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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.
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Keywords:
- self-resonance law /
- single wave train /
- deep-water capillary-gravity waves /
- Fourier-Stieltjes transform
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Newton I (translated by Wang K D) 2006 Mathematical Principles of Natural Philosophy (Beijing: Peking University Press) (in Chinese)
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
Huang H 2010 Acta Phys. Sin. 59 740
Google Scholar
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Google Scholar
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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}$ Figure 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. 9 193
Google Scholar
[3] Hasselmann K 1962 J. Fluid Mech. 12 481
Google Scholar
[4] Longuet-Higgins M S 1962 J. Fluid Mech. 12 321
Google Scholar
[5] Benney D J 1962 J. Fluid Mech. 14 577
Google Scholar
[6] Bretherton F B 1964 J. Fluid Mech. 20 457
Google Scholar
[7] Longuet-Higgins M S, Smith N D 1966 J. Fluid Mech. 25 417
Google Scholar
[8] McGoldrick L F, Phillips O M, Huang N E, Hodgson T H 1966 J. Fluid Mech. 25 437
Google Scholar
[9] Sun C, Jia S, Barsi C, Rica S, Picozzi A, Fleischer J W 2012 Nat. Phys. 8 470
Google 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 61
Google Scholar
[12] Davis G, Jamin T, Deleuze J, Joubaud S, Dauxois T 2020 Phys. Rev. Lett. 124 204502
Google Scholar
[13] Galtier S, Nazarenko S V 2017 Phys. Rev. Lett. 119 221101
Google 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 59
Google Scholar
[17] 黄虎 2013 物理学报 62 139201
Google Scholar
Huang H 2013 Acta Phys. Sin. 62 139201
Google Scholar
[18] Krasitskii V P 1994 J. Fluid Mech. 272 1
Google Scholar
[19] Dyachenko A I, Korotkevich A O, Zakharov V E 2004 Phys. Rev. Lett. 92 134501
Google Scholar
[20] Griffin A, Krstulovic G, L’vov V S, Nazarenko S 2022 Phys. Rev. Lett. 128 224501
Google Scholar
[21] Dias F, Kharif C 1999 Annu. Rev. Fluid Mech. 31 301
Google 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 244501
Google Scholar
[23] Aubourg Q, Mordant N 2015 Phys. Rev. Lett. 114 144501
Google Scholar
[24] Aubourg Q, Mordant N 2016 Phys. Rev. Fluids 1 023701
Google Scholar
[25] Madsen P A, Fuhrman D R 2006 J. Fluid Mech. 557 369
Google Scholar
[26] Madsen P A, Fuhrman D R 2012 J. Fluid Mech. 698 304
Google Scholar
[27] Hammack J L, Henderson D M 1993 Annu. Rev. Fluid Mech. 25 55
Google 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 140301
Google Scholar
Cui W, Yan Z Z, Mu R 2014 Acta Phys. Sin. 63 140301
Google 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 1048
Google Scholar
Mei F X 2003 Acta Phys. Sin. 52 1048
Google Scholar
[32] Zakharov V E 1968 J. Appl. Mech. Tech. Phys. 9 86
[33] McGoldrick L F 1965 J. Fluid Mech. 21 305
Google Scholar
[34] Krasitskii V P, Kozhelupova N G 1995 Oceanology 34 435
[35] Lin G B, Huang H 2019 China Ocean Eng. 33 734
Google 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 240501
Google Scholar
Ma Z Z, Yang Q C, Zhou R P 2021 Acta Phys. Sin. 70 240501
Google Scholar
[39] Yao L S 1999 J. Fluid Mech. 395 237
Google Scholar
[40] Hasselmann K 1963 J. Fluid Mech. 15 273
Google Scholar
[41] Wilton J R 1915 Phil. Mag. 29 688
Google 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. 96 191
Google 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 077102
Google Scholar
Ye P 2020 Acta Phys. Sin. 69 077102
Google Scholar
[46] Matsuno Y 1992 Phys. Rev. Lett. 69 609
Google Scholar
[47] 黄虎, 夏应波 2011 物理学报 60 044702
Google Scholar
Huang H, Xia Y B 2011 Acta Phys. Sin. 60 044702
Google Scholar
[48] 黄虎 2010 物理学报 59 740
Google Scholar
Huang H 2010 Acta Phys. Sin. 59 740
Google Scholar
[49] Artiles W, Nachbin A 2004 Phys. Rev. Lett. 93 234501
Google 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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