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研究生： 江文山
Chiang, Wen-Son
論文名稱： 深水區非線性波列調變之研究
A Study on Modulation of Nonlinear Wave Trains in Deep Water
指導教授： 黃煌煇
Hwung, Hwung-Hweng
學位類別： 博士
Doctor
系所名稱： 工學院 - 水利及海洋工程學系
Department of Hydraulic & Ocean Engineering
論文出版年： 2005
畢業學年度： 93
語文別： 英文
論文頁數： 144
中文關鍵詞： 主頻下移 、不穩定 、調變 、副頻 、再現性
外文關鍵詞： sideband, modulation, recurrence, instability, frequency downshift
相關次數： 點閱：90  下載：5 
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    • 　　本文於成功大學水工試驗所之大型波浪水槽(300m*5m*5.2m)進行一系列深水波實驗研究。針對三種典型非線性波列之演變進行觀察研究，分別是初始均勻波列、初始給定主頻及一組微弱副頻組成之弱調變波列與初始給定兩相等振幅而頻率存在微小差異成份波組成之雙頻波列。

    　　實驗結果顯示初始均勻波列之調變乃源自於造波機由靜止啟動至穩態運轉過程中所造出之非穩態波列，同時伴隨衍生出副頻能量。之後，波形調變及副頻能量隨此非穩態波列傳遞而增加，最終達到近似穩定調變，其中有一組(高、低)自然衍生之副頻因成長最快而特別顯著。前人對於此類初始均勻波列衍生之調變的研究侷限於大尖銳度之波列( )，而本文的研究範圍則涵蓋較大區域( )。初始階段一組副頻振幅呈現對稱指數成長，直到碎波後低副頻振幅超越原主頻形成所謂主頻下移現象。

    　　觀察分析一系列相同主頻而主頻與副頻之頻差不同的初始弱調變波列的實験結果，發現波列的演變終將會使得最不穩定副頻突顯出來。其中特別值得提出的是對頻差微小的波列，其最不穩定副頻是透過多次的主頻下移而得以顯露，這是首次被觀察到的現象。其隱含透過副頻不穩定機制，伴隨波列的傳遞主頻會逐漸往更低頻遷移。

    　　針對不同尖銳度之初始弱調變波列所作的實験，顯示在後碎波階段，波列呈現週期性調變與反調變，其中波列大部份能量在原主頻與給定之一組副頻間傳遞。因初始尖銳度的不同，後碎波階段之實驗結果分別呈現出永久性與暫時性主頻下移。亦即前人觀察到之碎波引起永久性主頻下移其實未必是永久性。

    　　雙頻波列演變的實驗分析結果，在非碎波情況，波列呈現初始狀態的近似再現性，對波列演變過程中出現碎波的情況，則發現波列碎波過程中低副頻呈現相對增強，這與初始弱調變波列的實驗結果一致。

    　　因為副頻不穩定機制，非線性波列演變過程中會演化出瞬間極大波乃至發生碎波。相較於Stokes波的碎波極限，觀察到的碎波其對應之尖銳度與波峰水粒子速度對相位速度的比值皆偏小。顯示基於均勻波列之假設推導得到之Stokes波的碎波極限與調變波列中碎波發生之條件並不一致。

    　To investigate the wave modulation induced by sideband instability in deep water. A series of experiments on the long time evolution of nonlinear wave trains in deep water were carried out in a super wave flume (300m x 5.0m x 5.2m) at Tainan Hydraulics Laboratory. In order to examine the wave modulations which evolve naturally and initially seeded, three typical wave trains, namely uniform wave, imposed sidebands wave and bichromatic wave were generated initially by a piston-type wave maker.

    　The modulation of wave trains increases with propagation and the corresponding wave spectra show the exponential growth of sideband amplitudes, which further increase due to wave reflection from the flume boundary. Eventually, a quasi steady state of wave train evolution is achieved and a pair (upper and lower) of fastest growth sidebands evolved. Initially, the amplitudes of fastest growth sidebands exhibit a symmetric exponential growth in spatial domain until the onset of wave breaking. The amplitude of lower sideband becomes larger than those of the carrier wave and upper sideband after wave breaking, which is so called frequency downshift. Present results on the evolution of initial uniform wave trains cover a wide range of initial wave steepness (kcac=0.12~0.30) and greatly extended previous researcher’s studies which only confine to the initial larger wave steepness region (kcac>0.20).

