the value of A that is found to freeze dispersion in §3b andis 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 asoriginally discussed by Boussinesq. There, energy conservation rather than massconservation is used to model the evolution of the wave as it gradually movesinto shallower water leaving a long low ‘hump’ of water behind, which contains anegligible proportion of the incoming energy.

(b) Approximate theory for A/S > 23/4

In this regime, the initial focused group is taller than a simple Gaussianapproximation for a single soliton, so it reasonable to assume that two solitarywaves will emerge together with a very weak tail. Thus, we assume thatthe strength of the two solitary waves can be estimated from I2 and I4conservation, neglecting the weak contributions to each conserved quantitycarried by the dispersive tail. Assuming that each of the emerging solitons has aheight/bandwidth A/S of 23/4, and that bandwidth of the initial group at focusis Sf = 1, then the strength of each soliton can be written as

As1,s2 = 14

[21/4A2

f ± 1√3

√−32

√2 + 32A2

f − √2A4

f

]. (4.5)

Although these are closed form approximations for the strength of the two solitarywaves that emerge, the physical basis of the behaviour is clearer if we perform aseries expansion around Af = 23/4. Then the solitons are

As1 = 23/4 + 2(Af − 23/4) −

(Af − 23/4

)2

3.23/4+ O

[A1 − 23/4]3

(4.6)

andAs2 = 2

321/4 (

Af − 23/4)2 + O[Af − 23/4]3

. (4.7)

Thus, perturbations upwards away from the single soliton limit Af = 23/4

produce linear variation in the size of the larger soliton and a quadratic birth ofa very weak second soliton, in a manner comparable to hump solitons emergingfrom a sech2-hump in the Korteweg–de Vries equation for waves on shallow water.Although this analysis is only appropriate for Af > 23/4, the series expansion for





As1 given here matches that for As given in the previous section expanded aboutthe same soliton value to O[Af − 23/4]5, giving confidence in both forms. It isexpected that as nonlinearity increases a third soliton would emerge.

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

(c) Comparison with numerical simulations

The spatial evolution of the wave envelope for Af/Sf > 21/4 is shown infigure 1d–f.

We can also track the evolution of the spectrum over time by fitting a Gaussianas was done for the dispersive groups. This is shown in figure 8 along with valuesof S2 and Ss. During the initial transient behaviour, the group shape approachesthe shape of a Gaussian with amplitude A2 and bandwidth S2, but does notreach this.

Comparison of the amplitude of the soliton produced in the numerical solutionagrees well with that predicted by the approximate analytical solution (equations(4.3) and (4.5)), as shown in figure 9. As expected, the greatest discrepancy is forvalues of Af/Sf close to 21/4, 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 solutionfor A/S > 23/4. We believe that this may be real, rather than an artefact of thenumerical scheme, and is probably linked to the prediction of two solitons. Unlikehump solitons of the Korteweg–de Vries equation, in which the propagation speedincreases with the height of the soliton, solitons in the NLSE do not show anynonlinear modifications to the group velocity. Therefore, the separation of largesolitons from small will not occur in the NLSE, unlike the behaviour of solitarywave envelopes in the full water wave equations that does exhibit such amplitude-dependent modifications (Bryant 1983). The oscillation at long times representsthe continued interaction of two wave groups, and this interaction slowly leaksenergy, 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 usingthe numerical scheme developed by Bateman et al. (2001). The evolution of initialGaussian wave group at −80 periods before focus was investigated.





0

0.2

0.4

0.6

0.8

1.0 (a)

0.5

0.6

0.7

0.8

0.9

1.0 (b)

0 500 1000 15000.90

0.95

1.00

1.05

1.10

time (periods)

S/S f

S/S f

(c)

0 500 1000 15001.0

1.2

1.4

1.6

1.8

2.0

time (periods)

(d)

Figure 8. The evolution of the bandwidth for non-dispersive cases: (a) Af/Sf = 1.05(21/4),(b) Af/Sf = 1.25(21/4), (c) Af/Sf = 23/4, and (d) Af/Sf = 1.68(21/4). Solid line, numerical data;dashed line, S2/Sf ; dotted line, Ss/Sf .

