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海洋深水表面张力波-重力波的单波列第n阶自共振定律

黄虎 田泽冰

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海洋深水表面张力波-重力波的单波列第n阶自共振定律

黄虎, 田泽冰

The nth-order self-resonance law of single wave train for surface capillary-gravity waves in deep water

Huang Hu, Tian Ze-Bing
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  • 波-波共振机制, 无论在微观物质还是在宏观物质的能量传播、分布进程中都起着一种根本、突显的作用. 对于地球上最为广阔、直观的海洋表面波运动, 势必更加如此而可观可知. 那么, 可否从中提炼出一般的波-波共振规律? 尤其是最为特殊、简要的单波列共振法则. 为此, 依据Phillips开创现代水波动力学而提出特定4-波共振条件的经典一整套方式方法, 从基本的海洋深水表面张力波-重力波控制方程组出发, 运用Fourier-Stieltjes变换和摄动方法依次给出自由表面位移的Fourier分量的第一阶微分方程和愈来愈复杂却趋于完整的第二、三、四阶积分微分方程, 在一套自创、自明而又简洁的符号体系下依次求解这些方程而求得其单波列第一阶自由表面位移和第二、三、四阶非共振与共振自由表面位移的Fourier系数以及第二、三、四阶共振条件, 从而顺势推断出一般的单波列第$n$阶自共振定律. 这就完整揭示了海洋表面张力波-重力波之单波列共振动力学的丰富内涵, 有效扩展了海洋表面重力波之经典Phillips共振单波列解的适用范围, 为刻画海洋表面波之双波列、更多波列的单重、多重共振相互作用机制奠定了基石, 因而在全部波动领域的单波列共振规律的探寻上提供了一种典型范例.
    Wave-wave resonance mechanism plays a fundamental and prominent role in the process of energy transfer and distribution, whether it is in microscopic or macroscopic matter. For the most extensive and intuitive ocean surface wave motion on earth, it is bound to be even more so. Can we extract the general wave-wave resonance law from it, especially the most special and brief resonance law for single wave train? To this end, according to a set of classical methods proposed by Phillips for initiating modern water wave dynamics with the specific 4-wave resonance conditions, and starting from the basic governing equations of ocean deep-water surface capillary-gravity waves, the first-order differential equation, and the second-, third- and fourth-order integral differential ones, which are becoming more and more complex but tend to be complete, of the Fourier components of free surface displacement are respectively given by the Fourier-Stieltjes transformation and perturbation method. Under a set of symbol system, which is self-created, self-evident and concise, these equations are solved in turn to obtain the first-order free surface displacement of single wave train, the Fourier coefficients of the second-, third- and fourth-order non-resonant and resonant free surface displacements, and the second-, third- and fourth-order resonant conditions, thus leading to the general nth-order self-resonance law of single wave train. This completely reveals the rich connotation of single wave resonance dynamics of ocean surface capillary-gravity waves, effectively expands the application range of the classical single wave resonance solutions given by Phillips for ocean surface gravity waves, lays the foundation for depicting single and multiple resonance interaction mechanisms of double and multi-wave trains of ocean surface waves, and so provides a typical example for the exploration of single-wave resonance law in all wave fields.
      通信作者: 黄虎, hhuang@shu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11772180)和海洋工程国家重点实验室开放课题基金(批准号: 1503)资助的课题
      Corresponding author: Huang Hu, hhuang@shu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11772180) and the State Key Laboratory of Ocean Engineering of China (Grant No. 1503).
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    Dyachenko A I, Korotkevich A O, Zakharov V E 2004 Phys. Rev. Lett. 92 134501Google Scholar

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    Cazaubiel A, Mawet S, Darras A, Grosjean G, van Loon J J W A, Dorbolo S, Falcon E 2019 Phys. Rev. Lett. 123 244501Google Scholar

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    Aubourg Q, Mordant N 2015 Phys. Rev. Lett. 114 144501Google Scholar

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    崔巍, 闫在在, 木仁 2014 物理学报 63 140301Google Scholar

    Cui W, Yan Z Z, Mu R 2014 Acta Phys. Sin. 63 140301Google Scholar

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    Gowers T 主编 (齐民友 译)2014 普林斯顿数学指南 (北京: 科学出版社) 第333—334页

    Gowers T (translated by Qi M Y) 2014 The Princeton Companion to Mathematics (Beijing: Science Press) pp333–334 (in Chinese)

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    梅凤翔 2003 物理学报 52 1048Google Scholar

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    Zakharov V E 1968 J. Appl. Mech. Tech. Phys. 9 86

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    McGoldrick L F 1965 J. Fluid Mech. 21 305Google Scholar

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    Krasitskii V P, Kozhelupova N G 1995 Oceanology 34 435

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    Lin G B, Huang H 2019 China Ocean Eng. 33 734Google Scholar

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    老子 2014 老子 (北京: 中华书局) 第165页

