摘要
小波变换是对非平稳信号进行时频分析的理想工具,目前该方法在许多科学和工程领域中得到了广泛的应用。然而,小波变换在波浪非线性特征分析中的应用研究却相对匮乏,有关方面的应用研究尚不深入,没有充分展示小波变换的优越性。本文的目的是将Morlet小波变换及其相关概念应用到波浪的非线性特征分析研究中,深入地挖掘Morlet小波变换在分析波浪非线性特征中的潜力,寻求基于Morlet小波变换的处理非线性波浪信号的新方法。
本文首先对数据处理方法进行了简单的回顾,并详细介绍了Morlet小波变换的概念、特性以及优越性。然后,基于Morlet小波变换,通过对几种典型的非线性波浪信号进行了分析研究,取得了创新性研究成果：
1.基于小波局部波能过程线提出了一个新的量化波浪群性程度的波群因数GF,与以往的GF的相比,本文提出的GF受计算时参数选取的影响几乎可以忽略,更加适合用来统一量化波浪的群性程度。
2.引入小波二阶相关谱分析短时／瞬态非平稳波浪的非线性特征,从而解决了传统傅里叶二阶相关谱方法不能应用于短时／瞬态信号非线性耦合分析的问题。并用该小波二阶相关谱研究了不规则波浪在曲线形潜堤上的非线性传播变形、波浪局部的非线性特征变化以及非线性聚焦波浪在等水深下的传播演化过程,取得了理想的研究结果。
3.提出用小波局部尺度平均波能来量化局部瞬态高阶谐波,从而解决了传统方法不能很好地量化局部瞬态谐波能量的问题。
4.采用小波脊方法提取波浪瞬时峰频并用其表示波浪的瞬时频率,从而解决了用传统方法定义复杂信号的瞬时频率所遇到的困难。并用该方法首次分析研究了波浪在沿空间变化逆水流中的非线性频率调制过程,对波浪在逆水流中频带下移的条件以及频带下移的特征和机理进行了详细的分析研究,取得了重要的研究进展。
5.根据Morlet小波变换的解析性质,提出了一种适合在变水深情况下分离非平稳入、反波浪的时域方法,从而解决了以往基于傅里叶变换的分离方法不能精确分离非平稳波浪场的问题。提出了一个新的分离低频波浪的时域方法,应用该方法能够准确地得到约束长波和入、反射自由长波的时间序列,为研究在变浅过程中约束长波滞后于短波包络的相位变化提供了很好的分析方法。
Wavelet transform is an ideal tool for time-frequency analysis of nonstational signals. In recent years, this method has been used successfully in many scientific and engineering applications. However, applications of wavelet transform in nonlinearity of ocean waves are rare. Even though some studies have used the new method to slove some issues, the advantages of it are not revealed in depth. The goal of this dissertation is to apply the Morlet wavelet transform and some related definitions for analyzing the nonlinear ocean waves, digging the advantages of the new method in analyzing the nonlinear signals deeply. Furthermore, some new methods, which are suitable to process signals in nonlinear ocean wave analysis, will be proposed based on the Morlet wavelet transform.
In this thesis, based on a brief introduction of the signal processing technology, the definition, properties and advantages of the Morlet wavelet transform are introduced carfully. Then, some typical nonstational signals in physical experimental models are analyzed in detail using the Morlet wavelet transform. The major creative contributions of this dissertation are summarized as follows:
1. A new groupiness factor(GF), which is the main parameter to measure groupiness of waves, is proposed based on the local wavelet energy history. The main advantage of the new GF is that the effect of the operational definition on it is much smaller than on other GF. Therefore, this new GF is much more suitable to unify wave groupiness.
2. To analyze the nonlinear phase coupling in short duration series, the wavelet bicoherence spectrum is introduced. Thus, the concept of bispectrum can be used to analysis series of physical models in ocean engineering. Then, the wavelet bicoherence spectrum is used to study the nonlinear evolutions of both irregular waves propagating over a submerged sill and focusing waves over a constant depth.