21/4 1.4 1.6 23/4 1.8 2.0 2.2 2.40

0.20.40.60.81.01.21.4

A/A

f

Af /Sf

Figure 9. Comparison of amplitude of the soliton formed in the non-dispersive case with thatpredicted. Note that As1/Af and As2/Af are not physical for Af/Sf < 23/4. Plus, numerical; greyline, As/Af ; solid line, As1/Af ; dotted line, As2/Af .

Gibbs (2004) found that the effect of nonlinearity was to increase both theamplitude at focus and the local bandwidth of the wave group as it focused froman initially dispersed initial condition. The NLSE produces similar changes in theshape of the wave group, although not all nonlinear effects are captured. A typicalcomparison is shown in figure 10. It can be seen that in this case the Gaussian





–1000 –500 0 500 10000

2

4

6

8

10

x (m)

U (

m)

Figure 10. Comparison of linearized envelope of fully nonlinear numerical results with NLSEnumerical results. Input ak = 0.16. The runs were started at −80 periods before linear focus. Theoutput is at t = −17.2 periods when the wave group has highest amplitude in the fully nonlinear run.Dotted line, fully nonlinear; solid line, NLSE.

is not an accurate model for the wave group, which has pronounced sidebands(Yuen & Lake 1980), which are symmetrical in the NLSE and non-symmetricalin the fully nonlinear model. Henderson et al. (1999) report similar differencesbetweeen the solutions of the NLSE and the full water wave equations in onedimension.

The Gaussian approximation may be used to predict the bandwidth changesobserved in the fully nonlinear simulations. The predictions are based oncomparing the values of the conserved quantities at −80 periods with thoseat focus.

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

For fully nonlinear outputs, data have to be ‘linearized’ to remove boundharmonics. This is performed by running the numerical model for two cases 180◦out of phase to produce ‘crest’ and ‘trough’ focused cases. The even harmonicsare removed using

ηodd = (ηcrest − ηtrough)

2. (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 numericalmodel. All three are in good agreement for the cases with input steepness lessthan ak = 0.1. For the more nonlinear cases, the nonlinear group is less wellmodelled 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 comparedwith the NLSE. This is partly owing to the deviation from the Gaussian shape inthe numerical model, but also owing to the group not focusing perfectly becauseof the nonlinear dynamics.





0 0.02 0.04 0.06 0.08 0.10 0.121.0

1.2

1.4

1.6

1.8

2.0

ak0

s f/s

t = –

80

Figure 11. Comparison of predicted bandwidth for a focused Gaussian with maximum s innumerical simulation of fully nonlinear and NLSE scheme. The ak 0 is the amplitude of the wavegroup at focus assuming simple linear evolution, based on the initial condition. Diamond, fullynonlinear; square, NLSE; solid line, analytical prediction.

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 therelative phases required for linear focusing, then there is no guarantee that, atthe instant when the highest peak envelope occurs, all the wave components willbe exactly in phase. In fact, this will not occur. However, it is possible to startfrom an exactly focused initial condition and then allow the group to de-focus. Inthe first case, the evolution from −80 periods, through focus (which may not beat the same time as linear focus), to +80 periods will not be precisely symmetricin time. In contrast, starting at perfect focus and allowing the simulations to runboth backwards to −80 periods and forwards to +80 periods will give exactlysymmetric evolution. This latter case is the problem that our simple Gaussianmodel is intended to address.

