DOCSLIB.ORG

Third-Order Stokes Wave Solutions of the Free Surface Capillary- Gravity Wave and the Interfacial Internal Wave

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Third-Order Stokes Wave Solutions of the Free Surface Capillary- Gravity Wave and the Interfacial Internal Wave China Ocean Eng., 2017, Vol. 31, No. 6, P. 781–787 DOI: 10.1007/s13344-017-0089-z, ISSN 0890-5487 http://www.chinaoceanengin.cn/ E-mail: [email protected] Third-Order Stokes Wave Solutions of the Free Surface Capillary- Gravity Wave and the Interfacial Internal Wave MENG Rui-jun†, CUI Ji-feng†, CHEN Xiao-gang*, ZHANG Bao-le, ZHANG Hong-bo College of Science, Inner Mongolia University of Technology, Hohhot 010051, China Received September 7, 2016; revised June 21, 2017; accepted August 10, 2017 ©2017 Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature Abstract Based on the Stokes wave theory, the capillary-gravity wave and the interfacial internal wave in two-layer constant depth’s fluid system are investigated. The fluids are assumed to be incompressible, inviscid and irrotational. The third-order Stokes wave solutions are given by using a perturbation method. The results indicate that the third-order solutions depend on the surface tension, the density and the depth of each layer. As expected, the first-order solutions are the linear theoretical results (the small amplitude wave theoretical results). The second-order and the third-order solutions describe the nonlinear modification and the nonlinear interactions. The nonlinear impact appears not only in the n (n≥2) times’ high frequency components, but also in the low frequency components. It is also noted that the wave velocity depends on the wave number, depth, wave amplitude and surface tension. Key words: surface tension, capillary-gravity waves, Stokes wave theory, third-order Stokes wave solutions Citation: Meng, R. J., Cui, J. F., Chen, X. G., Zhang, B. L., Zhang, H. B., 2017. Third-order Stokes wave solutions of the free surface capillary- gravity wave and the interfacial internal wave. China Ocean Eng., 31(6): 781–787, doi: 10.1007/s13344-017-0089-z 1 Introduction (2009), using elliptic integral of the first kind and so on. For In recent years, people have paid more attention to the internal waves, Stokes (1847) established a two-layer ocean importance of interfacial internal waves and capillary grav- internal wave theory; Benjamin (1966, 1967), Davis and ity waves. Firstly, investigation on them is very important Acrivos (1967), Ono (1975), Joseph (1977), Kubota et al. not only to understand the phenomena related to themselves (1978), Choi and Camassa (1996) derived the general evolu- but also to find out the knowledge about the interactions tion equations of two dimensional weakly nonlinear intern- between internal waves and many other multi-scale ocean al waves in two-fluid system. Song (2004) and Song and Sun waves and so on. Secondly, the remote sensing technology (2006) derived the second-order solutions for the random has become an effective means of the observation on sea interfacial waves in a two-layer fluid system, respectively. conditions. Owing to the Bragg scattering mechanisms, ca- Chen et al. (2005) studied the second-order Stokes solu- pillary waves have a significant effect on the sea microwave tions for internal waves in three-layer density-stratified