3. A new method to measure the instantaneous harmonic components of nonstational signals, which can not be resolved using the traditional methods, is proposed using the local wavelet scale-averaged energy.
4. The evolution of wave modulations on a spatially varying, opposing current has been studied experimentally. In this study, the wave frequency modulations are represented by the instantaneous peak frequencies which are extracted by the Morlet wavelet ridges, avoiding the difficulties for obtaining instaneaneous frequencies of complicated signals using the traditional methods. The criterion for the occurrence of frequency downshift in opposing currents has been investigated. Furthermore, the characteristic and the physical mechanism for frequency downshift have also been studied.
5. A new method for separating nonstationary waves into incident and reflected low-frequecy waves in time domain has been proposed based on the Morlet wavelet transform. Hence, the problem about separating nonstational wave fields accuately by the methods based on Fourier transform is sloved. A new method for separating low-frequency waves in time domain has also been proposed. The method can accurately obtain the time series of bound low-frequency waves, incident and reflected low-frequecy waves. Hence, this method can be used to investigate the cross-shore variations of phase lag of bound low-frequency waves behind short wave envelops.
引文
[1]冯士筰,李凤岐,海洋科学导论[M].北京：高等教育出版社,1999.
[2]李炎保,郄禄文,祝振宇,小波变换在随机海浪及相关课题中的应用与前景[J].力学进展,2003,33：541-547.
[3]俞聿修,随机波浪极其工程应用(第三版)[M].大连：大连理工大学,2003.
[4]李玉成,滕斌,波浪对海上建筑物的作用(第二版)[M].北京：海洋出版社,2002.
[5]邹志利,水波理论及其应用[M],北京：科学出版社,2005.
[6]Daubechies I, Ten lectures on wavelets [M]. Philadelphia, PA:Society for Industrial & Applied Mathematics,1992.
[7]Mallat S, A wavelet tour of signal processing (Second Edition). USA:Academic Press,1999.
[8]Goupillaud P, Grossmann A, Morlet J, Cycle-octave and related transforms in seismic signal analysis [J]. Geoexploration Amsterdam,1984,23(1):85-102.
[9]Farge M, Wavelet transforms and their applications to turbulence [J]. Annal Review of Fluid Mechanics, 1992,24:395-457.
[10]Liu P C, Is the wind wave frequency spectrum outdated? [J]. Ocean Engineering,2000,27(5):577-588.
[11]Liu P C, Hawley N, Wave grouping characteristics in nearshore great lakes ii [J]. Ocean Engineering,2002, 29(11):1415-1425.
[12]吴克俭,宋金保,楼顺里,风浪局域小波谱峰的涨落[J].海洋学报,1998,20：1-5.
[13]吴克俭,宋金宝,楼顺里,论风浪的局部结构Ⅰ.风浪的局部结构和局部小波能谱[J].海洋与湖沼,1998,29(4)：403-408.
[14]吴克俭,宋金宝,楼顺里,风浪的局部结构Ⅱ.风浪局部小波能谱的性质和应用[J].海洋与湖沼,1998,29(5)：488-492.
[15]Massel S R, Wavelet analysis for processing of ocean surface wave records [J]. Ocean Engineering,2001, 28(8):957-987.
[16]Huang M C, Wave parameters and functions in wavelet analysis [J]. Ocean Engineering,2004,31(1): 111-125.
[17]Huang M C, Wave parameters and functions in wavelet analysis with filtering [J]. Ocean Engineering,2004, 31(7):813-831.
[18]Rozynski G, Reeve D, Multi-resolution analysis of nearshore hydrodynamics using discrete wavelet transforms [J]. Coastal Engineering,2005,52(9):771-792.
[19]Liu E B, Liu P C, A discrete wavelet analysis of freak waves in the ocean [J]. Journal of Applied Mathematics 2004,5:379-394.