(b) Comparison with experimental results

Baldock et al. (1996) investigated the focusing of uni-directional wave groupsexperimentally. The wave-group parameters used were somewhat different fromthose used in this paper. The wave groups were generated by a paddle 8 maway from linear focus in a 20 m long tank. This equates to around 10 periodsbefore linear focus for wave components around the peak of the spectrum. Thewavetank was 0.7 m deep. A typical kd based on the mean wavenumber ofthe components was around 3, although the lowest wavenumber componentsgenerated had kd ∼ 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 lowwavenumber components, we do not present cases A and B (Baldock et al.1996), which are the most broadbanded cases. Case D has a very narrowbandedspectrum, and Vijfvinkel (1996), who used a fully nonlinear scheme based on





10 15 20 25 30 35 40 45 50 55 601.00

1.05

1.10

1.15

1.20

1.25

Af/

Ain

put

Ainput (mm)

Figure 12. Comparison of experimental data of Baldock et al (1996) (case C) with analyticalprediction. Solid line, analytical prediction; plus, experimental data.

Craig & Sulem (1993), found poor agreement for this case when starting thesimulations with a spatial profile at a fixed point in time. Our approach alsorequires an initial spatial profile at a point in time so we do not present resultsfor this case. Instead, we look at case C that has an initial spectrum over afrequency range of 0.77–1.42 Hz.

We calculate initial values for I2 and I4 using equations (3.5) and (3.6). Thevalue of A is found directly from the summation of the amplitude componentsof the initial spectrum. The spectral width, S , is found from the variance ofthe spectrum about the mean wavenumber. For the spectrum in case C, thisgives a mean wavenumber of 4.42 m−1 and sx = 1.54 m−1. The time τ is set as10 periods before linear focus. The equations may then be solved for Af and Sfat τ = 0. We can then compare the extra amplitude predicted with that foundexperimentally. The experimental data have been linearized by combining crestand trough focused groups (Baldock 1994), as for the fully nonlinear simulationsdescribed earlier.

A comparison is shown in figure 12. It can be seen that, despite the verydifferent spectra used, there is good agreement, except for the most nonlinearcase in which it is likely that the focal quality in the experimental results wasrelatively poor.

6. Discussion: group and field properties

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

While the recent work of Onorato et al. (2001, 2002a,b, 2003), Dysthe et al.(2003) and Janssen (2003) is clearly all of great importance for the explorationof freak waves, published results from random simulations concentrate rightly





on the statistics of the field, particularly looking for divergence from the normaldistribution for the surface, or the Rayleigh distribution for wave height. Thereis seldom any detailed discussion of the local temporal or spatial structure ofextreme events that may arise. Our work is perhaps a small contribution inthis direction.

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

The work of Janssen (2003) on the statistical properties of random fields isalso important in terms of both importance of four-wave interactions and inthe context of warning shipping of the increased likelihood of unusually severeextreme waves. Can a useful BF index be validated for directional spread orperhaps crossing seas? If so, it could then be a reliable indicator of the dangerposed by freak waves, when coupled with the standard predictions of significantwave height, wave period, wave directionality, and so on. In a subsequent paperextending the approach presented here to directional spread waves, we intend toaddress some of these issues.

7. Conclusions

In this paper, we show that it is possible to model the nonlinear changes to theshape 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 andto predict the shape of the most nonlinear group accessible from an initiallydispersed group. Our results are in good agreement with numerical solutions tothe NLSE. However, the agreement is not as good, although still qualitativelycorrect, when compared with solutions to the full water wave equations.

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

Overall, our approach of using the conserved quantities to derive approximateevolution equations for Gaussian wave groups captures much of the physics ofthe problem and allows robust predictions to be made. Thus in a subsequentpaper, we propose to extend this approach to directionally spread wave groups





using the two-dimensional NLSE. The two-dimensional NLSE cannot be solvedby inverse scattering, and the additional lateral dimension means wave-groupbehaviour cannot be simply categorized using the BF index, via a simple ratio ofwave steepness to bandwidth. The directionality of the wave field also needs tobe accounted for. However, the two-dimensional NLSE is a much better modelfor real ocean waves.

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

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