    　The investigations on the evolution of initial imposed sidebands wave trains with constant initial wave steepness but different normalized frequency difference between the carrier wave and the imposed sideband (sideband space) illustrate that the most unstable mode of initial wave train will manifest itself in the evolution of wave train. Especially, the most unstable mode of initial wave train develops through a multiple-downshift of wave spectrum for the wave train with smaller sideband space, which is observed the first time in literature. The evolution of wave train is a periodic modulation and demodulation at post breaking stages, in which most of the energy of wave train transfers cyclically between the carrier wave and two imposed sidebands. Meanwhile, the wave spectrum shows temporal and permanent frequency downshift respectively for different initial wave steepness, which suggests that the permanent frequency downshift induced by wave breaking observed by previous researchers may not be permanent.

    　Near recurrence of initial status of bichromatic wave train is demonstrated for non-breaking case. However, for breaking case, the amplitude of lower sideband is selectively amplified through the breaking process. The results confirm the findings on the evolution of initial uniform and imposed sidebands wave trains.

    　The local wave steepness and the ratio of horizontal particle velocity to linear phase velocity at wave breaking in modulated wave group indicate that the breaking criterion of Stokes wave, which is derived based on the uniform wave, is not consistent with the results of modulated wave trains.

    Contents CHAPTER 1 Introduction 1 1.1 Problem Statement 1 1.2 Literature Review 1 1.3 Outline of Contents 6 CHAPTER 2 Experiments and Data Analysis 10 2.1 Experimental facilities and setup 10 2.2 Experimental conditions 12 2.3 Experimental procedures 17 2.4 Data analysis 18 2.4.1 analysis of local maximum wave 18 2.4.2 analysis of wave spectrum 19 2.4.3 analysis of initial growth rate 19 2.5 noise characteristics 20 CHAPTER 3 Transformation of transient wave front 24 3.1 Transient-wave transformations 24 3.2 Effect of wave reflection 34 3.3 Summary 38 CHAPTER 4 Quasi steady modulation evolves from nonlinear uniform wave trains 39 4.1 Wave trains propagation without modulation 39 4.2 Quasi-steady modulation of wave trains 41 4.3 Fastest growth mode and initial growth rate 46 4.4 Spatial evolution of normalized wave amplitudes 49 4.5 Summary 51 CHAPTER 5 Quasi-steady modulation evolves from initial imposed sidebands wave trains 52 5.1 Evolution of non-breaking wave train 53 5.2 Evolution of wave trains with breaking 55 5.2.1 evolution with breaking and without downshifting 55 5.2.2 evolution with breaking and downshifting 58 5.2.3 evolution with breaking and multiple downshifting 60 5.3 Evolution of dimensionless amplitudes of one carrier and two pairs of sidebands 64 5.4 Summary 68 CHAPTER 6 Effect of wave steepness on the evolution of nonlinear wave trains 69 6.1 Evolution of wave trains without breaking (case T172) 69 6.2 Evolution of wave trains with breaking 73 6.2.1 initial wave steepness (case T091) 73 6.2.2 initial wave steepness (case T166) 78 6.2.3 initial wave steepness (case T168) 81 6.3 Breaking criterion in modulated wave trains 84 6.4 Summary 84 CHAPTER 7 Evolution of bichromatic wave trains 86 7.1. Evolution of wave groups for non-breaking type 87 7.2. Evolution of wave groups for breaking type 98 7.3. Summary 102 CHAPTER 8 Characteristics of large transient wave 103 8.1 Large transient wave on the wave front 103 8.2 Large transient wave at quasi steady state 114 8.3 Group velocity 117 8.4 Time-frequency features of transient large wave 119 8.5 Summary 122 CHAPTER 9 Conclusions and future works 124 9.1 Conclusions 124 9.2 Future works 126 Reference 128 Appendix A 134 Appendix B 136 List of tables Table 2.1 Experimental wave conditions of initial uniform wave trains 15 Table 2.2 Experimental wave conditions of initial bichromatic wave trains 15 Table 2.3 Experimental wave conditions of initial weakly modulated wave trains 15 Table 5.1 Detail information on wave conditions of initial imposed sidebands wave trains 52 Table 6.1 Experimental wave conditions for different initial wave steepness 69 Table 7.1 Summary of experimental conditions, given wave period, wave height, relative water depth, wave steepness of initial bichromatic wave trains 87 Table 8.1 The comparisons of local maximum wave steepness between present results and other’s work 115 Table 8.2 The ratio of estimated group velocity to the linear group velocity 118 List of figures Fig. 2.1 Experimental setup and the configuration of wave tank. 11 Fig. 2.2 The calibration of wave gauge 12 Fig. 2.3 The typical time series and related spectrum of wave maker displacement for initial uniform wave train. 