Journal of Applied Mathematics and Physics, 2015, 3, 1044-1050
Published Online August 2015 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2015.38129
How to cite this paper: Wang, L., Li, J.X. and Li, S.X. (2015) Numerical Simulation of Freak Wave Generation in Irregular
Wave Train. Journal of Applied Mathematics and Physics, 3, 1044-1050. http://dx.doi.org/10.4236/jamp.2015.38129
Numerical Simulation of Freak Wave
Generation in Irregular Wave Train
Lei Wang, Jinxuan Li*, Shuxue Li
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China
Received 15 July 2015; accepted 19 August 2015; published 26 August 2015
Abstract
In this paper, a numerical wave tank is developed based on High Order Spectral (HOS) method
considering the wave-maker boundary. The 2D irregular wave trains are simulated for a long time
by using this model. The freak wave is observed in the wave train, and its generation process is
analyzed via wavelet analysis. The results show that the numerical tank can accur ate ly simulate
the wave generation and propagation, even for the freak wave. From the analysis of freak wave
generation, it can be found that, two wave groups with different frequency components superpose
together to form a large wave group. The large wave group modulation generates the freak wave.
Keywords
Freak Wave, High Order Spectral Method, Numerical Wave Tank
1. Introduction
The freak wave (also called extreme wave, rouge wave) was defined as the wave whose height is more than
twice the significant wave height, which represents the extreme oceanic conditions. It has larger wave height
and stronger nonlinear, and can cause serious damage to the oceanic structures and ships. In recent years, as
frequent human activities move towards the deep ocean, the harsh marine environment becomes a thread to
oceanic engineering. Accurate estimation and understanding of severe extreme events are essential for human
safety and for cost-effective design.
In the past years, many researchers paid attentions to the studies of freak wave [1]-[3]. Some mechanisms
were presented to explain the generation of freak wave, such as, modulation instability (or Benjamin-Feir insta-
bility) [4] [5], wave focusing [6] [7], and wave-current or wave-wind interaction [8]-[10]. Based on these me-
chanisms, the freak wave was generated and studied through theoretical, experimental and numerical methods.
The theoretical analysis was mainly based on modulation instability. Benjamin-Feir [11] found that the Stokes
wave was unstable to small perturbations, and the unstable sideband components would grow exponentially.
This phenomenon can be described by the nonlinear Schrödinger equation. Dysthe &Trulsen [12] and Osborne
et al. [13] showed the breather solutions of nonlinear Schrödinger equation were considered as simple analytical
models of freak waves.
*
Corresponding author.

L. Wang et al.
1045
There were many physical experiments on the generation of freak wave. Onorato et al. [14 ] [15] studied freak
wave generation in long-crested sea conditions initialized as a JONSWAP spectrum. Li et al. [16] observed the
generation process of freak wave in unstable irregular wave train and analyzed the freak wave based on wavelet
anal ysis. Baldock et al. [17] investigated nonlinear surface water waves undertaken in 2-dimenional wave flume
by a large number of water waves were focused at one point in space and specific time to produce a large tran-
sient wave group. Chin and Yao [18] investigated the influence of current on the freak wave generation in 2D
wave flume.
Numerical model was also an important approach to study the freak wave generation and its hydrodynamics.
It can solve the problem that in physical experiment, the length of the wave tank cannot satisfy requirement of
the long-time evolution of wave surface. The most common method is based on nonlinear Schrödinger equation.
Onorato et al. [19] studied freak wave generation in long-crested sea conditions based on the cubic nonlinear
Schrödinger equation. While, the nonlinear Schrödinger equation is weak nonlinear equation, considering that
the freak wave generation is strong nonlinear process, many fully nonlinear numerical models were developed.
For example, Zhao et al. [20] simulated the freak wave generation based on VOF method. Yan and Ma [21] si-
mulated the interaction of freak wave and wind using finite element method.
In order to deeply understand the freak wave generation, a numerical wave model based on High Order Spec-
tral (HOS) method (Dommermuth and Yue [22]) is developed in this paper. Long-time irregular wave trains are
generated and the freak wave is observed and analyzed. The method solves the derivative terms by Fast Fourier
Transform (FFT) method. Compared with other methods, the spectral method has the properties of fast conver-
gence and low computational cost. It is convenient to the long-time simulation.
2. Numerical Model
2.1. Governing Equations
It is assumed that the fluid is incompressible and inviscid, and the flow is irrotational. The velocity potential Φ(x,
z, t) in the 2D fluid domain satisfies Laplace’s equation,
2
(,,) 0xzt∇Φ =
. (1)
Following Zakharov [23], the fully nonlinear free surface boundary conditions can be expressed in terms of
the velocity potential at the water surface as
2s
(1 ),
x xx
z
tz
ηη ηη ∂ ∂Φ
=+ ∇−∇Φ ∇=
∂∂
(2)
2
s22
s
11
g(1),
22 z
xx
z
t
η ηη
∂Φ ∂Φ

