On the statistical properties of waves measured at the Japan Sea

(1)

TE

1'VflITEIT

LE.'

- 'z cum voor

.;crn9chanica

Archef

Mc!z!wcg 2, 2E28 CD Deft

On the Statistical Properties

Measured at the Japan Sea

lIirofui Yoshioto, Shigeo Ohiatsu

Ship Research Institute, Ministry of Transport. Japan

Deformation of some deck plates of "POSPI-DON" has been observed and this phenomenon has

been investigating. Photo i shows an example

of the deformation of the bottom of the deck plate. A possible explanation of the deforma-tion is that waves may hit the bottom of' the

plate.

In the first place, wave amplitude

statis-tics has been investigated, especially in

stormy sea states. As a result, the measured

wave data in stormy sea states reveal that

high wave crests occur more frequently than the predicted by the Rayleigh distribution.

The non-linear theory, developed by Hinenol>,

resulting in a non-Rayleigh distribution of wave crest and trough amplitudes has been used to predict the highest 1/n amplitude and the

extreme statistics in stormy sea state. In

this paper, a comparison between the measure-ments and the non-linear theory is presented.

1. Wave data

2. Analysis

Fig.1 shows the definition sketch of the

wave crest amplitude A and the wave trough

amplitude At. A is defined to represent the

zero up-crossing wave crest height and At

represents the zero up-crossing wave trough excursion.

Fig. 2 shows the relation between the wave steepness and the average values of the

high-est 1/n amplitudes divided by the standard

Photo i Example of the deformation of the bottom of the deck plate

The analysis has been carried out using the 6 continuous data when low atmospheric pressures

L.A.P. ) in winter or a typhoon approached to the experimental site. The data have been divided

into 254 runs of 2048 seconds each. Table i shows statistical values of its highest significant

wave height H113 for each runs where _TFI1/3' H and _TFax are the significant wave period, the

maximum wave height and the maximum wave period respectively.

Table 1 Statistical values of the wave data

i

Fig. i Definition sketch of wave amplitudes

File Length(h) H1/3(m) T}11/3(s) H3x(m)

Tax(5)

D70831 21. 7 6. 690 11. 382 13. 115 10. 154 Typhoon D71124 23.8 5. 352 ii. 593 8.878 10. 458 L.A. P. D71217 22.4 7.340 10. 884 14. 328 li. 646 L. A. P. 080202 24.7 7.220 10.994 12.490 11.828 L.A.P. D81214 30.1 6.099 10. 525 11.013 11.241 L. A. P. D91119 47. 1 7.062 10. 996 12. 647 11. 138 L. A. P.

(2)

deviation of the surface elevation, where A11 is an average value of the highest 1/n c-est

amplitudes and At11, is an average value of the highest 1/n trough amplitudes. The wave steepiess

a is defined as

a =

H113/(gT2/2)

(1)

where g acceleration due to the gravity

T2 zero up-crossing mean period

2. 3

l.2

<2.1

2 1. 7 0. 01 3. 2 3. 1 2.9 2.8

<2.7

2. 6 2. 5 2.4 2. 3

<2.2

2. 1 2 0.01

i:

o + + +++--+

++

+ + 0+ + 0.03 0.05 *ave Steepness (H1/3/1.56Tz*2) (a)

Al/l

and

_Ati,a

0.87

0.03 0.05 0.'07

Wave Steepness (H1/3/1. S6Tz"2)

tD) Aol/lo and

Fig. 2 Relation between the wave steepness a and the average values of the highest 1/n amplitudes

The solid line shows the expected values of the highest 1/n amplitude predicted by the

Ray-leigh distribution (linear theory). It is clearly seen that the measured A11 increases as the

wave steepness increases, while the measured A111 become smaller, whereas the measured

becomes larger than the prediction by Rayleigh distribution, especially for the data on high

steepness condi t ions.

