応用力学研究所研究集会報告No.17SP1-2

「海洋巨大波の実態と成因の解明」（研究代表者　冨田宏）

Reports of RIAM Symposium No.17SP1-2

Study on features and generation mechanisms of freak waves

Proceedings of a symposium held at Chikushi Campus, Kyushu Universiy,Kasuga, Fukuoka, Japan, March 10 - 11, 2006

Research Institute for Applied Mechanics

Kyushu University

June, 2006

Article No. 04

Simulation of the ocean wavesand appearance of Freak waves

Igor Ten，冨田 宏（TOMITA Hiroshi）

（Received May 31, 2006）

Simulation of the Ocean Waves AndAppearance of Freak Waves

I. TEN and H. TOMITA

National Maritime Research InstituteMitaka, Tokyo, Japan

1 Introduction

Freak waves, one of the marine disasters, appear suddenly as an isolated majestic wave.Its height is more than twice greater than its surrounding companies. It was recognizedin earlier paper by Klinting [1] from the wave record taken at an observation site ofDenmark in the North Sea in 1981. After that many events like that features were foundin several sea areas worldwide including the North Pacific Ocean and the Sea of Japan.Those results were reported in the Rogue Wave 2000 Symposium [2] held in Brest. Inthe symposium, many papers were presented for the mechanism of Freak wave generationsuch as caused by currents, wind change and nonlinear wave interactions. The secondround of this symposium was held at the same place in 2004. Many sort of numericaltechniques to investigate such phenomena were presented and the practical problems suchas the prediction of Freak wave were proposed in this symposium. However any generalconsensus has not been made up to present.

In this paper we used Boundary Element Method to simulate Freak waves in the ocean.We confirmed that our numerical scheme is valid for generating highly nonlinear wavesby comparison numerical results with experimental and theoretical ones (the 3rd orderStokes’ waves).

We generated one of the exact solutions of Non Linear Shroedinger equation (breathersolution) in our Numerical Wave Tank (NWT). Next, we produced irregular waves inNWT by applying Pierson-Moskowitz and Swell spectra.

Finally, we studied behavior of solitons and random waves, and interaction of breathersolution waves with irregular ones.

2 Problem Statement

We consider a two-dimensional wave channel bounded by flat bottom at z = −h and twoend walls at x = ±L, where wave flume length is 2L, and the x-axis of the coordinatesystem coincides with undisturbed free surface and the z-axis positive upward. Fluid isassumed to be homogeneous, incompressible, inviscid and its motion irrotational. At theleft-hand side a wave-maker of plunger/flap/piston type is installed. The flow can bedescribed by the velocity potential, ϕ, which satisfies Laplace’s equation

∇2ϕ(t, x, z) = 0, in the fluid, (1)

1

with boundary conditions∂ϕ

∂z= 0, on z = −h, (2)

∂ϕ

∂x= 0, on x = L, (3)

∂ϕ

∂n= n · V = Vn, on x = −L, (4)

∂ϕ

∂t+

1

2∇ϕ · ∇ϕ + ρgz = 0, on free surface z = η(t, x), (5)

where n, and V are outward normal vector and velocity vector of the wave maker, respec-tively.

In the analysis all quantities are nondimensionalized in terms of a characteristic lengtha as the following

x = ax′; z = az′; ϕ = a√

agϕ′; t =√

a/gt′. (6)

Hereafter, the prime, which means nondimensional variable, will be omitted.For the fixed point (Lagrangian method) on the free surface P (x, z), time derivatives

of the x, z and ϕ areDx

Dt=

∂ϕ

∂x;

Dz

Dt=

∂ϕ

∂z, (7)

Dϕ

Dt=

∂ϕ

∂t+∇ϕ · ∇ϕ, (8)

where D/Dt is material (substantial) derivative. Substitution (8) into (5) yields

Dϕ

Dt= −z(t, P ) +

1

2∇ϕ · ∇ϕ (9)

Equations (1)-(4), (7), and (9) form mixed Euler-Lagrangian system of equations tobe solved to determine surface elevation.