=−− ∇Φ++∇=

∂∂

(3)
where Φs(x, t) = Φ(x, η, t) is the velocity potential on the water surface z = η.
On the fixed boundaries (the bottom, side and end walls of the tank), the conditions can be written simply as
0n∇Φ ⋅=

(4)
where
n

is a vector normal to the boundary considered.
If the wave maker boundary is assigned to the left of the wave basin, the corresponding boundary condition
can be written as
(,) on 0 X ytx
xt
∂Φ ∂
= =
∂∂
(5)
where X(t) is the displacement of the wave maker board.
2.2. Numerical Procedure
In order to solve this non-homogeneous problem, following Agnon and Bingham [24], the velocity potential can
be split into the sum of a prescribed non-periodic components Φw and an unknown period component Φf, i.e.,
fw
Φ=Φ +Φ
(6)

L. Wang et al.
1046
whe r e the prescribed non-periodic components Φw satisfies the wave maker boundary condition (5) and other
lateral boundary conditions (4). So, Equat io ns (2) and (3) can be rewritten as follows:
2w
ffw
(1)() ,
s
x xx
t zz
z
ηηη
η
∂Φ ∂Φ ∂=+ ∇+−∇Φ+Φ⋅∇
∂ ∂∂
=
(7)
2
22
ff
f
2
2
ww
fw w
11
(1 )
22
11
,
22
ss
xx
s
xx x
g
tz
tz
z
ηη
η

∂Φ ∂Φ
=−− ∇Φ++∇
∂∂

∂Φ ∂Φ

−∇Φ⋅∇Φ−−∇Φ−
∂∂

=
(8)
Therefore, determining Φw becomes the first key problem in establishing the numerical model using the HOS
method. Different expansions can be used, such as, Agnon and Bingham [24] used polynomials in 2D wave tank
simulation. Bonnefoy et al. [25] employed a spectral expansion in 2D and 3D wave tank simulation. In this pa-
per, we use Bonnefoy’s method. The non-periodic components can be written as:
0
cosh[ ()]cos[ ()],
cosh[ ]
z
Npx
wp
ppx
kL xkzh
kL
=
−
Φ= +
∑
(9)
in which kp = (2p − 1)π/4h, h is water depth.
The unknown period component Φf satisfies the free surface Equations (2), (3) and lateral boundary condition
(4). It can be solved by HOS method proposed by Dommermuth and Yue [22].
2.3. Verification of the Numerical Model
In order to verify the accuracy of the numerical model, the two-dimensional experimental focused wave data
was used. The focused wave experiment was conducted in a two-dimensional wave flume. The focusing method
was introduced in Li and Liu [26]. The two cases wave, with A = 0.03 m, f = 0.5 - 1.16 Hz, fc = 0.83 Hz and A =
0.06 m, f = 0.5 - 1.36 Hz, fc = 0.83 Hz, were adopted. The corresponding wave steepness kcA were 0.09 and 0.18,
respectively. The waves were assumed to be focused at a distance 11.4m from the wave-maker. The NJS ampli-
tude distribution was used, and the water depth was 0.5 m (kch = 1.54). The numerical wave tank is 40 m long.
Figure 1 and Figur e 2 present the comparisons of the numerical and experimental time histories of the wave at
the focusing point. The figures show that the numerically simulated focused wave surfaces agree well with the
experimental data. However, the numerical and experimental values of the focused wave amplitude are larger
than the assumed amplitude because of the nonlinear wave evolution.
Figure 1. Comparison between experimental data (dotted) and numerical results (solid lines)
for two-dimensional focused waves for the case with A = 0.03 m, f = 0.50 Hz - 1.16 Hz, fc =
0.83 Hz.