Fig. 3 also shows the relation between the wave steepness and the maximum amplitudes divided

by the standard deviation , where is the maximum crest amplitude and

Atax

is the maximum

trough amplitude. The solid line shows the expected value of the maximum amplitude

E[Aax/a]

(3)

<5 -2 0.01 E [Amax/o] = 2[(lnN)'12+ a/2(lnN)2] (2)

where N is the number of zero 2

up-crossing and the measured N

distributes around 209 - 591. It

is seen that the measured A< oc-curs more frequently than the

ex-pected value predicted by the

equation (2) for N = 10,000. In

the linear theory, the expected

value of Ataax becomes the same

10

value as that of Acax, but there

is no case for the measured Atmax 8

to exceed the expected value for N s

10, 000. Fig. 4 shows examples of recorded waves with high Acsax/ in Fig. 3 and they indicate that there are high crest-trough

dissymnmet-ries. These results suggest the

increased occurrence of high

-crests and flattened troughs

caused by non-linearities in ran-dom sea.

In order to account for the

observed non-linear effects, a

non-linear probability density function, developed by Hineno0, resulting

distribution of wave crest and trough amplitudes has been used to predict the the highest 1/n amplitudes and the maximum amplitude.

3. Non-linear probability density function

Wave Steepness (HI/a/i.56Tz2)a

Fig.3 Relation between the wave steepness a and the maximum wave amplitudes

24th Nov. '87

o-0 17th Dec. '87

JtJa

D0

c o

-::

D .p i1 0o1

óDUDqJ

+ 9 D++ +

°

\N-1000 -f-D U

+.0,i_

D++o_

..

- + + *+₊ * + _:2OO 0.03 0.105 0.07 N:10000

N2000

17thDec. '87 Wave Probe-2

H,14.62O

(m) H,,3:7.247 (m) A....,/cr:4.811 Probe-1 Wave Probe-3 87/11/24 line(s)omo Tine(s) 920 in a non-Rayleigh expected values of

Hinenol> developed a non-linear probability density function of wave amplitudes by using the

Vinje' s ¡nethod3 based on the narrow-band assumption. It includes the effect of the second-order wave and is given as

b

N.

43

750 770 790 D + m 87/12/17 24th Nov. '87

}L:13.593 (m)

H,,3:5.250 (m) :6.251 850 000 900

(4)

P(z)

= exp(-z2/2K20).

[ -K10/K20-K12/2K20K02+K 30/2K202+z/K20+ (K10+K12/2K02-K30/K20)z2/K202+K30/6K204z4 ]

K10 =

f

S()G2(, -)d,

₂₀ =

fs(jG1(

2da. K02

f

2S()

G1( 2d K12

=ff(

2+222)SC1)S(2)

(G1(1)G1(-2)G2(1. 2)+G1(-1)G1(-2)G2(1, 2)}d1d2 K33 =

3f/s

_Li)S

L)

(G1 (-)G1(-2) G2 L1.

'2)

G (-1)G1 (-2)G2' Li.

2) } d1d2

( Limits on

integrals, -toare omitted in this paper )

where P(z) : probability density function

circular frequency

S()

: double side wave spectrum

G1() : linear response amplitude operator

quadratic response amplitude operator

* : complex conjugate

Equation (3) shows that the non-linear probability density function can be calculated with the response amplitude operators ( RAO ) and wave spectrum. Before the theoretical predictions

by the equation (3) are compared with the measured data, the effect of spectrum width and water

depths on the probability density function has been investigated by numerical simulations.

4. Nuaerical Si.ulation

Numerical simulation has been carried out using four input wave spectra and the quadratic RAO both in shallow water and in deep water.

The input wave spectra are based on the following modified J0NSAP spectra4> with the

sig-nificant wave height H113 = 6 um and the peak frequency f = 0. 696 Hz. Fig. 5 shows the input wave

spectra, giving the spectral peak enhancement factor = 1, 2, 4, and 6 respectively to estimate

the effect of spectrum width.

exp[- (fpf)2/2a2fp2]

SLj(f) =

2 -1

S (f) = 0. 173. (i/3 f, ) (f/fY4exp(-2/3(f/fY6)

= 0.07 ( f f ).