[20]Panizzo A, Bellotti G, De Girolamo P, Application of wavelet transform analysis to landslide generated waves [J]. Coastal Engineering,2002,44(4):321-338.
[21]Liu P C, Babanin A V, Using wavelet spectrum analysis to resolve breaking events in the wind wave time series [J]. Annales Geophysicae,2004,22(10):3335-3345.
[22]Mori N, Yasuda T. Orthonormal wavelet analysis for deep-water breaking waves [C]. In Proceedings of the 24th International Conference on Coastal Engineering Kobe, Japan:ASCE,1994,412-426.
[23]宇宙文,丁平兴,孙孚,二阶谱理论及其在海浪研究中的应用Ⅰ：.二阶谱的性质[J].海洋与湖沼,1993,24：553-559.
[24]Milligen V B P, Sanchez E, Estrada T, et al., Wavelet bicoherence:A new turbulence analysis tool [J]. Physics of Plasmas,1995,2(8):3017-3032.
[25]Powers E J, Yi E J, Park I S, et al. Wavelet-based bicoherence analysis of irregular wave data [C]. In Proceedings of the 8th International Offshore and Polar Engineering Conference. Montreal, Canada:ISOPE, 1998,170-175.
[26]Elsayed M A K, A novel technique in analyzing non-linear wave-wave interaction [J]. Ocean Engineering, 2006,33(2):168-180.
[27]Elsayed M A K, Wavelet bicoherence analysis of wind-wave interaction [J]. Ocean Engineering,2006, 33(3-4):458-470.
[28]Donelan M A, Drennan W M, Magnusson A K, Nonstationary analysis of the directional properties of propagating waves [J]. Journal of Physical Oceanography,1996,26(9):1901-1914.
[29]Donelan M A. A new method for directional wave analysis based on wavelets. In Ocean wave measurement and analysis [C]. San Francisco:ASCE,2001,310-317.
[30]Krogstad H K, Magnusson A K, Donelan M A. Wavelet and local directional analysis of ocean waves [C]. In Proceedings of the 16th International Offshore and Polar Engineering Conference. San Francisco, USA: ISOPE,2006,66-73.
[31]Rey H. Wave group formation among storm waves [C]. In Procedding of 14th International Conference on Coastal Engineering. Copenhagen, Denmark:ASCE,1974,164-183.
[32]Nolte K G, Hus F H. Statistics of ocean wave groups [C]. In Proceedings of 4th Offshore Technology Conference. Houston, Texas,1972,637-644.
[33]Goda Y, Numerical experiments on wave statistics with spectral simulation.1970, Report of Port and Harbour Research Institution, Nagase, Yokosuka, Japan:3-57.
[34]Funke E R, Mansard E P D. On the synthesis of realistic sea states [C]. In Proceedings of 17th Coastal Engineering Conference:ASCE,1980,2974-2991.
[35]List J H, Wave groupiness variations in the nearshore [J]. Coastal Engineering,1991,15(5-6):475-496.
[36]俞聿修,桂满海,海浪的群性及其主要特征参数[J].海洋工程,1998,16：9-21.
[37]Medina J R, Hudspeth R T, Review of the analyses of ocean wave groups [J]. Coastal Engineering,1990, 14(6):515-542.
[38]Song J B, Banner M L, On determining the onset and strength of breaking for deep water waves. Part Ⅰ: Unforced irrotational wave groups [J]. Journal of Physical Oceanography,2002,32(9):2541-2558.
[39]Dysthe K, Krogstad H E, Muller P, Oceanic rogue waves [J]. Annual Review of Fluid Mechanics,2008,40: 287-310.
[40]Gibson R S, Swan C, The evolution of large ocean waves:The role of local and rapid spectral changes [J]. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences,2007,463(2077): 21-48.
[41]Elgar S, Herbers T H C, Okihiro M, et al., Observations of infragravity waves [J]. Journal of Geophysical Research-Oceans,1992,97(C10):15573-15577.
