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A Preliminary result on the long time evolution of non-linear waves --as a partial result of. The Characteristics of Nonlinear Wave Transformation on Sloping Bottoms A MOE Program for Promoting Academic Excellence of Universities, Taiwan under grant number A-91-E-FA09-7-3.

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A Preliminary result on the long time evolution of non-linear waves --as a partial result of


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    1. A Preliminary result on the long time evolution of non-linear waves--as a partial result of The Characteristics of Nonlinear Wave Transformation on Sloping Bottoms A MOE Program for Promoting Academic Excellence of Universities, Taiwan under grant number A-91-E-FA09-7-3

    2. Basic Information of the Project (1/2) • Objectives • The objectives of the project are to investigate the nonlinear wave transformation, modulation, and the characteristics of related flow structures during the shoaling process of wave propagation. 卓越計畫--斜坡上非線性波浪傳動特性之研究

    3. Nonlinear modulation of progressive wave The studies on wave breaking and related flow fields The generation of infragravity waves The transmission of wave envelops Under certain conditions Different I. C. Basic Information of the Project (2/2) • Sub-Projects 卓越計畫--斜坡上非線性波浪傳動特性之研究

    4. Previous work done by (1/3) : • Theoretical studies • Benjamin and Feir(1967) • weakly nonlinear deep water wavetrains were unstable to modulational perturbations • Whitham(1967), Chu and Mei(1970,1971) : • Zakharzov(1968) : Schrödinger Eq.(third-order of ka) • (1) deep water • (2) small ka • (3) slow modulation of amplitude(narrow band) • Dysthe(1979) : forth-order equation • (1) mean flow effect(mean water level variation) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    5. Previous work done by (2/3) • Numerical development • Fully Nonlinear Eq. : • Louguet-Higgins(1978) • Wang Pei(1995) • Wethuis(2001) • Cubic Schrödinger Eq. : • Yuen and Lake(1978) • Shemer et al.(1998) • Forth-order Eq. : • Janssen(1983) • Lo and Mei(1985); • Lo and Mei(1987) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    6. Previous work done by (3/3) : • Experimental Studies • Lake et. al. (1977)—F-P-U phenomena • Su(1982) , Kit et al.(2000), Wethuis(2001) • Melville(1982) • Shemer et al.(1998) : intermediate depth • Tulin and Waseda(1999) : breaking effect • Waseda and Tulin(1999) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    7. The approaches and methodology of the Project • The basic approaches are to utilize the facilities of the THL to investigate this topic experimentally followed by developing a numerical model that is able to describe the transformation of waves from a deep water region to shallow water region and even breaking taken place. • The methodology applied on this project is to investigate the subject under a relatively simple boundary condition first, then gradually proceed to complicated ones. 卓越計畫--斜坡上非線性波浪傳動特性之研究

    8. Overview of Experiments • Three phases • Phase I: • In the Super Tank(5m*5m*300m), THL • With a slopping bottom(1:80) • 39 wave gages • Bicromatic waves • Phase II • In a meso-flume(7m*1m*60m), THL • With a slopping bottom(1:40) • Seeded three waves system • Phase III • In the Super Tank(5m*5m*300m) • Constant water depth • 66+24 wave gages, 8+1 velocimetries • Seeded three waves system, un-seeded waves.. 卓越計畫--斜坡上非線性波浪傳動特性之研究

    9. EXP. I (1/2) • Tests in the Super Tank , THL • To ensure the capabilities of the facilities • Wave generator • Two waves system (bichromatic waves) • Boundary conditions • Relative dimensions, sidewall effects, reflection… • Instrumentation • using 39 wave sensors • Data acquisition and analysis 卓越計畫--斜坡上非線性波浪傳動特性之研究

    10. IPC1 IPC1 EXP. I (2/2) ~ WH01 WH03 WH04 WH05 WH06 WH07 WH08 WH09 WH10 WH11 WH12 WH13 WH14 WH15 WH16 WH17 WH18 WH19 WH20 WH21 WH22 WH23 WH24 WH25 WH39 ~ 3.5m 3.15 45 285m MNDAS A Schematic Diagram of the Experimental set-up(phase I, III) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    11. (1/10)Results-Exp. Phase I kx/2=3.5 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/2=4.3 kx/2=4.6 kx/2=8.5 卓越計畫--斜坡上非線性波浪傳動特性之研究