15 Fig. 2.4 The typical time series and related spectrum of wave maker displacement for initial bichromatic wave train 16 Fig. 2.5 The typical time series and related spectrum of wave maker displacement for initial modulated wave train 16 Fig. 2.6 The implicit mode functions analyzed by HHT for a given analytical signal 22 Fig. 2.7 The contours of wave amplitude versus time and frequency analyzed by HHT for a given analytical signal 22 Fig. 2.8 The measured data and decomposed implicit mode functions 23 Fig. 2.9 The contours of wave amplitude versus time and frequency analyzed by HHT for the measured wave data 23 Fig. 3.1 The measured surface elevations of the wave front of initial uniform wave train (non-breaking) 27 Fig. 3.2 The evolution of dimensionless wave height and wave steepness of the wave front for initial uniform wave train with (non-breaking) 27 Fig. 3.3 The measured surface elevations of the wave front of initial uniform wave train (fully developed without breaking) 28 Fig. 3.4 The evolution of dimensionless wave height and wave steepness of the wave front for initial uniform wave train with (fully developed without breaking) 28 Fig. 3.5 The measured surface elevations of the wave front of initial uniform wave train (fully developed with breaking) 29 Fig. 3.6 The evolution of dimensionless wave height and wave steepness of the wave front for initial uniform wave train with . (fully developed with breaking) 29 Fig. 3.7 The measured surface elevations of wave front of the initial imposed sidebands wave train and (fully developed without breaking) 30 Fig. 3.8 The evolution of dimensionless wave height and wave steepness of the wave front for initial modulated wave train with (fully developed without breaking) 30 Fig. 3.9 The measured surface elevations of wave front of the initial imposed sidebands wave train and (fully developed with breaking) 31 Fig. 3.10 The evolution of dimensionless wave height and wave steepness of the wave front for initial modulated wave train with (fully developed with breaking) 31 Fig. 3.11 The measured surface elevations of the wave front of initial bichromatic wave train and (fully developed without breaking) 32 Fig. 3.12 The evolution of dimensionless wave height and wave steepness of the wave front of initial bichromatic wave train, (fully developed without breaking) 32 Fig. 3.13 The measured surface elevations of the wave front of initial bichromatic wave train, and (fully developed with breaking) 33 Fig. 3.14 The evolution of dimensionless wave height and wave steepness of the wave front of initial bichromatic wave train, and (fully developed with breaking) 33 Fig. 3.15 Photos of the breaking events for initial imposed sidebands wave train with 34 Fig. 3.16 The contours of amplitude spectra for initial uniform wave train versus frequency and non-dimensional fetch at four different instants 36 Fig. 3.17 The contours of amplitude spectra for initial modulated wave train versus frequency and non-dimensional fetch at four different instants 37 Fig. 4.1 The measured surface elevations and related wave spectra at quasi steady state for initial uniform wave train, , 40 Fig. 4.2 The evolution of dimensionless maximum crest and minimum trough for initial uniform wave train, and 40 Fig. 4.3 The amplitude contours versus dimensionless frequency difference and fetch for initial uniform wave train Hz, and 41 Fig. 4.4 The measured surface elevations and related wave spectra at quasi steady state for initial uniform wave train, 44 Fig. 4.5 The evolution of dimensionless maximum crest and minimum trough for initial uniform wave train, 44 Fig. 4.6 The amplitude contours versus dimensionless frequency difference and fetch for initial uniform wave train Hz, and 45 Fig. 4.7 The measured surface elevations and related wave spectra at quasi steady state for initial uniform wave train, 45 Fig. 4.8 The evolution of dimensionless maximum crest and minimum trough for initial uniform wave train and 46 Fig. 4.9 The amplitude contours