Statistial Mehanis of the Frequeny Modulation of Sea Waves Hiroshi Tomita and Takafumi Kawamura Ship Researh Institute, Shinkawa 6-8-, Mitaka, Tokyo 8-, Japan tomita@srimot.go.jp Abstrat. The longtime stationary oean wave data taken from the Sea of Japan are analyzed in partiular for the stohasti property of their wave periods. The probability density funtion (PDF) of wave heights and periods are determined. A quasi-linear wave propagation model is examined to simulate the reation and annihilation of Rogue like wave. This simple model reprodues the atual feature of this phenomenon very well. It suggests that the frequeny modulation in random sea waves is a possible ause of suh an abnormal wave. A statistial mehanial tehnique is applied to the sequene of wave periods. The flutuation of wave period sequene has its spetrum inversely proportional to the frequeny of the variation, whih is fairly ommon feature in many natural phenomena. Introdution The ause of Rogue (Freak) waves in atual oean area has long been investigated by many researhers, and many hypothetial mehanisms of their ourrene have been proposed from different point of views and orresponding tehniques. These are lassified roughly as follows:. Non-linear effets of water waves.. External influenes, varying urrent and/or bottom topography.. Superposition of wave system, wave groups, multidiretional waves. In this paper, we pay attention to the statistial properties of wave periods and their role in the generation and annihilation of abnormal waves like Rogue waves. Firstly, atual oean wave data from the Sea of Japan are reexamined in detail. Several existing formulae for the probability distribution of wave periods in stohasti proesses are omp ared with the data of,7 waves, whih were taken in almost same sea ondition. The agreement in the distribution of wave periods is not as good as in the distribution of wave heights in general. However, for this large number of wave samples, lassial Weibull distribution with index is found to be in good agreement. The autoorrelation between suessive wave periods is.5, whih is slightly less than those of preeding results in the storm field. Seondly, we attempt to predit the propagation harateristis of atually observed Rogue waves by assuming the non-linear dispersion relation for all wave
frequeny omponents in a stohasti wave field. Referring to the results of this analysis, we perform a simple numerial simulation of generating and annihilating a Rogue wave by superposing a small frequeny modulated wave group. Distribution of Wave Period Many types of probability density funtion (PDF) of sea waves have been proposed. They were dedued from the assumption of the linear stohasti proess, that is, Gaussian proess. The most typial formula is presented here as equation. ν m p m p p( T) dt = + + νt T ν / dt () In this formula, narrow banded spetrum of the proess is also assumed and symbols and m p represent the bandwidth and the mean value of the period respetively. Otherwise we an make use of some empirial formula like T p( T) dt = T exp dt m p m p () whih is alled the Weibull probability density distribution of index, and m p means the root mean square of the periods. Its exess distribution is integrated to be T PT ( ) = p( T) dt = exp m () p T Analysis of Data from the Sea of Japan The most ruial ondition for the statistis of sea waves is that we an not obtain enough long time data of stationary sea state to be ompared with stationary random theory if non-linear effet of surfae waves is not pronouned in atual oean. In this paper, we deal with an available longtime (7 hours) wave data from the Sea of Japan on January 9 th and th, 988. The loation of the observation site is shown in Fig..