[42]Longuet-Higgins M S, Stewart R W, Radiation stress and mass transport in gravity waves, with application to surf-beats. [J]. Journal of Fluid Mechanics,1962,13:481-504.
[43]Doering J C, Bowen A J, Parametrization of orbital velocity asymmetries of shoaling and breaking waves using bispectral analysis [J]. Coastal Engineering,1995,26(1-2):15-33.
[44]Hasselmann K, Munk W, MacDonald G, Bispectra of ocean waves, in Time series analysis [M]. Wiley: New York,1963:125-139.
[45]Elgar S, Guza R T, Observations of bispectra of shoaling surface gravity-waves[J]. Journal of Fluid Mechanics,1985,161:425-448.
[46]Elgar S, Herbers T H C, Chandran V, et al., Higher-order spectral-analysis of nonlinear ocean surface gravity-waves [J]. Journal of Geophysical Research-Oceans,1995,100(C3):4977-4983.
[47]Eldeberky Y, Nonlinear transformation of wave spectra in the nearshore zone [D]. Delft:Delft University of Technology,1996.
[48]Young I R, Eldeberky Y, Observations of triad coupling of finite depth wind waves [J]. Coastal Engineering,1998,33:137-154.
[49]陈子燊,李志强,戴志军,et al.,近岸带三组成波耦合作用的观察与分析[J].热带海洋学报,2003,22(6)：46-53.
[50]Peng Z, Zou Q P, Reeve D, et al., Parameterisation and transformation of wave asymmetries over a low-crested breakwater [J]. Coastal Engineering,2009,56:1123-1132.
[51]Elgar S, Sebert G, Statistics of bicohrence and biphase [J]. Journal of Geophysical Research,1989,94: 10993-10998.
[52]Rapp R J, Melville W K, Laboratory measurements of deep-water breaking waves [J]. Philosophical Transactions of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences, 1990,331(1622):735-800.
[53]Kway J H L, Loh Y S, Chan E S, Laboratory study of deep-water breaking waves [J]. Ocean engineering, 1998,25(8):657-676.
[54]柳淑学,洪起庸,三维极限波浪的产生方法及特性[J].海洋学报,2004,6：133-142.
[55]Baldock T E, Swan C, Taylor P H, A laboratory study of nonlinear surface waves on water [J]. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences,1996,354(1707):649-676.
[56]Baldock T E, Swan C, Extreme waves in shallow and intermediate water depths [J]. Coastal Engineering, 1996,27(1-2):21-46.
[57]Johannessen T B, Swan C, On the nonlinear dynamics of wave groups produced by the focusing of surface-water waves [J]. Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences,2003,459(2032):1021-1052.
[58]Brown M G, Jensen A, Experiments on focusing unidirectional water waves [J]. Journal of Geophysical Research-Oceans,2001,106(C8):16917-16928.
[59]Johannessen T B, The effect of directionality on the nonlinear behavior of extreme transient ocean waves [D]. London:University of London,1997.
[60]Benjamin T B, Feir J E, The disintegration of wave trains on deep water [J]. Journal of Fluid Mechanics, 1967,27:417-430.
[61]Phillips O M, Theoretical and experimental studies of gravity wave interactions [J]. Proceedings of the Royal Society of London Series A-Mathematical and Physical Sciences,1967,299:104-119.
[62]Longuethiggins M S, Modulation of the amplitude of steep wind-waves [J]. Journal of Fluid Mechanics, 1980,99(AUG):705-713.
[63]Huang N E, Long S R, Shen Z, The mechanism for frequency downshift in nonlinear wave evolution, in Advances in applied mechanics [M]. Elsevier:1996:59-117.
[64]Lake B M, Yuen H C, Rungaldier H, et al., Nonlinear deep-water waves-theory and experiment.2. Evolution of a continuous wave train [J]. Journal of Fluid Mechanics,1977,83(Nov):49-74.
[65]Melville W K, The instability and breaking of deep-water waves [J]. Journal of Fluid Mechanics,1982,115: 165-185.