To avoid reflection from the right-hand side wall an absorbing beach should be used.In this paper we used the following method

Dx

Dt=

∂ϕ

∂x

Dz

Dt=

∂ϕ

∂z− 2νz on wave absorbing beach, (10)

which is similar to suggested in [3]. Here ν is nonzero inside the beach and given by

ν = 3.6(x− xf )3/C3

w for x > xf , (11)

where xf and Cw denote the starting point and the length of the wave absorbing beach,respectively.

3 Numerical scheme

Application of the Green’s second identity to the Laplace’s equation (1) yields

c(Q)ϕ(Q) =

∫

S

[ϕ(P )

∂ ln r(P,Q)

∂n− ∂ϕ(P )

∂nln r(P, Q)

]dS, (12)

2

where boundary S is common boundary, r = ((xP − xQ)2 + (zP − zQ)2)1/2, and C(Q)equals 1 if the point Q locates inside domain and 1/2 on the boundary. Integrals overrigid bottom, left-hand side wall give zero contribution, so the above equation transformsinto the following form

−c(Q)ϕ(Q) +

∫

Sf+Swm

[ϕ(P )

∂ ln r(P, Q)

∂n− ∂ϕ(P )

∂nln r(P, Q)

]dS = 0. (13)

To solve Eq. (13) we applied 2-point collocation method. The boundary was divided ontoN elements, where unknown variables can be presented as

(x, z, ϕ,∂ϕ

∂n) =

2∑i=1

Ni(ξ)(xi, zi, ϕi,

(∂ϕ

∂n

)

i

) (14)

with the shape functions N1 = 1− ξ, N2 = ξ, ξ = ((x− x1)2 + (z − z1)

2)1/2/((x2− x1)2 +

(z2 − z1)2)1/2. Then, we have

−1

2ϕj +

N∑i=1

2∑

k=1

[ϕ

(k)i

∫

Si

Nk∂ ln rj

∂ndS −

(∂ϕ

∂n

)(k)

i

∫

Si

Nk ln rjdS

], (15)

where Si = [(xi, zi), (xi+1, zi+1)], ϕ(k)i is value of ϕ at node point k of interval Si, and

rj = ((x−xj)2+(z−zj)

2)1/2. The integrals along interval Si can be calculated analytically,e.g. [4]. Let us define Tk,ij and Lk,ij as follows

Tk,ij =

∫ (xi+1,zi+1)

(xi,zi)

Nk∂ ln rj

∂ndS, Lk,ij =

∫ (xi+1,zi+1)

(xi,zi)

Nk ln rjdS. (16)

Substitution (16) into (15) yields

−1

2ϕj +

N∑i=1

2∑

k=1

[ϕ

(k)i Tk,ij −

(∂ϕ

∂n

)(k)

i

Lk,ij

]= 0 for j = 1, 2, .., N + 1. (17)

Taking into consideration that velocity potential and its derivative are continuous func-tions, we may regroup above equation to obtain system of simultaneous equations to besolved numerically for each time step.

Once velocity potential and its normal derivative are found we can proceed to the nexttime step by application of 4th order Runge-Kutta-Gill method.

For long-time simulations, re-arranging of nodes on the free surface (referred to as theregridding) is indispensable to avoid the cluster of nodes. In this paper we adopt theregridding scheme described in [5].

The above described numerical scheme was confirmed by comparing some numericalresults with experimental ones [6] and a good agreement were found. In addition, weillustrate the possibility of generating very steep nonlinear waves and transient part ofgenerated waves.