versus dimensionless frequency difference and fetch for initial uniform wave train Hz, and 46 Fig. 4.10 The comparison of initial growth rate between the experimental results for initial uniform wave train and Tulin&Waseda’s (1999) prediction based on Krasitskii’s (1994) theory. The error bar denoted the standard deviation 48 Fig. 4.11 The comparisons of normalized frequency difference between the carrier wave and the fastest growth sideband versus initial wave steepness between present experiments and previous experiments and theoretical predictions. : Melville(1982); ○: Waseda&Tulin(1999); ◇: present experiments; thick line: Tulin and Waseda (1999) calculated based on Krasitskii’s (1994) theory; dash line: Dysthe(1979); dot line: Longuet-Higgins (1980); thin line: Benjamin&Feir(1967) 48 Fig. 4.12 Evolution of normalized amplitudes of one carrier wave and a pair of fastest growth sidebands versus dimensionless fetch. (a) (b) (c) (d) . circle: carrier wave; diamond: lower sideband; cross: upper sideband; solid line: growth curve of sidebands predicted by Benjamin and Feir(1967); dash line: growth curve of sidebands predicted by Tulin and Waseda(1999) 50 Fig. 5.1 The measured surface elevations and related wave spectra at quasi steady state for initial modulated wave train , and 54 Fig. 5.2 The amplitude contours of wave train at quasi steady state versus dimensionless frequency difference and fetch for initial modulated wave train, and 55 Fig. 5.3 The evolution of dimensionless maximum crest and minimum trough elevations at quasi steady state for initial modulated wave train, and 55 Fig. 5.4 The measured surface elevations and related wave spectra at quasi steady state for initial modulated wave train, and 57 Fig. 5.5 The amplitude contours of wave train at quasi steady state versus dimensionless frequency difference and fetch for initial imposed sidebands wave train, and 57 Fig. 5.6 The evolution of dimensionless maximum crest and minimum trough elevations at quasi steady state for initial imposed sidebands wave train, and 58 Fig. 5.7 The measured surface elevations and related wave spectra at quasi steady state for initial modulated wave train, and 59 Fig. 5.8 The amplitude contours of wave train at quasi steady state versus dimensionless frequency difference and fetch state for initial modulated wave train, and 60 Fig. 5.9 The evolution of dimensionless maximum crest and minimum trough elevations at quasi steady state for initial modulated wave train, and 60 Fig. 5.10 The measured surface elevations and related wave spectra at quasi steady state for initial modulated wave train, and 62 Fig. 5.11 The amplitude contours of wave train versus dimensionless frequency difference and fetch for initial modulated wave train, and . (a) analyzed by using the measured data in 10 minutes after wave generation. (b) analyzed by using the measured data in 20 minutes after wave generation 63 Fig. 5.12 The evolutions of dimensionless amplitudes including one carrier wave and two pairs of sidebands for wave trains with the same wave steepness ( ) but different normalized sideband space. (a) (b) (c) (d) . circle: ; diamond: ; cross: ; square: ; triangle: 66 Fig. 5.13 The evolutions of dimensionless amplitudes including one carrier wave and two pairs of sidebands for wave trains with the same wave steepness ( ) but different normalized sideband space. (a) (b) (c) (d) .circle: ; diamond: ; cross: ; square: ; triangle: 67 Fig. 6.1 (a) The measured surface elevations for initial wave train case T172. (b) The corresponding wave spectrum. (c) The amplitude contours versus dimensionless frequency difference and fetch 71 Fig. 6.2 Evolution of local maximum wave characteristics for case T172 (a) normalized crest elevation (b) normalized wave height (c) local maximum wave steepness based on maximum wave height ( ) and crest elevation ( ) (d) the ratio of horizontal velocity at maximum wave crest to linear phase velocity 72 Fig. 6.3 The spatial evolution of dimensionless amplitudes including one carrier wave (circle) and two imposed sideband components (lower sideband : diamond; upper sideband : cross) which roughly corresponded to the most unstable mode for case T172 72 Fig. 6.4 The measured surface elevations and the corresponding wave spectrum for case T091 75 Fig. 6.5 The evolution of dimensionless amplitudes of one carrier wave and two imposed sideband components which roughly corresponded to