[66]Su M Y, Evolution of groups of gravity waves with moderate to high steepness [J]. Physics of Fluids Physics of Fluids,1982,25(12):2167-74.
[67]. Melville W K, Wave modulation and breakdown [J]. Journal of Fluid Mechanics,1983,128(Mar):489-506.
[68]Ramamonjiarisoa A, Mollochristensen E, Modulation characteristics of sea-surface waves [J]. Journal of Geophysical Research-Oceans and Atmospheres,1979,84(NC12):7769-7775.
[69]Chereskin T K, Mollochristensen E, Modulational development of nonlinear gravity-wave groups [J]. Journal of Fluid Mechanics,1985,151(Fer):337-365.
[70]Tulin M P, Waseda T, Laboratory observations of wave group evolution, including breaking effects [J]. Journal of Fluid Mechanics,1999,378:197-232.
[71]Hwung H H, Chiang W S, Hsiao S C, Observations on the evolution of wave modulation [J]. Proceedings of the Royal Society a-Mathematical Physical and Engineering Sciences,2007,463(2077):85-112.
[72]Chiang W S, Hwung H H, Steepness effect on modulation instability of the nonlinear wave train [J]. Physics of Fluids,2007,19:1-13.
[73]Goda Y, Y. S. Estimation of incident and reflected waves in random wave experiments [C]. In Procedding of 15th International Conference on Coastal Engineering. Honolulu, Hawaii:ASCE,1976,828-865.
[74]Morden D B, Richy E P, Christensen D R. Decomposition of coexisting random wave energy [A]. In Procedding of 15th International Conference on Coastal Engineering [C]. Honolulu, Hawaii,1976, 846-865.
[75]孙昭晨,王利生,斜向入射波和反射波的分离[J].海洋学报,1999,21(4)：114-120.
[76]Mansand E, Funk E. The measurement of incident and reflected spectra using a least squares method [C]. In Proceeding of 17th Interactional Conference on Coastal Engineering. Sydney, Australia:ASCE,1980, 154-172.
[77]Zelt J A, Skjelbreia J E. Estimating incident and reflected wave fields using an arbitrary number of wave gauges [C]. In Procedding of 23th International Conference on Coastal Engineering. Venice Italy:ASCE, 1992,777-789.
[78]Frigaard P, Brorsen M, A time-domain method for separating incident and reflected irregular waves [J]. Coastal Engineering,1995,24(3-4):205-215.
[79]Baldock T E, Simmonds D J, Separation of incident and reflected waves over sloping bathymetry [J]. Coastal Engineering,1999,38(3):167-176.
[80]Chang H K, Hsu T W, A two-point method for estimating wave reflection over a sloping beach [J]. Ocean Engineering,2003,30(14):1833-1847.
[81]Wang S K, Hsu T W, Weng W K, et al., A three-point method for estimating wave reflection of obliquely incident waves over a sloping bottom [J]. Coastal Engineering,2008,55(2):125-138.
[82]Sun H Q, Wang Y X, Peng J P, Hilbert transform applied to separation of waves [J]. China Ocean Engineering,2002,16(2):239-248.
[83]邵利民,入反射波浪的分离与反射系数研究[D].大连：大连理工大学,2003.
[84]Stamos D G, Hajj M R, Reflection and transmission of waves over submerged breakwaters [J]. Journal of Engineering Mechanics,2001,127(2):99-105.
[85]Kostense J K. Measurements of surf beat and set-down beneath wave groups [C]. In Procedding of 19th International Conference on Coastal Engineering. Houston:ASCE,1984,724-740.
[86]Bakkenes H J, Observation and separation of bound and free low-frequency waves in the nearshore zone [D]. Delft:Delft University of Technolog,2002.
[87]Coo ley J W, Tukey J W, An algorithm for the machine calculation of complex fourier series [J]. Mathematics of Computation,1965,19:297-301.