0.09 (f> f

where S(f) : modified J0NS'AP spectrum

F1 coefficient to equate the area of S1(f) with that

of

S1(f)

f

peak frequency of wave spectrum

'y : peak enhancement factor

140 130 100 110 100 IO 011 70 00 50 40 30 20 00 m'sec 'Y o2

Spectra for Siriletica

Freqoency(flz)

Fig. 5 Input wave spectra for simulation

4

1.0 2.0 3.0 3.3 4.0 5.0 6.0

(5)

The linear RAO is simply G1 = 1, while the following expressions5 are found for the quadratic RAO in shallow water.

G2C, )=K(3coth3Kh-cothKh)exp(-2iK. x)/2

G2(. - )

= o.

G2 (i 12) = Kexp(-iKt. x)/2. { () (1+cothK1h. cothKh)

+(1cothK1h+2cothK2h) cothKh} /(gK1- () cothKh)

G2 (ei. -2) =lCexp(-ilÇ. x)/2. ((t)2+o12 (l-cothK1h. cothK2h)

- (1cothK1h+2cothK2h) cothKTh} /(gK- () 2cothKTh)

I2'

12'

K4=K1+K2, IC=K1-k2

where K wave number

h : water depth

We can (5).

0. 6

0. 7

also obtain the quadratic RAO in deep water by taking the limit h -> in the equation

z//R 0.8 0.7 0. 1 L0.5 0. 4 0. 3 0. 2 0. 1 -0. 1 0 Rayleiqh

Crest Distribution ( Shallow

0,0 . 0.0

0. Trouqfl Ditribut,o, ( Deo 0. 7

0.6 - 0.6

[0.5

' [Q.5

2 4 _z//R'5 0 2 _z/1k0

(a) Deep water (b) Shallow water

Fig.6 Probability density distribution of wave amplitudes, whereJo defined by the equation (3)

equals to the standard deviation of the surface elevations.

-

--- y=2

r=4

--Fig. 6 shows examples of the effect of spectrum width and water depths on the probability

density function for crest and trough amplitudes through the above expressions from (3) to (5).

The water depth in the shallow water case equals to the water depth of the experimental site 43 in ). Fig. 7 shows the expected values of the highest 1/n amplitude and the maximum amplitude

predicted by the results shown in Fig.6.

It is found that the water depth has a great effect on both the peak and the tail of the

distribution leading to the increased probabilities of high wave crest amplitudes, while the

effect of spectrum width is small. lt is also found that the equation (4) gives the negative tail

probabilities for trough amplitudes and the expected values of in sallow water cannot be

shown due to the inadequate tail behavior. This suggests that there is a possibility that it produces anomalous results and an alternative statistical model has to be adopted.

(5) 0. 6 0.5 N 0.4 0. 3 0. 2 0. 1 -0. 1

crest DistributionfDeep RayIeqh

(6)

3. 5 0 2.5 N 2 1. 5

<4.5

W4

3. 5 2

Crest (with shallow effect) Crest dee. water

- 71.0

72.0 _74.0

-- 7=6.0

Crest (with shallow effect)

Crest (deep water)

Trou.h dee. water

- 7_4

r=2

-- y=6

50 40 30 20 10 -. FkaLLuQ

6

-0.06 0.2 0.3 frequency (Hz)

Fig.8 Input wave spectra for the each wave steepness

o 4 8 12 16

1/n

20

(a) Expected values of Al/ and A11

0 100 200 N 300

(b) Expected values of and Atsax

Fig.7 Expected values of the highest 1/n aniplitude and the inaxinium amplitude, whereft0 defined

by the equation (3) equals to the standard deviation of the surface elevations.

5. Couparison between non-linear theory and ucasured data

Bearing the features of the probability density function developed by Hineno in our mind, the measured data have been compared with the predictions by the non-linear theory.

(7)

Fig.8 shows the input wave spectra with the values of the parameters given by Table 2, where

,

H, and f

are average values of the measured data in the range of the wave steepness. The

quadratic RAO in shallow water is used and the water depth is 43 in.