Fig.. Observation site at Yura m(m/s) 8 6 8 6 Data Number SignifiantWaveHeight AverageWindSpeed Fig.. Long term sea state of analyzed data The sea ondition during this observation period (average wind speed and signifiant wave height of every minutes) is shown also in Fig.. Wind speed was around 5m/se, and signifiant wave height was almost onstant at m throughout. This ondition satisfies that of fully arisen sea and wind waves are to be found without swell. It is very rare ase to have data for suh many waves of amount to, under suh a stationary sea ondition. We an ertify the law of large numbers in statistis of wave height and period analyzed by Zero Up Crossing method from these data. At first, we examine the well-known result of wave height distribution with established theoretial formula of Rayleigh distribution in equation (). H H p( H) dh = exp dh m 8m ()
The result is shown in Fig.. The agreement of the observed PDF with that of the theory by the parameter m is exellent as expeted. Next, we examine the PDF of sea wave period. A omparison is made with a theoretial formula () [] in Fig.. The agreement seems rather poor partiularly in the longer period region. It is not easy to explain this sort of disrepany beause it is not neessarily aused by the bandwidth effet. Atually, the bandwidth parameter for every minutes reord is found to be between.6 and.7 that is onsidered intermediate. On behalf of the unreliable theory, we adopt an empirial PDF of Weibull type of index (), whih is an extension of Rayleigh PDF. Atually it redues to Rayleigh distribution when =. For identifiation of the index, we integrate it to have its exess probability distribution (), whih is desribed in Fig. 5. Double logarithm of the both side of equation () is shown in Fig. 6. In this figure, we find the index is very lose to an integer (atually.98). This result is in aordane with a formerly published result []. In Fig. 7, we onfirm the exellent agreement of this formula with observed data. For referene, we show the joint PDF for normalized wave height and period of the whole 7 hour reord in Fig. 8. The orrelation of wave height and period seems high, whih is mentioned in a latter setion. PDF of Wave Height.6....8.6.. 5 6 8 m Observation Rayleigh Dist Fig.. Comparison of observed wave height with theory PDF of Period. 5 5.5..5 Observation RL-H Dist 6 8 6 se Fig.. Comparison of observed wave period with theory
Exess Probability Distribution..8.6.. 5 5 se Fig. 5. Exess probability distribution of the period Exess PDF of Period y =.98x - 8.6.5.5.5 - - Observation linear approx - - -5-6 Fig. 6. Linear regression of the log-log plot PDF of Weibull Dist. 5 5.5..5 Observation Weibull Dist 6 8 6 se Fig. 7. Comparison of the observed period with Weibull distribution
8 7 D imensionless Height 6 5 Fig. 8. Joint PDF of wave height and period Dimensionless Period Transformation of Wave Reord Oean wave data are aquired at a fixed position as time series. On the other hand, we an desribe the wave elevation like the following form. ω ω η( x,t) = a( ω) os ωt x dω+ b( ω ) sin ωt x dω g g (5) where the dispersion relation of deep water wave is taken into aount. Moreover, we an onsider the nonlinear dispersion relation of eah omponent by replaing both sinusoidal terms as follows: x x osω t osω t L NL (6) x x sinω t sinω t L NL (7) From equation (5), one an reprodue the observed time series simply by setting x=. ( ) = ( ) ( ) + ( ) ( ) η,t a ω os ωt dω b ω sin ωt dω (8)
The Fourier oeffiients a(ø) and b(ø) are determined easily from the observed reord. We obtain the transformation of aquired reord at x= to arbitrary point x by substituting a(ø) and b(ø) to equation (5) and integrate them by ø. If x <, ( x, t ) represents the wave reord at x m upstream of the observed one. Note that we neglet the bounded wave omponents and the diretional spreading of random wind waves in the above quasi-linear model. The former is justified in the atual oean however the latter is not always permissible. 5 Appliation to Rogue Waves As was written in the preeding papers of the present authors [], we found several ases of minutes data in eah of whih a typial Rogue wave is inluded. We pik one of them up here and examine the transformation of its hange of waveform at up and down-stream virtual observation sites. An original data is shown in Fig. 9 as x =. The transformation proedure explained in setion is applied to the data to have the reords eah 8m up and down side. It is lear that there is no suh a pronouned peak at all in the both alulated reords. It suggests that rogue wave itself does not propagate as a rest preserved isolated wave. Instead, we an reognize that there is a omparatively small amplitude wave train, of whih period is gradually inreasing in time at x = -8m (see Fig. ). Similarly, gradually dereasing frequeny modulated wave train appears at the reord at x=8m (see Fig. ). It means that one an experiene an appearane of large wave rest only within the interval of several times as long as the wavelength. ROGUE WAVE ( X=8m) m 8 6-6 8 - -6 se Fig. 9. Wave modulation at 8m downstream virtual site