fluid. (Wilton, 1915; Hogan, 1979, 1980, 1981; Maxworthy, In this paper, we investigate the capillary-gravity wave 1979; Song and Li, 1989, Cummins et al., 2003; Duda, and the interfacial internal wave in two-layer constant dep- 2004). th’s fluid system. Capillary-gravity waves and internal waves have very interesting properties. For example, for capillary-gravity 2 Basic equation and boundary conditions waves, the evidence of multiple solutions was first shown The interfacial waves propagating at the interface of two-layer fluid system are shown in Fig. 1. by Harrison (1909) and Wilton (1915) who included sur- The upper surface of the system is assumed to be free and face tension in Stoker’s classical expansion for pure gravity with the fluid in each layer homogeneous, inviscid, incom- waves. It was later extended by Lamb (1932), Defant pressible, and irrotational. The velocity potential Φ(1) in the up- (1961), Umeyama (2000, 2002) and others. Steady periodic per and Φ(2) in the lower layer satisfy the Laplace equations: capillary-gravity waves were calculated by Aider and Debi- 2 (1) 2 (1) ane (2006) using Neutrino’s method. Small-amplitude capil- ∂ Φ ∂ Φ + = 0; z1 + η 6 z 6 ζ; (1) lary-gravity waves were considered by Ionescu-Kruse ∂x2 ∂z2 Foundation item: The project was financially supported by the Science Research Project of Inner Mongolia University of Technology, China (Grant No. ZD201613). *Corresponding author. E-mail: [email protected] †The authors contributed equally. 782 MENG Rui-jun et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 781–787 ∂2Φ(2) ∂2Φ(2) where k is the wave number, ω is the angular frequency, and + = 0; z 6 z 6 z + η; (2) ∂x2 ∂z2 2 1 H is the characteristics of amplitude. where the horizontal coordinate axis x is fixed on the undis- Substituting Eq. (9) into Eqs. (1)–(8), the dimensionless turbed free surface. The vertical axis z is positive upwards equations can then be obtained as follows (for simplicity, “′” (Fig. 1); t is time; h1 and h2 are the depth of the upper and is neglected in the dimensionless quantity): lower layers, respectively; η(x, t) is the vertical displace- ∂2Φ(1) ∂2Φ(1) ment of the density interface; and ζ(x, t) is the vertical dis- + = 0; z1 + εη 6 z 6 εζ; (10) ∂x2 ∂z2 placement of the free surface measured from z=z0. ∂2Φ(2) ∂2Φ(2) + = 0; z 6 z 6 z + εη; (11) ∂x2 ∂z2 2 1 ∂Φ(1) ∂ζ ∂ζ ∂Φ(1) = ω + ε ; z = εζ; (12) ∂z ∂t ∂x ∂x 8 2 ! ! 39 > (1) 6 (1) 2 (1) 27> < ∂Φ 1 6 ∂Φ ∂Φ 7= ρ >ζ + ω + ε46 + 57> 1 : ∂t 2 ∂x ∂z ; 2 ! 3− 3 ∂2ζ 6 ∂ζ 27 2 = Γ 461 + ε 57 ; z = εζ; (13) ∂x2 ∂x Fig. 1. Sketch map of the two-layered density-stratified fluid system. ∂Φ(2) The boundary conditions at the free surface and the = 0; z = z ; (14) ∂z 2 density interface are 8 2 ! ! 39 (1) (1) > (1) 6 (1) 2 (1) 27> ∂Φ ∂η ∂η ∂Φ <∂Φ 1 6 ∂Φ ∂Φ 7= = ω + ε ; z = z1 + εη; (15) ρ > + gζ + 46 + 57> ∂z ∂t ∂x ∂x 1 : ∂t 2 ∂x ∂z ; (2) (2) 2 ! 3− 3 ∂Φ ∂η ∂η ∂Φ 2 2 = + ; = + ∂2ζ 6 ∂ζ 7 ω ε z z1 εη; (16) − Γ 461 + 57 = 0; z = ζ; (3) ∂z ∂t ∂x ∂x ∂x2 ∂x 8 2 ! ! 39 > (2) 6 (2) 2 (2) 27> < ∂Φ 1 6 ∂Φ ∂Φ 7= ∂Φ(1) ∂ζ ∂ζ ∂Φ(1) ρ >η + ω + ε46 + 57> = + ; z = ζ; (4) 2 : ∂t 2 ∂x ∂z ; ∂z ∂t ∂x ∂x 8 2 ! ! 39 <> (1) 6 (1) 2 (1) 27=> (1) (1) = + ∂Φ + 1 6 ∂Φ + ∂Φ 7 ; = + ; ∂Φ ∂η ∂η ∂Φ ρ1 >η ω ε4 5> z z1 εη = + ; z = η + z1; (5) : ∂t 2 ∂x ∂z ; ∂z ∂t ∂x ∂x ∂Φ(2) ∂η ∂η ∂Φ(2) (17) = + ; z = η + z ; (6) ∂z ∂t ∂x ∂x 1 kH 8 2 ! ! 