the most unstable mode (a) Tulin and Waseda’s (1999) Fig. 13 in which hatched area indicates breaking region (b) case T091 (c) case T164 76 Fig. 6.6 Evolution of local maximum wave parameters for case T091. (a) normalized crest and trough elevation (b) normalized wave height (c) local maximum wave steepness based on maximum wave height ( ) and crest elevation ( ) (d) the ratio of horizontal particle velocity at maximum wave crest to linear phase velocity 77 Fig. 6.7 Evolution of local maximum wave parameters for case T164. (a) normalized crest and trough elevation (b) normalized wave height (c) local maximum wave steepness based on maximum wave height ( ) and crest elevation ( ) (d) the ratio of horizontal particle velocity at maximum wave crest to linear phase velocity 77 Fig. 6.8 The measured surface elevations and the corresponding wave spectrum for case T166 79 Fig. 6.9 The evolution of dimensionless amplitudes of one carrier wave and two imposed sideband components which roughly corresponds to the most unstable mode for case T166 80 Fig. 6.10 Evolution of local maximum wave parameters for case T166. (a) normalized crest and trough elevation (b) normalized wave height (c) local maximum wave steepness based on maximum wave height ( ) and crest elevation ( ) (d) the ratio of horizontal particle velocity at maximum wave crest to linear phase velocity 80 Fig. 6.11 The measured surface elevations and the corresponding wave spectra for case T168 82 Fig. 6.12 The evolution of dimensionless amplitudes of one carrier wave (circle) and two imposed sideband components which roughly corresponded to the most unstable mode (lower sideband : diamond; upper sideband : cross) for case T168 83 Fig. 6.13 Evolution of local maximum wave parameters for case T168. (a) normalized crest and trough elevation (b) normalized wave height (c) local maximum wave steepness based on maximum wave height ( ) and crest elevation ( ) (d) the ratio of horizontal particle velocity at maximum wave crest to linear phase velocity 83 Fig. 6.14 The ratio of horizontal particle velocity at maximum crest elevation to linear phase velocity as a function of initial wave steepness 84 Fig. 7.1 Comparisons between experimental data and numerical results for Case B11 (a) NLS model. (b) three-layer Boussinesq model (solid line, numerical results; dotted line, experimental data) 88 Fig. 7.2 Comparisons of amplitude spectra between experimental data and numerical results for Case B11 (a) NLS model. (b) three-layer Boussinesq model (solid line, numerical results; dotted line, experimental data) 90 Fig. 7.3 The amplitude contours of experimental results for Case B11 versus frequency and fetch 91 Fig. 7.4 The amplitude contours of three –layer Boussinesq model results versus frequency and fetch for Case B11 91 Fig. 7.5 The amplitude contours of NLS model results versus frequency and fetch for Case B11 91 Fig. 7.6 The evolution of dimensionless amplitudes of two-layer Boussinesq model results for Case B11. Circle (solid line): ; diamond (dashed line): ; Triangle (dashed-dotted line): ; Asterisk (dotted line): . The total amplitude is defined by 92 Fig. 7.7 The evolution of dimensionless amplitudes of three-layer Boussinesq model results for Case B11. symbols description are the same as Fig. 7.6 93 Fig. 7.8 The evolution of dimensionless amplitudes of NLS model results for Case B11. symbols description are the same as Fig. 7.6 93 Fig. 7.9 Comparisons between experimental data and numerical results for Case B39 (a) NLS model. (b) three-layer Boussinesq model (solid line, numerical results; dotted line, experimental data) 95 Fig. 7.10 Comparisons of amplitude spectra between experimental data and numerical results for Case B39 (a) NLS model (b) three-layer Boussinesq model (solid line, numerical results; dotted line, experimental data) 96 Fig. 7.11 The amplitude contours of experimental results versus frequency and fetch for Case B39 97 Fig. 7.12 The amplitude contours of three –layer Boussinesq model results versus frequency and fetch for Case B39 97 Fig. 7.13 The amplitude contours of NLS model results versus frequency and fetch for Case B39 97 Fig. 7.14 The evolution of dimensionless amplitudes of NLS model results for Case B39. Circle (solid line): ; diamond (dashed line): ; Triangle (dashed-dotted line): ; Asterisk (dotted line): 98 Fig. 7.15 The evolution of dimensionless amplitudes of three-layer Boussinesq model results