    Lao Z 2014 Lao Zi (Beijing: Zhonghua Book Company) p165 (in Chinese)

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    Bender C M, Orszag S A 1978 Advanced Mathematical Methods for Scientists and Engineers (Berlin: Springer)

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    马召召, 杨庆超, 周瑞平 2021 物理学报 70 240501Google Scholar

    Ma Z Z, Yang Q C, Zhou R P 2021 Acta Phys. Sin. 70 240501Google Scholar

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    Yao L S 1999 J. Fluid Mech. 395 237Google Scholar

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    Hasselmann K 1963 J. Fluid Mech. 15 273Google Scholar

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    Wilton J R 1915 Phil. Mag. 29 688Google Scholar

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    牛顿 (王可迪 译) 2006 自然哲学之数学原理 (北京: 北京大学出版社)

    Newton I (translated by Wang K D) 2006 Mathematical Principles of Natural Philosophy (Beijing: Peking University Press) (in Chinese)

    [43]

    Yang C N, Mills R L 1954 The Phys. Rev. 96 191Google Scholar

    [44]

    Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry (Berlin: Springer )

    [45]

    叶鹏 2020 物理学报 69 077102Google Scholar

    Ye P 2020 Acta Phys. Sin. 69 077102Google Scholar

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    Matsuno Y 1992 Phys. Rev. Lett. 69 609Google Scholar

    [47]

    黄虎, 夏应波 2011 物理学报 60 044702Google Scholar

    Huang H, Xia Y B 2011 Acta Phys. Sin. 60 044702Google Scholar

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    黄虎 2010 物理学报 59 740Google Scholar

    Huang H 2010 Acta Phys. Sin. 59 740Google Scholar

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    Artiles W, Nachbin A 2004 Phys. Rev. Lett. 93 234501Google Scholar

    [50]

    Huang H 2009 Dynamics of Surface Waves in Coastal Waters: Wave-Current-Bottom Interactions (Beijing, Berlin: Higher Education Press, Springer)

  • 图 1  深水海洋表面张力波-重力波的单波自共振定律, ρ = 1000 kg/m3, $g = 9.81{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{m}}/{{\text{s}}^2}$, $T' = 0.074{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {{\text{N}}} /{{\text{m}}} $, $T = 7.4 \times $$ {10^{ - 5}}~{{{\text{m}}} ^3}/{{{\text{s}}} ^2}$

    Fig. 1.  Self-resonance law of one wave for ocean surface waves in deep water: $\rho = 1000\;{{\text{kg}}}/{{{\text{m}}} ^{3}}$ , $g = 9.81{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{m}}/{{\text{s}}^2}$ , $T' = $$ 0.074~{{\text{N}}} /{{\text{m}}}$, $T = 7.4 \times {10^{ - 5}}~{{{\text{m}}} ^3}/{{{\text{s}}} ^2}$.

  • [1]

    Whitham G B 1974 Linear and Nonlinear Waves (New York: Wiley) pp2–4

    [2]

    Phillips O M 1960 J. Fluid Mech. 9 193Google Scholar

    [3]

    Hasselmann K 1962 J. Fluid Mech. 12 481Google Scholar

    [4]

    Longuet-Higgins M S 1962 J. Fluid Mech. 12 321Google Scholar

    [5]

    Benney D J 1962 J. Fluid Mech. 14 577Google Scholar

    [6]

    Bretherton F B 1964 J. Fluid Mech. 20 457Google Scholar

    [7]

    Longuet-Higgins M S, Smith N D 1966 J. Fluid Mech. 25 417Google Scholar

    [8]

    McGoldrick L F, Phillips O M, Huang N E, Hodgson T H 1966 J. Fluid Mech. 25 437Google Scholar

    [9]

    Sun C, Jia S, Barsi C, Rica S, Picozzi A, Fleischer J W 2012 Nat. Phys. 8 470Google Scholar

    [10]

    Dyachenko S, Newell A C, Pushkarev A, Zakharov V E 1992 Phys. D 57 96

    [11]

    Nazarenko S, Lukaschuk S 2016 Annu. Rev. Condens. Matter. 7 61Google Scholar

    [12]

    Davis G, Jamin T, Deleuze J, Joubaud S, Dauxois T 2020 Phys. Rev. Lett. 124 204502Google Scholar

    [13]

    Galtier S, Nazarenko S V 2017 Phys. Rev. Lett. 119 221101Google Scholar

    [14]

    Zakharov V E, L’vov V S, Falkovich G 1992 Kolmogorov Spectra of Turbulence I: Wave Turbulence (Berlin: Springer-Verlag )

    [15]

    Nazarenko S 2011 Wave Turbulence (Berlin: Springer)

    [16]

    Newell A C, Rumpf B 2011 Annu. Rev. Fluid Mech. 43 59Google Scholar

    [17]