39 where ε = ≪ 1 (H ≪ L, H is the characteristics of wave > (2) 6 (2) 2 (2) 27> 2 < ∂Φ 1 6 ∂Φ ∂Φ 7= ρ >gη + + 46 + 57> height, and L is the characteristics of wavelength). 2 : ∂t 2 ∂x ∂z ; 8 2 ! ! 39 > (1) 6 (1) 2 (1) 27> 3 Third-order Stokes-wave solutions with expansion < ∂Φ 1 6 ∂Φ ∂Φ 7= = ρ >gη+ + 46 + 57>; z = η + z1; (7) technology 1 : ∂t 2 ∂x ∂z ; Φ(i) (i=1, 2), ζ, η and ω in power series of an ordering where g is the acceleration of gravity, and Γ is the surface parameter ε (similar to Song and Sun (2006) and Umeyama tension which is a constant. (2000)) are expanded: The boundary condition at the bottom is n o X3 n o ( ) (2) (i); ; ; = (i); ; ; n−1 + 3 ; ∂Φ Φ ζ η ω Φn ζn ηn ωn ε O ε (18) = 0; z = z2: (8) ∂z n=1 Dimensionless quantity is introduced by choosing x′ as where O is the order symbol, and subscripts 1, 2 and 3 de- the unit horizontal length, z′ as the unit vertical length and t′ note quantities corresponding to the first-order, second-or- as the unit time. Now we define: der, and third-order perturbation solutions. 8 0 = ; 0 = ; 0 = ; > x kx z kz t sωt By substituting Eq. (18) into Eqs. (10)–(17), the govern- > (i) > 0 ing equations for each Φ , ζ , η , ω (i=1, 2, 3) are ob- <> (i) 0 0 0 2 k (i) 0 n n n n Φ (x ;z ;t ) = Φ (x;z;t); hi = khi(i = 1;2); > H g (9) tained by equating power of ε as follows. > > 0 0 0 2 0 0 0 2 The first-order equations and the boundary conditions : ζ (x ;t ) = ζ(x;t); η (x ;t ) = η(x;t): H H are MENG Rui-jun et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 781–787 783 (1) (1) (2) (2) ∂2Φ ∂2Φ ∂Φ ∂2Φ ∂η ∂η 1 + 1 = 0; z 6 z 6 ζ; (19) 2 + η 1 = ω 2 + ω 1 ∂x2 ∂z2 1 ∂z 1 ∂z2 1 ∂t 2 ∂t (2) 2 (2) 2 (2) ∂Φ ∂ Φ ∂ Φ + ∂η1 1 ; = 1 + 1 = 0; z 6 z 6 z ; (20) z z1; (33) ∂x2 ∂z2 2 1 ∂x ∂x (1) 8 ∂Φ > (2) (2) 2 (2) 1 = ∂ζ1 ; = <> ∂Φ ∂Φ ∂ Φ ω1 z 0; (21) > + 2 + 1 + 1 ∂z ∂t ρ2 :>η2 ω1 ω2 ω1η1 2 3 ∂t ∂t ∂t∂z 6 (1) 7 2 20 1 0 1 39 6 ∂Φ 7 ∂ ζ 6 (2) 2 (2) 27> 6 + 1 7 = 1 ; = 6B∂Φ C B∂Φ C 7> ρ1 46ζ1 ω1 57 Γ z 0; (22) 1 6B 1 C B 1 C 7= ∂t ∂x2 + 6@B AC + @B AC 7> 2 4 ∂x ∂z 5;> (2) 8 > (1) (1) 2 (1) ∂Φ1 <> ∂Φ ∂Φ ∂ Φ = 0; z = z ; (23) 2 1 1 2 = ρ >η + ω1 + ω2 + ω1η ∂z 1 : 2 ∂t ∂t 1 ∂t∂z (1) 20 1 0 1 39 ∂Φ 6 (1) 2 (1) 27> 1 = ∂η1 ; = 6B∂Φ C B∂Φ C 7=> ω1 z z1; (24) 1 6B 1 C B 1 C 7 ∂z ∂t + 6@B AC + @B AC 7>; z = z1: (34) 2 4 ∂x ∂z 5;> (2) ∂Φ ∂η 1 = ω 1 ; z = z ; (25) The third-order equations and boundary conditions are ∂z 1 ∂t 1 2 3 2 3 ∂2Φ(1) ∂2Φ(1) 6 ∂Φ(2) 7 6 ∂Φ(1) 7 3 + 3 = 0; z 6 z 6 z ; (35) 6 1 7 6 1 7 2 2 1 0 ρ 46η + ω1 57 = ρ 46η + ω1 57; z = z1: (26) ∂x ∂z 2 1 ∂t 1 1 ∂t ∂2Φ(1) ∂2Φ(1) The second-order equations and boundary conditions are 3 + 3 = 0; z 6 z 6 z ; (36) 2 2 2 1 2 (1) 2 (1) ∂x ∂z ∂ Φ2 ∂ Φ2 + = 0; z1 6 z 6 z0; (27) (1) (1) (1) (1) ∂x2 ∂z2 ∂Φ ∂2Φ ∂2Φ 1 ∂3Φ 3 + ζ 2 + ζ 1 + ζ2 1 1 2 2 2 1 3 ∂2Φ(2) ∂2Φ(2) ∂z ∂z ∂z 2 ∂z 2 + 2 = 0; z 6 z 6 z ; (28) (1) 2 2 2 1 ∂ζ ∂ζ ∂ζ ∂ζ ∂Φ ∂x ∂z = ω I + ω 2 + ω 3 + 1 2 3 ∂t 2 ∂t 1 ∂t ∂x ∂x (1) 2 (1) ∂Φ ∂ Φ (1) (1) 2 + 1 = ∂ζ2 ∂ζ ∂Φ ∂ζ ∂2Φ ζ1 ω1 + 2 1 + 1 1 ; = z 2 t ζ1 z 0; (37) ∂ ∂z ∂ ∂x ∂x ∂x ∂x∂z (1) ∂ζ1 ∂ζ1 ∂Φ1 + ω2 + ; z = 0; (29) 2 0 1 ∂t ∂x ∂x 6 ∂Φ(1) B ∂2Φ(1) ∂2Φ(1) ∂3Φ(1) C 8
Load more

Recommended publications
  • Accurate Modelling of Uni-Directional Surface Waves