for Case B39. Circle (solid line): ; diamond (dashed line): ; Triangle (dashed-dotted line): ; Asterisk (dotted line): 98 Fig. 7.16 Comparisons of amplitude spectra between experimental data and numerical results for Case B46 (a) NLS model (b) three-layer Boussinesq model (solid line, numerical results; dotted line, experimental data) 100 Fig. 7.17 The comparisons of evolution of dimensionless amplitudes between experimental data and NLS model results for Case B46. Circle (solid line): ; diamond (dashed line): ; Triangle (dashed-dotted line): ; Asterisk (dotted line): 101 Fig. 7.18 The amplitude contours of experimental results versus frequency and fetch for Case B46 101 Fig. 7.19 The amplitude contours of NLS model results versus frequency and fetch for Case B46 101 Fig. 8.1 Evolution of local maximum wave parameters for initial uniform wave train and (a) normalized crest and trough elevations (b) normalized wave height (c) local maximum wave steepness (d) ratio of horizontal particle velocity at maximum wave crest to linear phase velocity 106 Fig. 8.2 Evolution of local maximum wave parameters for initial uniform wave train and (a) normalized crest and trough elevations (b) normalized wave height (c) local maximum wave steepness (d) ratio of horizontal particle velocity at maximum wave crest to linear phase velocity 107 Fig. 8.3 Evolution of local maximum wave parameters for initial uniform wave train and (a) normalized crest and trough elevations (b) normalized wave height (c) local maximum wave steepness (d) ratio of horizontal particle velocity at maximum wave crest to linear phase velocity 108 Fig. 8.4 Evolution of local maximum wave parameters for initial uniform wave train and (a) normalized crest and trough elevations (b) normalized wave height (c) local maximum wave steepness (d) ratio of horizontal particle velocity at maximum wave crest to linear phase velocity 109 Fig. 8.5 Evolution of local maximum wave parameters for initial bichromatic wave train , and , (a) normalized crest and trough elevations (b) normalized wave height (c) local maximum wave steepness (d) ratio of horizontal particle velocity at maximum wave crest to linear phase velocity 110 Fig. 8.6 The calculated local maximum wave steepness ( ) versus initial wave steepness for initial uniform wave trains. symbols: experimental data; solid line is the best fitted curve 111 Fig. 8.7 The calculated local maximum wave steepness ( ) versus initial wave steepness for initial uniform wave trains. symbols: experimental data; solid line is the best fitted curve 111 Fig. 8.8 The calculated local maximum wave steepness ( ) versus initial wave steepness for initial bichromatic wave trains. symbols: experimental data; solid line is the best fitted curve 112 Fig. 8.9 The calculated local maximum wave steepness ( ) versusinitial wave steepness for initial weak modulated wave trains. symbols: experimental data; solid line is the best fitted curve 112 Fig. 8.10 The calculated local maximum wave steepness ( ) versus initial wave steepness for initial weak modulated wave trains. symbols: experimental data; solid line is the best fitted curve 113 Fig. 8.11 The calculated local maximum wave steepness ( ) versus initial wave steepness for initial weak modulated wave trains. symbols: experimental data; solid line is the best fitted curve 113 Fig. 8.12 The estimated local maximum wave steepness versus initial wave steepness for initial modulated wave train. circle: ; square: at quasi-steady state; cross: ; diamond: on the wave front. The solid line is the breaking criterion of Stokes wave 116 Fig. 8.13 Normalized maximum crest elevation versus initial wave steepness 116 Fig. 8.14 The ratio of horizontal particle velocity at maximum crest elevation to the phase velocity versus initial wave steepness. ◇: ; +: ; ○: and △: 117 Fig. 8.15 The locations where the maximum ratio of horizontal particle velocity at wave crest to the phase velocity is estimated, versus initial wave steepness. ◇ : ; +: ; ○: and △: 117 Fig. 8.16 Time series of surface elevation and the related time-frequency energy distribution (a) (b) (c) 121 Fig. 8.17 Photos of wave breaking in strongly modulated wave train. (a) two-dimensional breaker (b) three-dimensional breaker, which appears on the center of the wave flume (c) three-dimensional breaker, which appears on the sidewall of the wave flume 122

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