[88]Torrence C, Compo G P, A practical guide to wavelet analysis [J]. Bulletin of the American Meteorological Society,1998,79(1):61-78.
[89]Kijewski T, Kareem A, Wavelet transforms for system identification in civil engineering [J]. Computer-Aided Civil and Infrastructure Engineering,2003,18(5):339-355.
[90]Yan B F, Miyamoto A, Bruhwiler E, Wavelet transform-based modal parameter identification considering uncertainty [J]. Journal of Sound and Vibration,2006,291(1-2):285-301.
[91]Cohen L, Time frequency analysis:Theory and applications [M]. Englewood Cliffs, New Jersey:Prentice Hall,1995.
[92]Kijewski-Correa T, Kareem A, Efficacy of hilbert and wavelet transforms for time-frequency analysis [J]. Journal of Engineering Mechanics,2006,132(10):1037-1049.
[93]Huang N E, Long S R, Tung C C, et al., The local properties of ocean surface-waves by the phase-time method [J]. Geophysical Research Letters,1992,19(7):685-688.
[94]Delprat N, Escudie B, Guillemain P, et al., Asymptoic wavelet and gobor analysis:Extraction of instantaneous frequency [J]. IEEE Transaction on Information Theory,1992,38(2):644-664.
[95]Hsu F H, Blenkarn, Analysis of peak mooring forces caused by slow vessel drift oscillations in random seas [J]. Society of Petroleum Engineer Journal,1972 (Aug):329-344.
[96]Pinkster J A, Low frequency phenomena associated with vessels moored at sea [J]. Society of Petroleum Engineer Journal,1975 (May):487-494.
[97]Johnson R R, Mansard E P D, Plocg J. Effects of wave grouping on breakwater stability [C]. In The 16th Coastal Engineering Conference. Hamburg,Germany:ASCE,1978,2228-2243.
[98]Symonds G, Huntley D A, Bowen A J, Two-dimensional surf beat:Long wave generation by a time-varying breakpoint [J]. Journal of Geophysical Research,1982,87(C1):492-498.
[99]Thompson E F, Seelig W N, High wave grouping in shallow water [J]. Journal of Waterway, Port, Coastal and Ocean Engineering,1984,110(2):139-157.
[100]Veltcheva A D. Wave groupiness in the nearshore area by Hilbert spectrum [C]. In Proceeding of Ocean wave measurement and analysis, San Francisco, CA:ASCE,2001,367-376.
[101]Kuznetsov S Y, Saprykina Y Y, An experimental study of the near-shore evolution of wave groups [J]. Oceanology,2002,42(3):336-343.
[102]Ewing J A, Mean length of runs of high waves [J]. Journal of Geophysical Research,1978,78:1933-1936.
[103]Kimura A. Statistical properties of random wave groups [C]. In Proceedings of 17th Coastal Engineering Conference:ASCE,1980,2955-2973.
[104]Battjes J A. Surf similarity [C]. In Proceeding of the 14th Coastal Engineering Conference. New York: ASCE,1974,1419-1438.
[105]Liu P C, Wave grouping characteristics in nearshore great lakes [J]. Ocean Engineering,2000,27(11): 1221-1230.
[106]Phillips O M, On the dynamics of unsteady gravity waves of finite amplitude, part1. The elementary interactions [J]. Journal of Fluid Mechanics,1960,9:193-217.
[107]Elgar S, Guza R T, Observations of bispectra of shoaling surface gravity waves [J]. Journal of Fluid Mechanics,1985,161:425-448.
[108]Hasselmann K, On the non-linear energy transfer in a gravity-wave spectrum part 1. General theory [J]. Journal of Fluid Mechanics,1962,12:481-500.
[109]Chung J, Powers E J, The statistics of wavelet-based bicoherence [J]. Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis,1998:141-144.
[110]Larsen Y, Hanssen A, Wavelet-polyspectra:Principles and properties [J]. Signal Processing X Theories and Applications.2000 (2):797-800.