    Journal of Computational and Applied Mathematics 234 (2010) 1747–1756 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Accurate modelling of uni-directional surface waves E. van Groesen a,b,∗, Andonowati a,b,c, L. She Liam a, I. Lakhturov a a Applied Mathematics, University of Twente, The Netherlands b LabMath-Indonesia, Bandung, Indonesia c Mathematics, Institut Teknologi Bandung, Indonesia article info a b s t r a c t Article history: This paper shows the use of consistent variational modelling to obtain and verify an Received 30 November 2007 accurate model for uni-directional surface water waves. Starting from Luke's variational Received in revised form 3 July 2008 principle for inviscid irrotational water flow, Zakharov's Hamiltonian formulation is derived to obtain a description in surface variables only. Keeping the exact dispersion Keywords: properties of infinitesimal waves, the kinetic energy is approximated. Invoking a uni- AB-equation directionalization constraint leads to the recently proposed AB-equation, a KdV-type Surface waves of equation that is also valid on infinitely deep water, that is exact in dispersion for Variational modelling KdV-type of equation infinitesimal waves, and that is second order accurate in the wave height. The accuracy of the model is illustrated for two different cases. One concerns the formulation of steady periodic waves as relative equilibria; the resulting wave profiles and the speed are good approximations of Stokes waves, even for the Highest Stokes Wave on deep water. A second case shows simulations of severely distorting downstream running bi-chromatic wave groups; comparison with laboratory measurements show good agreement of propagation speeds and of wave and envelope distortions.
    [Show full text]
  • Arxiv:2002.03434V3 [Physics.Flu-Dyn] 25 Jul 2020
    APS/123-QED Modified Stokes drift due to surface waves and corrugated sea-floor interactions with and without a mean current Akanksha Gupta Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, U.P. 208016, India.∗ Anirban Guhay School of Science and Engineering, University of Dundee, Dundee DD1 4HN, UK. (Dated: July 28, 2020) arXiv:2002.03434v3 [physics.flu-dyn] 25 Jul 2020 1 Abstract In this paper, we show that Stokes drift may be significantly affected when an incident inter- mediate or shallow water surface wave travels over a corrugated sea-floor. The underlying mech- anism is Bragg resonance { reflected waves generated via nonlinear resonant interactions between an incident wave and a rippled bottom. We theoretically explain the fundamental effect of two counter-propagating Stokes waves on Stokes drift and then perform numerical simulations of Bragg resonance using High-order Spectral method. A monochromatic incident wave on interaction with a patch of bottom ripple yields a complex interference between the incident and reflected waves. When the velocity induced by the reflected waves exceeds that of the incident, particle trajectories reverse, leading to a backward drift. Lagrangian and Lagrangian-mean trajectories reveal that surface particles near the up-wave side of the patch are either trapped or reflected, implying that the rippled patch acts as a non-surface-invasive particle trap or reflector. On increasing the length and amplitude of the rippled patch; reflection, and thus the effectiveness of the patch, increases. The inclusion of realistic constant current shows noticeable differences between Lagrangian-mean trajectories with and without the rippled patch.