    黄虎 2013 物理学报 62 139201Google Scholar

    Huang H 2013 Acta Phys. Sin. 62 139201Google Scholar

    [18]

    Krasitskii V P 1994 J. Fluid Mech. 272 1Google Scholar

    [19]

    Dyachenko A I, Korotkevich A O, Zakharov V E 2004 Phys. Rev. Lett. 92 134501Google Scholar

    [20]

    Griffin A, Krstulovic G, L’vov V S, Nazarenko S 2022 Phys. Rev. Lett. 128 224501Google Scholar

    [21]

    Dias F, Kharif C 1999 Annu. Rev. Fluid Mech. 31 301Google Scholar

    [22]

    Cazaubiel A, Mawet S, Darras A, Grosjean G, van Loon J J W A, Dorbolo S, Falcon E 2019 Phys. Rev. Lett. 123 244501Google Scholar

    [23]

    Aubourg Q, Mordant N 2015 Phys. Rev. Lett. 114 144501Google Scholar

    [24]

    Aubourg Q, Mordant N 2016 Phys. Rev. Fluids 1 023701Google Scholar

    [25]

    Madsen P A, Fuhrman D R 2006 J. Fluid Mech. 557 369Google Scholar

    [26]

    Madsen P A, Fuhrman D R 2012 J. Fluid Mech. 698 304Google Scholar

    [27]

    Hammack J L, Henderson D M 1993 Annu. Rev. Fluid Mech. 25 55Google Scholar

    [28]

    Stokes G G 1847 Trans. Camb. Phil. Soc. 8 441

    [29]

    崔巍, 闫在在, 木仁 2014 物理学报 63 140301Google Scholar

    Cui W, Yan Z Z, Mu R 2014 Acta Phys. Sin. 63 140301Google Scholar

    [30]

    Gowers T 主编 (齐民友 译)2014 普林斯顿数学指南 (北京: 科学出版社) 第333—334页

    Gowers T (translated by Qi M Y) 2014 The Princeton Companion to Mathematics (Beijing: Science Press) pp333–334 (in Chinese)

    [31]

    梅凤翔 2003 物理学报 52 1048Google Scholar

    Mei F X 2003 Acta Phys. Sin. 52 1048Google Scholar

    [32]

    Zakharov V E 1968 J. Appl. Mech. Tech. Phys. 9 86

    [33]

    McGoldrick L F 1965 J. Fluid Mech. 21 305Google Scholar

    [34]

    Krasitskii V P, Kozhelupova N G 1995 Oceanology 34 435

    [35]

    Lin G B, Huang H 2019 China Ocean Eng. 33 734Google Scholar

    [36]

    老子 2014 老子 (北京: 中华书局) 第165页

    Lao Z 2014 Lao Zi (Beijing: Zhonghua Book Company) p165 (in Chinese)

    [37]

    Bender C M, Orszag S A 1978 Advanced Mathematical Methods for Scientists and Engineers (Berlin: Springer)

    [38]

    马召召, 杨庆超, 周瑞平 2021 物理学报 70 240501Google Scholar

    Ma Z Z, Yang Q C, Zhou R P 2021 Acta Phys. Sin. 70 240501Google Scholar

    [39]

    Yao L S 1999 J. Fluid Mech. 395 237Google Scholar

    [40]

    Hasselmann K 1963 J. Fluid Mech. 15 273Google Scholar

    [41]

    Wilton J R 1915 Phil. Mag. 29 688Google Scholar

    [42]

    牛顿 (王可迪 译) 2006 自然哲学之数学原理 (北京: 北京大学出版社)

    Newton I (translated by Wang K D) 2006 Mathematical Principles of Natural Philosophy (Beijing: Peking University Press) (in Chinese)

    [43]

    Yang C N, Mills R L 1954 The Phys. Rev. 96 191Google Scholar

    [44]

    Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry (Berlin: Springer )

    [45]

    叶鹏 2020 物理学报 69 077102Google Scholar

    Ye P 2020 Acta Phys. Sin. 69 077102Google Scholar

    [46]

    Matsuno Y 1992 Phys. Rev. Lett. 69 609Google Scholar

    [47]

    黄虎, 夏应波 2011 物理学报 60 044702Google Scholar

    Huang H, Xia Y B 2011 Acta Phys. Sin. 60 044702Google Scholar

    [48]

    黄虎 2010 物理学报 59 740Google Scholar

    Huang H 2010 Acta Phys. Sin. 59 740Google Scholar

    [49]

    Artiles W, Nachbin A 2004 Phys. Rev. Lett. 93 234501Google Scholar

    [50]

    Huang H 2009 Dynamics of Surface Waves in Coastal Waters: Wave-Current-Bottom Interactions (Beijing, Berlin: Higher Education Press, Springer)

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出版历程
  • 收稿日期:  2022-06-30
  • 修回日期:  2022-11-25
  • 上网日期:  2022-12-29
  • 刊出日期:  2023-03-05

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