    12. (2/10)Results-Exp. Phase I kx/2=12.5 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/=16.0 kx/2=19.8 kx/2=23.6 1:80 slope 卓越計畫--斜坡上非線性波浪傳動特性之研究

    13. (3/10)Results-Exp. Phase I kx/2=27.3 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/2=31.1 kx/2=34.9 kx/2=38.6 卓越計畫--斜坡上非線性波浪傳動特性之研究

    14. (4/10)Results-Exp. Phase I kx/2=43.4 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/2=46.2 kx/2=50.0 kx/2=53.7 卓越計畫--斜坡上非線性波浪傳動特性之研究

    15. (5/10)Results-Exp. Phase I kx/2=57.5 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/2=58.8 kx/2=59.2 kx/2=59.5 卓越計畫--斜坡上非線性波浪傳動特性之研究

    16. (6/10)Results-Exp. Phase I kx/2=59.7 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/2=60.2 kx/2=60.7 kx/2=61.2 卓越計畫--斜坡上非線性波浪傳動特性之研究

    17. (7/10)Results-Exp. Phase I kx/2=61.6 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/2=61.9 kx/2=62.3 kx/2=62.7 卓越計畫--斜坡上非線性波浪傳動特性之研究

    18. (8/10)Results-Exp. Phase I kx/2=62.8 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/2=62.9 kx/2=63.0 kx/2=63.1 卓越計畫--斜坡上非線性波浪傳動特性之研究

    19. (9/10)Results-Exp. Phase I kx/2=63.2 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/2=63.29 kx/2=63.3 kx/2=63.4 卓越計畫--斜坡上非線性波浪傳動特性之研究

    20. (10/10)Results-Exp. Phase I kx/2=63.5 • (a) Wave profiles showing the modulational wave train.(b)Spectral evolution.(c)amplitude envelope of surface elevation. kx/2=63.6 kx/2=63.7 卓越計畫--斜坡上非線性波浪傳動特性之研究

    21. Experiments-Phase II 卓越計畫--斜坡上非線性波浪傳動特性之研究

    22. Power spectrum of waveboard motion (EXP.II) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    23. Data Analysis EXP. II • Bandpass filter : 0.01 Hz ~ 5 Hz • data low passed by Hilbert transform • power spectrum were calculated under : • Data length=3072 • NFFT=1024 • Window=Hanning(1024) • Overlap=0 • Frequency resolution=1.953*10-2 卓越計畫--斜坡上非線性波浪傳動特性之研究

    24. Test conditions EXP. II • Three waves system ka ~>0.14~0.16 breaking, ka ~>0.29 three dimensional 卓越計畫--斜坡上非線性波浪傳動特性之研究

    25. Spatial evolution of wave profiles(EXP. II, case 2) kx/2=3.2 kx/2=3.7 kx/2=3.8 kx/2=4.5 kx/2=5.8 卓越計畫--斜坡上非線性波浪傳動特性之研究

    26. Spatial evolution of wave profiles(EXP. II, case 2) kx/2=7.0 kx/2=8.3 kx/2=9.6 kx/2=10.9 kx/2=12.2 卓越計畫--斜坡上非線性波浪傳動特性之研究

    27. Spatial evolution of wave profiles(EXP. II, case 2) kx/2=13.5 kx/2=14.0 kx/2=14.5 kx/2=15.0 kx/2=16.1 卓越計畫--斜坡上非線性波浪傳動特性之研究

    28. Spatial evolution of wave profiles (EXP. II, case 3) kx/2=3.2 kx/2=3.7 kx/2=3.8 kx/2=4.5 kx/2=5.8 卓越計畫--斜坡上非線性波浪傳動特性之研究

    29. Spatial evolution of wave profiles (EXP. II, case 3) kx/2=7.0 kx/2=8.3 kx/2=9.6 kx/2=10.9 kx/2=12.2 卓越計畫--斜坡上非線性波浪傳動特性之研究