In Fig. 1 it is shown the dependence of generated wave height on wave maker strokesignal. As it is known, the Stokes’ waves do not exceed the maximum steepness, see forexample [7], H/λ = 1/7. The calculation has been done for the frequency f = 1 Hz, thepaddle type wave maker fixed at position 1.5 m beneath the free surface, and installed in

3

0 0.05 0.1 0.15 0.20

0.05

0.1

0.15

0.2

0.25

Stroke, m

Wa

ve

hei

ght,

m

Linea

r

Theoretical maximum

Figure 1: Nonlinear dependence of wave height with respect to wave maker stroke. Paddle typewave maker fixed at 1.5 m beneath the free surface, water depth is 4.5 m, f = 1 Hz.

the 4.5 m in deep wave tank. In this figure we can see that for the waves of small steepnesstheir height proportional to the stroke signal. The larger the stroke, curve declines fromthe line and tends to the maximum possible wave height.

In Fig. 2 it is shown the comparisons for the regular waves for the plunger type(left, depth: 0.5 m, f = 0.5 Hz, stroke: 0.04 m) and paddle type (right, depth: 4.5 m,f ≈ 1.25 Hz, stroke: 0.0188 m) wave makers including transient wave front. Wellagreements are found.

0 3 6 9 12 15-0.02

-0.01

0

0.01

0.02

Time, sec

Fre

e su

rfa

ce e

lev

ati

on

, m

Numerical computation

Experiment

15 20 25-0.04

-0.02

0

0.02

0.04

Time, sec

Numerical computation

Experiment

Fre

e su

rfa

ce e

lev

ati

on

, m

Figure 2: Comparison between computed and measured results for the plunger type (left, depth:0.5 m, f = 0.5 Hz, stroke: 0.04 m) and paddle type (right, depth: 4.5 m, f ≈ 1.25 Hz, stroke:0.0188 m) wave makers.

Finally, we compared the 3rd order Stokes’ waves with calculated wave of high steepness(H/λ ≈ 0.128). The results are shown in Fig. 3. We can notice a good agreement betweentwo curves.

4

24 25 26 27 28 29 30-0.2

-0.1

0

0.1

0.2

Time, sec

Fre

e su

rfa

ce e

lev

ati

on

, m

3-rd order Stokes’s wavesNumerical computation

Figure 3: Comparison of the free surface waves of high steepness (H/λ ≈ 0.128) with the 3rd

order Stokes’ waves. Paddle type wave maker, water depth 4.5 m, f = 1 Hz.

4 Possible creation mechanisms of waves of high steep-

ness

We suggested several mechanism of waves of high steepness (wave interference, Benjamin-Feir instability, solitons) [6]. Here we discuss about breather waves, random waves andinteraction between them.

4.1 Breather waves

The second group (the first one is solitons) of solutions of Nonlinear Shrodinger equationis breather type solutions. The first one is the Akhmediev breather [8] (Soliton on FiniteBackground [9], [10]) nondimensional form of which can be written as the following

qA = e2iT cosh(ΩT − 2iϕ)− cos(ϕ) cos(pX)

cosh(ΩT )− cos(ϕ) cos(pX), (18)

where Ω = 2 sin(2ϕ), p = 2 sin(ϕ) for a real parameter ϕ. This solution, as shown in Fig.4 (left), in plane of nondimensional variables (X, T ) is space-periodic function.

The next solution, which is time-periodic solution in plane of nondimensional variables(X,T ), see Fig. 4 (right), is the Ma breather [11], or Ma-soliton, solution. Its presentationis given as follows

qM = e2iT cos(ΩT − 2iϕ)− cosh(ϕ) cosh(pX)

cos(ΩT )− cosh(ϕ) cosh(pX), (19)

where Ω and p as same as for Akhmediev solution.In the limited case when the parameter ϕ tends to zero, the Akhmediev and Ma

breathers tend to

qP = e2iT

[1− 4(1 + 4iT )

1 + 4X2 + 16T 2

], (20)

which is called as Peregrine breather [12] (algebraic, isolated Ma soliton, rational breather,explode-decay solitary wave [13]).

5

Figure 4: Akhmediev breather (left), which is space periodic solution, and Ma breather (right),which is time periodic solution of self-focusing Nonlinear Shrodinger equation.