L. Wang et al.
1047
Figure 2. Comparison between experimental data (dotted) and numerical results (solid lines)
for two-dimensional focused waves for the case with A = 0.06 m, f = 0 .50 Hz - 1.16 Hz, fc =
0.83 Hz.
3. The Simulation of Freak Wave Generation in Random Trains
In this paper, long-time irregular wave trains are simulated to generate and observe the freak wave. The incident
irregular waves are modeled by a superposition of regular waves with different frequencies with random phases.
The frequency spectrum adopts the JONSWAP spectrum with the peak enhancement factor
γ = 3.3. The peak
frequency is fp = 1.0 Hz. In order to avoid the numerical calculation instability owing to the wave breaking, the
significant wave height of the wave train was adopted 0.037 m. The corresponding BFI value is 0.58. Although
the value of BFI is moderate, the freak wave can be generated during long time simulation (see Li et al. [16]).
The wave tank is 100 m long, and the water depth is 1.2 m.
From long time simulation, the freak wave occurrence can be observed. Figure 3 gives short segments of time
series of wave surface elevation at different locations. The freak generation process can be observed from the
figure. At the location near the wave-maker, there are no signs of freak wave. With the wave propagation, some
large wave generates. At 29.45 m from the wave maker, owing to the modulation of wave group, these large
waves focus together and form a freak wave. The freak wave is similar to the typical form of “New Year Wave”.
After the location 29.45 m, the freak wave disappears, these waves are dispersed and the number of wave in
group increases. Comparing to physical experiment, the long distance evolution of freak wave can be conve-
niently observed in the numerical model simulation.
In order to analyze the process of freak wave generation, the evoluti on of the wavelet ener g y spectra at se-
lected positions fro m the wave-ma ker is shown in F igur e 4. In the figure, thesurfaceelevationofthewavesis-
sho wnbythewhitecurvesuperimposedontheenergyisolines.Two successive wave groups are seen at 9.0 m from
the wave-mak er. A group with comparatively low-frequency components is lagging behind the group with high-
frequency components. Then, at the 19 m location, the group with low-frequency components, travelling at
greater speed, is merging with the higher-frequency group. At 29.45 m from the wave-maker, owing to the
modulation of wave group, the freak wave occurs, the wavelet spectrum isolines show an almost symmetrical
triangular form and higher-frequency components and higher energy values are evident, indicative of stronger
nonlinear interactions at the time of the freak wave generation. Then, at 50 m to 60 m from the wave-maker, the
surfaces begin the group demodulation, the wavelet spectra show that the low-frequency component is at the
front of the wave group.
4. Conclusion
In order to deeply understand the freak wave generation, a 2D numerical wave tank is developed based on high
order spectral method considering the non-periodic boundary condition in this paper. Long time irregular wave
simulation is conducted. The freak wave generation process is analyzed by wavelet analysis. The simulation re-
sults show that the numerical model is valid to simulate the freak waves. From the process of freak wave gener-
ation in the simulation, it can be found that, two wave groups with different frequency components superpose
together to form a large wave group. The large wave group modulation generates the freak wave.

L. Wang et al.
1048
Fig ure 3. The time series of wave surface elevation at different locations.
Figure 4. Wavelet energy spectra at different distances.
Acknowledgements
This work was financially supported by the National Natural Science Foundation of China (Grant Nos.
51309050, 51221961 and 51179028). Their supports are gratefully acknowledged.
References
[1] Kharif, C. and Pelinovsky, E. (2003) Physical Mechanisms of the Rogue Wave Phenomenon. European Journal of