Auto orrelation (H,H)..8.6.. -. 5 5 5 time lag Fig.. Autoorrelation of the sequene of wave height Auto orrelation (T,T)..8.6.. -. 5 5 5 time lag Fig.. Autoorrelation of the sequene of wave period 6 Flutuation Property of Sea Wave Period In the statistial studies on oean waves as random proesses, PDF of wave height and period were investigated from various point of views. In ontrast, the study on the nature of time sequene of wave height and period is so far somehow sare. So we
present here briefly the time variation of these quantities and their stohasti natures. From the sequene of wave heights, we alulate their autoorrelation shown in Fig.. In this figure, one an see that the orrelation of suessive wave height is., whih is almost the medium of the values of atual measurements by Goda []. In his results, the values are.6 in swell and.6 in sea. For the sequene of wave periods, autoorrelation between suessive periods is.5 as is seen in Fig.. The autoorrelation of wave height is slightly higher than that of wave period in a fully arisen sea. As for the wave period, the orrelation of suessive wave is rather lower than that mentioned in []. The disrepany is partly explained by the fat that data adopted here is under the limati ondition of long lasting low atmospheri pressure in the winter season while their data were taken during a severe storm. In Fig., we present the ross orrelation of wave height to period. At the origin of time lag, say for a same ZUC wave, orrelation oeffiient is.6 omparatively higher than those of non-stationary and is not disrepant to the result shown formerly in the ontour lines from the ontingeny table of joint distribution. A loser study of the flutuation of wave period is performed by the statistial mehanial tehnique. We alulated the power spetra of every sequene of 5 wave periods. Averaging 5 samples extrated fro m the stationary wind sea, the spetral density is obtained in Fig. 5 in log-log sale. Note that the ordinate S and absissa f (reurrene frequeny of periods) are in arbitrary sales. For the higher end of Fig. 5, we have the linear regression of the oeffiient.98 shown in Fig. 6. This means the power law S? / f, whih is the famous relation in many branhes of siene. Cross orrelation (H,T).7.6.5.... -. - - - time lag Fig.. Cross orrelation of the sequene of wave height and period
Spetrum of Period Flutuations.5.5.5.5.5.5.5.5 Power Fig.. Averaged power spetrum of the flutuation of wave period Spetrum of Period Flutuations.8.6...8 Power linear regression.6.. y = -.985x + 5.9.....5 Fig.. Averaged power spetrum of the flutuation of wave period and its linear regression Spetrum of Period Flutuations.5.5.5.5.5.5.5.5 Power Fig. 5. Averaged power spetrum of the flutuation of wave period
Spetrum of Period Flutuations.8.6...8.6 y = -.985x + 5.9.......5 Power linear regression Fig. 6. Averaged power spetrum of the flutuation of wave period and its linear regression 7 Conlusions The nature of stohasti properties of a fully arisen wind sea are investigated by use of large number of wave data inluding up to, waves whih were taken under the almost stationary sea ondition. The preise analysis on PDF and temporal variation of wave period are performed. The results are onsidered to be statistially reliable beause of the law of large numbers. A simple quasi-linear method of wave reord transformation is examined. It is applied to a typial example of Rogue wave in the atual oean. The results suggest that the frequeny-modulated wave train is a possible ause of reation and annihilation of suh an abnormal wave in the oean. Nevertheless, more observational data and more strit non-linear theory onerning wave period in a random seaway is needed. We must identify the isolated Rogue wave from the theoretial point of view and distinguish it from the Abnormal or Freak wave (wave height is times larger than signifiant wave height), whih has ever been defined for the onveniene of pratial use. Referenes. Longuet-Higgins M. S.: On the Joint Distribution of Wave Periods and Amplitudes in a Random Wave Field, Pro. Roy. So. London, Ser. A., Vol.89 (98). Myrhaug D. and H. Rue: Note on a Joint Distribution of Suessive Wave Periods, J. Ship Researh, Vol.7 (99). Tomita H. and T. Kawamura: Statistial Analysis and Inferene from the In-Situ Data of the Sea of Japan with Referene to Abnormal and/or Freak Waves, Proeedings ISOPE, Vol. Seattle USA (). Goda Y.: Random Seas and Design of Maritime Strutures, University of Tokyo Press (985)