[111]Larsen Y, Hanssen A, Pecseli H L, Analysis of non-stationary mode coupling by means of wavelet-bicoherence [C].2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings,2001(6):3581-3584
[112]Gurley K, Kijewski T, Kareem A, First-and higher-order correlation detection using wavelet transforms [J]. Journal of Engineering Mechanics,2003,129:188-201.
[113]Yi E J, Applications of wavelets to nonlinear wave analysis and digital communication [D]. The University of Texas at Austin,2000.
[114]Elgar S, Guza R T, Freilich M, Observations of nonlinear interactions in directionally spread shoaling surface gravity waves [J]. Journal of Geophysical research,1993,98:20299-20305.
[115]Schaffer H A, Second-order wavemaker theory for irregular waves [J]. Ocean engineering,1996,23(1): 47-88.
[116]Madsen P A, Sorensen O R, Bound waves and triad interactions in shallow water [J]. Ocean Engineering, 1993,20:259-388.
[117]Madsen P A, Schaffer H A, Higher-order bossinesq-type equations for surface gravity waves:Derivation and analysis [J]. Proceedings of the Royal Society of London Series A-Mathematical and Physical Sciences, 1998, A356:3123-3184.
[118]Gobbi M F, Kirby J T, Ge W, A fully nonlinear boussinesq model for surface waves. Part 2:extension to O(kh)4 [J]. Journal of Fluid Mechanics,2000 (405):181-210.
[119]Osborne A R, Onorato M, Serio M, The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains [J]. Physics Letters-A,2000,275(5-6):386-393.
[120]Mori N, Onorato M, Janssen P, et al., On the extreme statistics of long-crested deep water waves:Theory and experiments [J]. Journal of Geophysical Research-Oceans,2007,112.
[121]Chien H, Kao C C, Chuang L Z H, On the characteristics of observed coastal freak waves [J]. Coastal Engineering Journal,2002,44(4):301-319.
[122]Slunyaev A, Kharif C, Pelinovsky E, et al., Nonlinear wave focusing on water of finite depth [J]. Physica D-Nonlinear Phenomena,2002,173(1-2):77-96.
[123]Smith S F, Swan C, Extreme two-dimensional water waves:An assessment of potential design solutions [J]. Ocean Engineering,2002,29(4):387-416.
[124]Kharif C, Pelinovsky E, Physical mechanisms of the rogue wave phenomenon [J]. European Journal of Mechanics B-Fluids,2003,22(6):603-634.
[125]Yao A F, Wu C H, Energy dissipation of unsteady wave breaking on currents [J]. Journal of Physical Oceanography,2004,34:2288-2304.
[126]Baldock T E, Long wave generation by the shoaling and breaking of transient wave groups on a beach [J]. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences,2006,462(2070): 1853-1876.
[127]Johannessen T B, Swan C, A laboratory study of the focusing of transient and directionally spread surface water waves [J]. Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences,2001,457(2008):971-1006.
[128]Wu C H, Yao A F, Laboratory measurements of limiting freak waves on currents [J]. Journal of Geophysical Research-Oceans,2004,109(C12):1-18.
[129]Yao A F, Wu C H, Incipient breaking of unsteady waves on sheared currents [J]. Physics of Fluids,2005, 17(8).
[130]Pelinovsky E, Talipova T, Kharif C, Nonlinear-dispersive mechanism of the freak wave formation in shallow water[J]. Physica-D,2000,147(1-2):83-94.
[131]Gibbs R H, Taylor P H, Formation of walls of water in 'fully' nonlinear simulations [J]. Applied Ocean Research,2005,27(3):142-157.
[132]Fochesato C, Grilli S, Dias F, Numerical modeling of extreme rogue waves generated by directional energy focusing [J]. Wave Motion,2007,44(5):395-416.
[133]Li J X, Liu S X, Hong K Y, Numerical study of two-dimensional focusing waves [J]. China Ocean Engineering,2008,22(2):253-266.