There is one more solution, which is double-periodic solution [8]

qAp =k√2eiT

cd(X

√1+k2

,√

1−k1+k

)cn(t, k) + i

√1 + ksn(t, k)

√1 + k − cd

(X

√1+k2

,√

1−k1+k

)dn(t, k)

, (21)

where sn, cn, dn, and cd are Jacobi elliptic functions, and the parameter k is restrictedby 0 6 k 6 1.

The maximum of Peregrine breather approaches at T = 0, and X = 0 and equals|qPmax| = 3 which means that the peak is three times higher than its surrounded waves.In the case of Ma breather the maximum |qMmax| = 2 cosh ϕ + 1. It means, the larger theparameter ϕ, the steeper waves at the wave peak. On the other hand the number of wavesin group which are steeper their surrounded waves becomes smaller when the parameterincreases. Fig. 5 illustrates these behaviors. We performed simulation of the Peregrine

-5 0 5-0.1

0

0.1

0.2

0.3

Time, sec

En

velo

pe p

rofi

le, m

φ = 0.0 (Peregrine breather)φ = 1.0

φ = 1.5

φ = 2.0

-5 0 5-0.1

0

0.1

0.2

0.3

time, sec

Wav

e p

ack

ets

, m

φ = 2.0

φ = 0.0

Figure 5: The larger the parameter ϕ, the steeper the waves, and the smaller the number ofwaves at maximum. The steepness of surrounding waves is A0k = 0.1, f = 1 Hz.

6

40 60 80 100

x = 5.6 m

x = 2.0 m

x = 38.0 m

x = 34.4 m

x = 30.8 m

x = 27.2 m

x = 23.6 m

x = 20.0 m

x = 16.4 m

x = 12.8 m

x = 9.2 m

time, sec

Su

rface

ele

vati

on

, m

0.1

8 m

Figure 6: Evolution of Peregrine breather. A0k ≈ 0.1, λ = 1 m, tB = 70 m, and xB = 15 m.

breather waves by the generating the free surface elevation by the following equation

η(x, t) = A0Re

exp

[(1 +

A20k

2

2

)iωt1

]1− 4(1− iA2

0k2ωt1)

1 + 8A20k

2(xB + ω2k

t1)2 + A40k

4ω2t21

(22)

at the position x = 0, with initial disturbance. Here is xB and tB (t1 = t − tB) are thespace and time where the maximum appears. The results are shown in Fig. 6 for the caseof wave length λ = 1 m, A0k ≈ 0.1, tB = 70 sec, and xB = 15 m.

4.2 Random waves

Waves in the sea have many aspects. They appear as the wind starts to blow, grow intomountainous waves amid storms, and almost disappear after the wind ceases blowing, soonly swell waves rest. Such changeability is one aspect of sea waves. An observer on aboat in the offshore region easily recognized the pattern of wave forms as being made upof a large and small waves moving in many directions. The irregularity of wave form isan important feature of waves in the sea.

Sea waves, which at first sight appear very random, can be analyzed by assuming thatthey consists of an infinite number of wavelets with different frequencies and directions.The distribution of the energy of these wavelets with respect to frequency is called wavespectrum.

The typical form for a wind-waves is the Pierson-Moskowitz spectrum defined as fol-lows

S(ω) = 0.0081g2 1

ω5exp

−0.74

( g

U

)4 1

ω4

, (23)

where U is a wind speed at a height 19.5 m above sea level. Statistical analysis allows usto obtain significant wave height Hs, average frequency ω, and bandwidth ν.

Hs = 4

√ζ2 = 4

√m0 = 0.2092

U2

g, (24)

ω =m1

m0

= 1.1366g

U, (25)

ν =

√m2m0

m21

− 1 = 0.4247, (26)

7

where mr are the r-th momentums. Fig. 7 shows two kind of spectra for the three differentsignificant wave heights Hs = 0.08 m, 0.10 m, and 0.12 m.