    30. Spatial evolution of wave profiles (EXP. II, case 3) kx/2=13.5 kx/2=14.0 kx/2=14.5 kx/2=15.0 kx/2=16.1 卓越計畫--斜坡上非線性波浪傳動特性之研究

    31. kx/2=6.5 kx/2=7.4 kx/2=7.8 kx/2=9.1 kx/2=11.7 Spatial evolution of wave profiles (EXP. II, case 1) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    32. kx/2=14.3 kx/2=16.9 kx/2=19.5 kx/2=22.1 kx/2=24.7 Spatial evolution of wave profiles(EXP. II, case 1) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    33. kx/2=27.3 kx/2=28.3 kx/2=29.3 kx/2=30.2 kx/2=32.5 Spatial evolution of wave profiles(EXP. II, case 1) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    34. kx/2=6.5 kx/2=7.4 kx/2=7.8 kx/2=9.1 kx/2=11.7 Spatial evolution of wave profiles and spectrum (EXP. II, case 1) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    35. kx/2=14.3 kx/2=16.9 kx/2=19.5 kx/2=22.1 kx/2=24.7 Spatial evolution of wave profiles and spectrum(EXP. II, case 1) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    36. kx/2=27.3 kx/2=28.3 kx/2=29.3 kx/2=30.2 kx/2=32.5 Spatial evolution of wave profiles and spectrum(EXP. II, case 1) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    37. Side band amplitude evolutions (EXP. II, case 1) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    38. Growth Curve 卓越計畫--斜坡上非線性波浪傳動特性之研究

    39. Wave steepness effects • The spatial evolution of wave profiles is not only related to the initial wave steepness • The experiment (ka=0.1) is similar to the condition of the evolution experiment on a weak wave reported in Lake et. al.(1977) and Tulin & Waseda(1999). • ka ~>0.14~0.16 breaking, ka ~>0.29 three dimensional 卓越計畫--斜坡上非線性波浪傳動特性之研究

    40. Instrumentation EXP. III(1/2) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    41. Instrumentation EXP. III(2/2) 卓越計畫--斜坡上非線性波浪傳動特性之研究

    42. test conditions were determined mainly based on Initial growth rate of the sideband disturbance, based on Krasitskii’s reduced four-wave interaction (Tulin & Washeda 1999). 卓越計畫--斜坡上非線性波浪傳動特性之研究

    43. Data Analysis EXP. III • 1. following that of EXP. II • 2. the bi-spectrum analysis 卓越計畫--斜坡上非線性波浪傳動特性之研究

    44. Spatial evolution of wave profiles(EXP. III, case 5) kx/2=4.6 kx/2=5.4 kx/2=5.6 kx/2=5.9 kx/2=7.8 卓越計畫--斜坡上非線性波浪傳動特性之研究

    45. Spatial evolution of wave profiles(EXP. III, case 5) kx/2=9.6 kx/2=11.5 kx/2=13.4 kx/2=15.2 kx/2=17.7 卓越計畫--斜坡上非線性波浪傳動特性之研究

    46. Spatial evolution of wave profiles(EXP. III, case 5) kx/2=19 kx/2=20.9 kx/2=22.7 kx/2=24.6 kx/2=26.5 卓越計畫--斜坡上非線性波浪傳動特性之研究

    47. Spatial evolution of wave profiles(EXP. III, case 5) kx/2=28.4 kx/2=30.2 kx/2=31.2 kx/2=32.1 kx/2=33 卓越計畫--斜坡上非線性波浪傳動特性之研究

    48. Spatial evolution of wave profiles(EXP. III, case 5) kx/2=34 kx/2=34.9 kx/2=35.9 kx/2=36.8 kx/2=37.7 卓越計畫--斜坡上非線性波浪傳動特性之研究

    49. Spatial evolution of wave profiles(EXP. III, case 5) kx/2=62.1 kx/2=63 kx/2=64 kx/2=64.9 kx/2=65.8 卓越計畫--斜坡上非線性波浪傳動特性之研究

    50. Spatial evolution of wave profiles(EXP. III, case 5) kx/2=66.8 kx/2=67.7 kx/2=68.6 kx/2=69.6 kx/2=70.5 卓越計畫--斜坡上非線性波浪傳動特性之研究