[134]Janssen P, Nonlinear four-wave interactions and freak waves [J]. Journal of Physical Oceanography,2003, 33(4):863-884.
[135]Tromans P S, Anatruk A H R, Hagemeijer P. New model for the kinematics of large ocean waves application as a design wave [C]. Edinburgh, Scotl:In Proceeding of the 1st International Society of Offshore and Polar Engineerns (ISOPE), Golden, CO, USA,1991,64-71.
[136]Taylor P H, Williams B A, Wave statistics for intermediate depth water-newwaves and symmetry [J]. Journal of Offshore Mechanics and Arctic Engineering-Transactions of the ASME,2004,126(1):54-59.
[137]Chaplin J R, Rainey R C T, Yemm R W, Ringing of vertical cylinder in waves [J]. Journal of Fluid Mechanics,1997,350:119-147.
[138]Grue J, Huseby M, Higher-harmonic wave force and ringing of vertical cylinder [J]. Applied Ocean Research,2002,24(4):203-214.
[139]Goda Y, A comparative review on the functional forms of directional wave spectrum [J]. Coastal Engineering Journal,1999,41(1):1-20.
[140]Sharma J N, Dean R G, Second-order directional seas and associated wave forces [J]. SPEJ, Society of Petroleum Engineers Journal,1981,21(1):129-140.
[141]Walker D A G, Taylor P H, Taylor R E, The shape of large surface waves on the open sea and the draupner new year wave [J]. Applied Ocean Research,2004,26(3-4):73-83.
[142]Peregrine D H, Interaction of water waves and currents [J]. Advance in Applied Mechanics,1976,16: 10-117.
[143]Turpin F, Benmoussa C, Mei C C, Effects of slowly varying depth and current on the evolution of a stokes wavepacket [J]. Journal of Fluid Mechanics,1983,132:1-23.
[144]Gerber M, The benjamin-feir instability of a deep-water stokes wavepacket in the presence of a non-uniform medium [J]. Journal of Fluid Mechanics,1987,176:311-332.
[145]Chawla A, Kirby J T, Monochromatic and random wave breaking at blocking points [J]. Journal of Geophysical Research,2002,107(C7):4-1-19.
[146]Duin V C A, The effect of non-uniformity of modulated wavepackets on the mechanism of benjamin-feir instability [J]. Journal of Fluid Mechanics,1999,399:237-249.
[147]Lai R J, Long S R, Huang N E, Laboratory studies of wave-current interaction-kinematics of the strong interaction [J]. Journal of Geophysical Research-Oceans,1989,94(C11):16201-16214.
[148]Chawla A, An experimental study on the dynamics of wave blocing and breaking on opposing currents [D]. University of Delaware,1999.
[149]Brethert F P, Garrett C J R, Wavetrains in inhomogeneous moving media [J]. Proceedings of the Royal Society of London Series A-Mathematical and Physical Sciences,1968,302(1471):529-554.
[150]Kamphuis J W. Designing for low frequency waves [C]. In Proceeding of 27th International Conference on Coastal Engineering. Sydney:ASCE,2000,1434-1447.
[151]Roelvink J A, Surf beat and its effect on cross-shore profiles [D].1993, Delft:Delft University of Technology.
[152]Bowers E C, Harbour resonance due to set-down beneath wave groups [J]. Journal of Fluid Mechanics, 1977,79(1):71-92.
[153]Longuet-Higgins M S, Stewart R W, Changes in the form of short gravity waves on long waves and tidal currents [J]. Journal of Fluid Mechanics,1960,8(4):565-583.
[154]Dong G, Ma X, Perlin M, et al., Experimental study of long waves generation on sloping bottoms [J]. Coastal Engineering,2009,56:82-89.
[155]Battjes J A, Bakkenes H J, Janssen T T, et al., Shoaling of subharmonic gravity waves [J]. Journal of Geophysical Research,2004,109(C2):C02009,doi:10.1029/2003JC001863.