H_s = 0.08 m

H_s = 0.10 m

H_s = 0.12 m

f, Hz

S(f

),m

^2/s

ec

0 1 2 30

0.0005

0.001

0.0015

0.002

0.0025

H_s = 0.08 m

H_s = 0.10 m

H_s = 0.12 m

f, Hz

S(f

),m

^2/s

ec

0 1 2 30

0.0005

0.001

0.0015

0.002

0.0025

Figure 7: Pierson-Moskowitz spectra (left) and Swell spectra (right). Hs = 0.08 m, 0.10 m,and 0.12 m.

For ocean swell the Swell spectrum is used [14] and it is given as the following

S(ω) =α√ω

exp

−n

2

(βω +

1

βω

), (27)

where α, β and n are constants. Simple algebraic manipulations gives

ζ2 =

√2πα√βn

e−n, ω =1 + 1/n

β, ν =

√n + 2

n + 1, (28)

where the spectral width parameter ν for ocean waves approximately equals 0.15, [14].Knowing ν we can simply determine n. Specifying average frequency and wave height wecan determine parameters α and β.

The surface elevation can be found as the superposition of all wave components

ζ(t) =N∑

j=1

√2S(fj)∆fj cos(2πfjt + ϕj), (29)

where phases ϕj are given as random numbers. The selection of frequencies was done in thefollowing wave, [15]. First, the range of frequencies from the lowest, fmin, to the highest,fmax, was divided into (N − 1) sub-ranges with the dividing frequencies constituting apower series of

f ′1 = fmin +fmax − fmin

N − 1, f ′j = CNf ′j−1, (30)

where

CN =

[fmax

f ′1

]1/(N−2)

. (31)

Then, the secondary dividing frequencies f ′′j were chosen at random in respective sub-ranges. The initial frequency f ′′0 was set equal to fmin and the last one was f ′′N = fmax.Finally the component frequency fj and its band width ∆fj were calculated as

fj =1

2(f ′′j−1 + f ′′j ), ∆fj = f ′′j − f ′′j−1. (32)

8

In Fig. 8 it is shown the propagation of waves generated by the wave maker using theSwell spectra and measured at several locations. We used the following parameters togenerate random waves:

Hs = 4

√ζ2 = 0.10 m, f = 1.0 Hz, ν = 0.15. (33)

We remark several waves in this calculation. At (t, x) ≈ (93.0 sec, 23.6 m) thewave height approach H ≈ 0.152 m, at (115.0 sec, 18.8 m) – H ≈ 0.163 m, at(180.0 sec, 23.6 m) – H ≈ 0.171 m. The calculation was performed for the numeri-cal wave tank of the following sizes: length = 30.0 m depth = 0.5 m.

0

0.1

8S

urf

ace

ele

va

tio

n,

m

20 40 60 80 100 120 140 160 180 200 220

x = 4.4 m

x = 2.0 m

x = 6.8 m

x = 9.2 m

x = 11.6 m

x = 14.0 m

x = 16.4 m

x = 18.8 m

x = 21.2 m

x = 23.6 m

x = 26.0 m

time, sec

Figure 8: Random waves generated by wave maker using the Swell spectra.

The same signal for the wave maker displacement was used for both the physicalexperiment and numerical computation. Wave gauge was set at position x = 6.1 m.Obtained measurements were compared with each other and results are shown in Fig. 9(Pierson-Moskowitz spectra) and in Fig. 10 (Swell spectra). Here we used the followingparameters:

Hs = 0.08 m(Pierson-Moskowitz and Swell spectra) (34)

f = 0.915816 Hz(output result for P-M spectra and input for Swell spectra) (35)

ν = 0.15. (36)

In addition, it was used the same generated sequences of frequencies and phases.In Fig. 9 and 10, left, the solid red lines show the experimental measurements of

the free surface and the solid black lines shows numerical results. One can notice thatagreement very good. In Fig. 9 and 10, right, the solid lines show the input spectrawhich we expect to obtain as output. Filled triangles are results obtained by analysis ofmeasured free surface at experiment and, on the other hand, open circles are the resultsof analysis of numerical experiment. For statistical analysis the series more than 100 wereused and average values were plotted.