L. Wang et al.
1049
Mechan ics—B/Fluids, 22, 603-634 . http://dx.doi.org/10.1016/j.euromechflu.2003.09.002
[2] D yst he , K., Krogstad , H.E. and Muller, P. (2008 ) Oceanic Ro guewa ves. Annu. Rev. Fluid Mech., 40, 287-310.
http://dx.doi.org/10.1146/annurev.fluid.40.111406.102203
[3] Slun yaev, A., Didenkulova, I. and Pelinovsky, E. (2011) Roguewat ers. Contemp. Phys., 52, 571-590.
http://dx.doi.org/10.1080/00107514.2011.613256
[4] Janssen, P. (2003) Nonlinear Four-Wave Interaction and Freak Waves. J Phys Oceanogr, 22, 863-884.
http://dx.doi.org/10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2
[5] Clamond, D. and Grue, J. (2002) Interaction between Envelope Solitons as a Model for Freak Wave Formations. Part I:
Long Time Interaction. C R Mec, 330, 575-580. http://dx.doi.org/10.1016/S1631-0721(02 )01 496 -1
[6] Johannessen, T. and Swan, C. (2001) A Laboratory Study of the Focusing of Transient and Directionally Spread Sur-
face Water Waves. Philos Trans R SocLond, Ser A, 4 57, 971 -10 06.
http://dx.doi.org/10.1098/rspa.2000.0702
[7] Liu, S. and Hong, K. (2005) Physical Inve stigation of Directional Wave Focusing and Breaking Waves in Wave Basin.
China Ocean Eng, 19, 21-35 .
[8] Hjelmervi, K.B. and Trulsen, K. (2009) Freak Wave Statistics on Collinear Currents. Journal of Fluid Mechanics, 637,
267-284. http://dx.doi.org/10.1017/S0022112009990607
[9] Toffoli, A., Cavaleri, L., Babanin, A.V., Benoit, M., Bi tn er -Gregersen, E.M., Monbaliu, J., Onorato, M., Osborne, A.R.
and Stansberg, C.T. (2011) Occurrence of Extreme Waves in Thre e -Dimensional Mechanically Generated Wave Fields
Propagating over an Oblique Current. Natural Hazards and Earth System Sciences, 11, 895-903.
http://dx.doi.org/10.5194/nhess-11-895 -2011
[10] Kharif, C., Giovanangeli, J.P., Touboul, J., Grare, L. and Pelinovsky, E. (2008) Influence of Wind on Extreme Wave
Event s: Experimental and Numerical Approaches. Journal of Fluid Mechanics, 594, 209 -247.
http://dx.doi.org/10.1017/S0022112007009019
[11] Benjamin, T.B. and Feir, J.E. (1967) The Disintegration of Wavetrains on Deep Water. Part 1: Th e ory. J. Fluid Mech.,
27, 417-430. http://dx.doi.org/10.1017/S002211206700045X
[12] Dysthe, K.B. and Trulsen, K. (1999) Note on Breather Type Solutions of the NLS as a Model for Freak Waves. Phys i-
ca Scripta, T82, 48-52 . http://dx.doi.org/10.1238/Physica.Topical.082a00048
[13] Osborne, A.R., Onorato, M. and Serio, M. (20 00) The Nonlinear Dynamics of Rogue Waves and Holes in Deep-Water
Gravity Wave Train. Phys. Letters A, 27 5 , 386 -393. http://dx.doi.org/10.1016/S0375-9601(00)00575-2
[14] Onorato, M., Osborne, A.R., Serio, M. and Cavaleri, L. ( 2005 ) Modulational Instability and Non -Gaussian Statistics in
Experimental Random Water-Wave Trains. Physics of Fluids, 17, Article ID: 078101.
http://dx.doi.org/10.1063/1.1946769
[15] Onorato, M., Osborne, A.R., Serio, M., Cavaleri, L., Brandini, C. and Stansberg, C.T. ( 2004 ) Observation of Strongly
Non-Gaussian Statistics for Random Sea Surface Gravity Waves in Wave Flume Experiments. Phys Rev E, 70, Article
ID: 067302. http://dx.doi.org/10.1103/physreve.70.067302
[16] Li, J., Yang, J., Liu, S. and Ji, X. (2015) Wave Groupiness Analysis of the Process of 2D Freak Wave Generation in
Random Wave Trains. Ocean Engin eeri ng, 1 04, 4 80-48 8. http://dx.doi.org/10.1016/j.oceaneng.2015.05.034
[17] Baldock, T., Swan, C. and Taylor, P. (1996) A Laboratory Study of Nonlinear Surface Waves on Water. Philos. Roy.
Soc. London, 452A, 649 -676. http://dx.doi.org/10.1098/rsta.1996.0022
[18] Chin, H. and Yao, A. (2004) Laboratory Measurement of Limiting Freak Waves on Currents. Journal of Geophysical
Research , 109 , Article ID: C12002.
[19] Onorato, M., Osborne, A.R., Serio, M. and Bertone, S. (2001 ) Freak Waves in Random Oceanic Sea States. Phys Rev
Lett, 86 , 5831-5834. http://dx.doi.org/10.1103/PhysRevLett.86.5831
[20] Zhao, X.Z., Hu, C.H. and Sun, Z.C. (2010) Numerical Simulation of Extreme Wave Generation Using VOF Method.
Journal of Hydrodynamics, 22, 466 -477. http://dx.doi.org/10.1016/S1001-6058(09)60078-0
[21] Yan, S. and Ma, Q.W. (2010) Numerical Simulation of Interaction between Wind 2D Freak Waves. European Journal
of Mechanics, B/Fluids, 29, 18-31. http://dx.doi.org/10.1016/j.euromechflu.2009.08.001
[22] Dommermuth, D.G. and Yue, D.K.P. (1987) A High-Order Spectral Method for the Study of Nonlinear Gravity Waves.
Journal Fluid Mechanics, 184, 267 -288. http://dx.doi.org/10.1017/S002211208700288X
[23] Zakharov, V.E. (1968) Stability of Periodic Waves of Finite Amplitude on the Surface of a Deep Fluid. J. Appl. Mech.
Tech. Phys, 9, 190-194 (English Tra n sl .). http://dx.doi.org/10.1007/BF00913182
[24] Agnon, Y. and Bingham, H. (1999) A Non-Periodic Spectral Method with Application to Nonlinear Water Waves.
European Journal Mech. B/Fluids, 18, 527-534. http://dx.doi.org/10.1016/S0997-7546(99)80047-8

L. Wang et al.
1050
[25] Bonnefoy, F., Touzé, D. and Le Ferrant, P. (2004) Generation of Fully-Nonlinear Prescribed Wave Fields Using a
High -Order Spectral Model. P ro c. 14th International Offshore and Polar Engineering Conference, Toulon.
[26] Li, J. and Liu, S. (2015) Foused Wave Properties Based on a High Order Spectral Method with a Non-Periodic Boun-
dary. China Ocean Engineering, 29, 1-16. http://dx.doi.org/10.1007/s13344-015-0001-7