Table 2 Parameters of Modified JONSYiAP spectra

In Fig. 9, the measured data are compared to the expected values of Ad/fl predicted by the

non-linear theory, where the expected values of Ati,ì cannot be shown due to the negative tail

probabilities for trough amplitudes. lt is found that the theoretical curves for the non-linear

theory increase with the increase of the wave steepness a. the measured data has also the sanie

tendency. However, there is a slight difference between the theoretical predictions by the

non-linear theory and the measured data. lt may come from the inadequate tail behavior of the

probability density function developed by Hineno. 2. 5 2. 4

b

c2.2

-2. 1

-2 1. 9

-b 3. 6 3. 4 3.2 2. 8 2. 6 2.4 9

<2.2

\-.-

oç Q +

++J

7 Wave Steepness (111/3/1. 56Tz.2) a

(a) Expected values of ACl/3

Non-Iinear(2nd) o t. - U

i

_: +

++

+ + ++ +

+1+JyI

+++ o ++ + Non-tinear(2nd o r

w1I?; 'O iD

o

.

B _.yleigh

+ + +

+1+

+ Steepness . H1,3 0.03 2.0 2.410 0.0850 0. 04 2. 0 3. 933 0. 0768 0. 05 2. 0 5. 264 0. 0743 0. 06 2. 0 5. 552 0. 0792 0. 01 0.03 0.05 0. hi

Wave Steepness (111/3/1. 56Tzi'2) cr

(b) Expected values of

_Aii10

Fig. 9 Expected values of Aii

0.07

(8)

In Fig. 10, the measured data are compared to the expected values of Acax predicted by the

non-linear theory for N = 200, 1000, 2000, and 10,000. It is found that the non-linear theory shows

about lo X increase of the expected values as compared to the Rayleigh distribution. However, we must note that there are still several cases exceeding the theoretical predictions by the

non-linear theory for N =10,000. The Yasuda's proposal6 that non-linear interactions higher than

third order cause the large Acm can be considered a possible explanation of this phenomena. It

may suggest the necessity to develop a new nonlinear theory including non-linearities of thirc

order. 7

2

6. Conclusion

The conclusions from the present investigation are summarized as follows

- The measured Ad/fl increases as the wave steepness increases, while the measured

become smaller, whereas the measured Au becomes larger than the prediction by Rayleigh distribution, especially for the data on high steepness conditions.

- The non-linear probability density function includes the effect of second order wave has been used to predict the highest 1/n amplitude and the extreme statistics in stormy

sea state. As a result, the expected values predicted by the non-linear theory increase

with the increase of the wave steepness a. the measured data has also the same tendency.

However, the non-linear probability density function gives the negative tail probabili-ties for trough amplitudes.

- The water depth has a great effect on both the peak and the tail of the distribution leading to the increased probabilities of high wave crest amplitudes, while the effect of spectrum width is small.

The prediction method presented in this paper are clearly not valid for the maximum

amplitude, especially AC(. Furthermore, an alternative statistical model has to be adopted due to the limitation of the non-linear probability density function developed by Hineno.

After making the above problems clear, this paper will be submitted to the OCEAN 93

Conference. 0. 01

¡ I I I

0.03 0.05

wave Steepness (H1/3/1. 5STz*+2) Q Fig. 10 Expected values of

(9)

References

1)Hineno, M. : A calculation of the Statistical distribution of the Maxima of Non-linear

responses in irregular Waves, Journal of the Society of Naval Architects of Japan, Vol. 156,

Dec. 1984, ( n Japanese

2)Cartwright & Longuet-Higgins, M.S : The Statistical distribution of the Maxima of a random

function, Proceedings of The Royal Society A, Vol.237, 1956

3)Vinje, T. : On the Calculation of Maxima of Non-linear Wave Forces and Wave induced Motion,

international Shipbuilding Progress, Vol.23, No.268, 1976

4)Yoshimoto, H. et.al. : At-sea Experiment of a Floating Offshore Structure -Characteristics of

Directional Wave Spectra Measured at Japan Sea-, Journal of the Society of Naval Architects of

Japan, Vol.168, 1990

5)Hamada, T. : The Secondary interaction of Surface Wave, Report of Port and Harbor Technical

Research institute, Report No. 10, 1965, pp. 1-28

6)Yasuda, T. et.al. : Non-linear Effects on the wave Height Distribution of Unidirectional

irregular Waves, Proceeding of Japan Society of Civil Engineers, No.443/-18, 1992, pp.83-92,

in Japanese

(10)