4.3 Solitons/breather solution and random waves

The most interesting part is to observe the behavior of solitons, breather solutions in realsituation. We simulate real ocean by random waves of swell spectrum and generated two

9

Su

rfa

ce e

lev

ati

on

, m

time, sec

0.2

0.1

0.0

-0.1

-0.2

0.2

0.1

0.0

-0.1

-0.2

0.2

0.1

0.0

-0.1

-0.2

0.2

0.1

0.0

-0.1

-0.2

0 10 20 30 40 50

60 70 80 90 100 110

220210200190180170

120 130 140 150 160

CalculationExperiment

0.0012

0.0010

0.0008

0.0006

0.0004

0.0002

0.0000

0 0.5 1 1.5 2 2.5 3f, Hz

S(f

), m

/se

c2

InputOutput (Calc)Output (Exp)

Figure 9: Random waves generated by using the Pierson-Moskowitz spectra. Comparisonbetween numerical and physical measurements at x = 6.1 m. In the both cases the same signalfor the wave maker displacement was used. Comparison of surface elevation (left) and spectraof input signal, and output signals of computation and physical experiment (right).

Su

rfa

ce e

lev

ati

on

, m

time, sec

0.2

0.1

0.0

-0.1

-0.2

0.2

0.1

0.0

-0.1

-0.2

0.2

0.1

0.0

-0.1

-0.2

0.2

0.1

0.0

-0.1

-0.2

0 10 20 30 40 50

60 70 80 90 100 110

220210200190180170

120 130 140 150 160

CalculationExperiment

0.0012

0.0010

0.0008

0.0006

0.0004

0.0002

0.0000

0 0.5 1 1.5 2 2.5 3f, Hz

S(f

), m

/se

c2

InputOutput (Calc)Output (Exp)

Figure 10: Random waves generated by using the Swell spectra. Comparison between numericaland physical measurements at x = 6.1 m. In the both cases the same signal for the wave makerdisplacement was used. Comparison of surface elevation (left) and spectra of input signal, andoutput signals of computation and physical experiment (right).

signals of soliton and random waves combined and breather solution with random waves.The following figures shows our results.

In Fig. 11 it is shown the time histories of the free surface measured at severallocations. We used mixed signal, combination of random waves (swell spectrum) andsolitons of different steepnesses. Looking on this figure one can say that solitons almostdo not interact with random waves and propagate with almost keeping their form.

Interaction between breather solution and random waves is shown in Fig. 12. It showsthe time records of the free surface at several positions of the wave gauges. We can noticethat there are several waves greater than its surrounded components. Next Fig.13. showspart of time records done at the same location for the three different signals: pure breathersolution (bottom), pure random waves (middle), and combination of two previous signalsby superposing them (top). We can say that large peaks appeared for mixed (combined)signal are not exist in records for both pure signals. The reason is possibly the interaction

10

between breather solution and random waves.

0.04

Surfa

ce el

evat

ion,

m

time, sec0 50 100 150 200

x = 38.0 m

x = 34.4 m

x = 30.8 m

x = 27.2 m

x = 23.6 m

x = 20.0 m

x = 16.4 m

x = 12.8 m

x = 9.2 m

x = 5.6 m

x = 2.0 m

Figure 11: Time records of the free surface done at several positions of the wave gauges. Thecombined signal (random waves and solitons) was used.

Surfac

e eleva

tion, m

time, sec0 50 100 150 200

0.18

x = 38.0 m

x = 34.4 m

x = 30.8 m

x = 27.2 m

x = 23.6 m

x = 20.0 m

x = 16.4 m

x = 12.8 m

x = 9.2 m

x = 5.6 m

x = 2.0 m

Figure 12: Time records of the free surface done at several positions of the wave gauges. Thecombined signal (random waves and breather solution) was used.

11

60 80 100 120time, sec

Figure 13: Time record of the free surface at x = 12.8 m from wave maker. Comparisonbetween three signals: mixed signal (breather solution and random waves), pure random waves(swell spectra), and pure breather solution.

5 Conclusion

In this paper we used Boundary Element Method (BEM) to simulate Freak wave eventsand their appearance in the ocean. We examined our numerical method for highly nonlin-ear waves by comparing numerical results with experimental ones and theoretical (Stokes’waves of 3rd order). Our results show that numerical scheme reproduce nonlinear andtransient part of waves with very good accuracy.

Next, we suggested breather type solution as one more mechanism (previous are dis-cussed in [6]) of Freak waves creation.

To simulate ocean waves we used two spectra: Pierson-Moskowitz and Swell. Weexamined several computations of generated random waves and found that there wereappearance of few waves the wave heights of which greater than their surrounding com-ponents almost twice.

Finally, we did simulations of mixed waves, which were combinations of solitons andrandom waves, and breather waves and random waves. We found that solitons couldpass through surrounded irregular waves almost without changing their shapes (solitonenvelope profile), i.e. they kept solitons’ property.

On the other hand, interaction of breather solution waves with irregular ones producedsome extreme waves. The comparison of time records of the free surface taken at the samelocation of the wave gauges between simulations of pure random waves, pure breatherwaves, and mixed signal waves showed that these extreme waves were results of theinteractions. This phenomenon will be studied in more details in the next work.

12

References

[1] Klinting P and Sand SE, “Analysis of Prototype Freak Waves”,Proc. Spec. Conf.Near-shore Hydrodynamics, pp. 618-632, 1987

[2] Proceedings of “Rogue Waves 2000”, Brest, France, 2000

[3] Kashiwagi, M. “Full-Nonlinear Simulations of Hydrodynamic Forces on a HeavingTwo-Dimensional Body”, J. Soc. Nav. Arch. of Japan, Vol. 180, pp. 373-381, 1996

[4] “Hydrodynamics of a floating body. Part 1: A numerical computation method ofmotion problem”, Textbook, 2003 (Japanese)

[5] Tanizawa, K.,“The State of The Art On Numerical Wave Tank”, Proc. 4th OsakaColloquium on Seakeeping Performance of Ships, Osaka, Japan, October, 2000, pp.95-114

[6] I. Ten, H. Tomita, “Creation and Annihilation of Extremely Steep Transient Wave”,Proc. of ISOPE 2006

[7] Newman, “Marine Hydrodynamics”

[8] Akhmediev, N.N., Eleonskii V.M. and Kulagin, N.E., Theor. Math. Phys. (USSR)72, 809, 1987

[9] Karjanto, Natanael and van Groesen, E., Relation Among Breather Type Solutions ofthe Nonlinear Schrodinger Equation, Proc. of the Mathematics National Conference,Univ. of Udayana, Bali, Indonesia, 23-27 July, 2004

[10] Karjanto, Natanael, van Groesen, E., and Peterson, Pearu, Investigation of theMaximum Amplitude Increase From the Benjamin-Feir Instability, J. of IndonesianMathematical Society (Majalah Ilmiah Himpunan Matematika Indonesia, MIHMI),8, 4:39-47, 2002

[11] Ma, Yan-Chow, The Perturbed Plane-Wave Solutions of the Cubic Shcrodinger Equa-tion, Studies in Applied Mathematics, 60, 43-58, 1979

[12] Peregrine, D.H. Water Waves, Nonlinear Schrodinger Equations and Their Solutions,J. Austral. Math. Soc. Ser. B 25, 16, 1983

[13] Nakamura Akira, Hirota Ryogo, A New Example of Explode-Decay Solitary Wavesin One-Dimension, J. of the Phys. Soc. of Japan, Vol. 54, No. 2, February, 1985, pp.491-499

[14] Longuet-Higgins MS, “Statistical Properties of Wave Groups In a Random SeaState”, Phil. Trans. R. Soc. Lond. A, Vol. 312, pp. 219-250, 1984

[15] Goda Y., “Numerical Experiments on Wave Statistics with Spectral Simulation”,Report of The Port and Harbour Research Institute, Vol. 9, No. 3, September, 1970

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