    [Show full text]
  • Part II-1 Water Wave Mechanics
    Chapter 1 EM 1110-2-1100 WATER WAVE MECHANICS (Part II) 1 August 2008 (Change 2) Table of Contents Page II-1-1. Introduction ............................................................II-1-1 II-1-2. Regular Waves .........................................................II-1-3 a. Introduction ...........................................................II-1-3 b. Definition of wave parameters .............................................II-1-4 c. Linear wave theory ......................................................II-1-5 (1) Introduction .......................................................II-1-5 (2) Wave celerity, length, and period.......................................II-1-6 (3) The sinusoidal wave profile...........................................II-1-9 (4) Some useful functions ...............................................II-1-9 (5) Local fluid velocities and accelerations .................................II-1-12 (6) Water particle displacements .........................................II-1-13 (7) Subsurface pressure ................................................II-1-21 (8) Group velocity ....................................................II-1-22 (9) Wave energy and power.............................................II-1-26 (10)Summary of linear wave theory.......................................II-1-29 d. Nonlinear wave theories .................................................II-1-30 (1) Introduction ......................................................II-1-30 (2) Stokes finite-amplitude wave theory ...................................II-1-32
    [Show full text]
  • Variational Principles for Water Waves from the Viewpoint of a Time-Dependent Moving Mesh
    Variational principles for water waves from the viewpoint of a time-dependent moving mesh Thomas J. Bridges & Neil M. Donaldson Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK May 21, 2010 Abstract The time-dependent motion of water waves with a parametrically-defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic PDEs. The aim is to study the effect of transformation on variational principles for water waves such as Luke’s Lagrangian formulation, Zakharov’s Hamiltonian formulation, and the Benjamin-Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian, and some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves. 1 Introduction In the theory of water waves, the fluid domain shape is changing with time. Therefore it is appealing to transform the moving domain to a fixed domain. This approach, first for steady waves and later for unsteady waves, has been widely used in the study of water waves. However, in all cases the transformation is either algebraic (explicit) or a conformal mapping. In this paper we look at the equations governing the motion of water waves from the viewpoint of an arbitrary time-dependent transformation of the form x(µ,ν,τ) (µ,ν,τ) y(µ,ν,τ) , (1.1) 7→ t(τ) where (µ,ν) are coordinates for a fixed time-independent rectangle D := (µ,ν) : 0 µ L and h ν 0 , (1.2) { ≤ ≤ − ≤ ≤ } with h, L given positive parameters.
    [Show full text]
  • Waves on Deep Water, I
    Lecture 14: Waves on deep water, I Lecturer: Harvey Segur. Write-up: Adrienne Traxler June 23, 2009 1 Introduction In this lecture we address the question of whether there are stable wave patterns that propagate with permanent (or nearly permanent) form on deep water. The primary tool for this investigation is the nonlinear Schr¨odinger equation (NLS). Below we sketch the derivation of the NLS for deep water waves, and review earlier work on the existence and stability of 1D surface patterns for these waves. The next lecture continues to more recent work on 2D surface patterns and the effect of adding small damping. 2 Derivation of NLS for deep water waves The nonlinear Schr¨odinger equation (NLS) describes the slow evolution of a train or packet of waves with the following assumptions: • the system is conservative (no dissipation) • then the waves are dispersive (wave speed depends on wavenumber) Now examine the subset of these waves with • only small or moderate amplitudes • traveling in nearly the same direction • with nearly the same frequency The derivation sketch follows the by now normal procedure of beginning with the water wave equations, identifying the limit of interest, rescaling the equations to better show that limit, then solving order-by-order. We begin by considering the case of only gravity waves (neglecting surface tension), in the deep water limit (kh → ∞). Here h is the distance between the equilibrium surface height and the (flat) bottom; a is the wave amplitude; η is the displacement of the water surface from the equilibrium level; and φ is the velocity potential, u = ∇φ.
    [Show full text]
  • 0123737354Fluidmechanics.Pdf
    Fluid Mechanics, Fourth Edition Founders of Modern Fluid Dynamics Ludwig Prandtl G. I. Taylor (1875–1953) (1886–1975) (Biographical sketches of Prandtl and Taylor are given in Appendix C.) Photograph of Ludwig Prandtl is reprinted with permission from the Annual Review of Fluid Mechanics, Vol. 19, Copyright 1987 by Annual Reviews www.AnnualReviews.org. Photograph of Geoffrey Ingram Taylor at age 69 in his laboratory reprinted with permission from the AIP Emilio Segre` Visual Archieves. Copyright, American Institute of Physics, 2000. Fluid Mechanics Fourth Edition Pijush K. Kundu Oceanographic Center Nova Southeastern University Dania, Florida Ira M. Cohen Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, Pennsylvania with contributions by P. S. Ayyaswamy and H. H. Hu AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400 Burlington, MA 01803, USA Elsevier, The Boulevard, Langford Lane Kidlington, Oxford, OX5 1GB, UK © 2008 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).
    [Show full text]
  • Answers of the Reviewers' Comments Reviewer #1 —- General
    Answers of the reviewers’ comments Reviewer #1 —- General Comments As a detailed assessment of a coupled high resolution wave-ocean modelling system’s sensitivities in an extreme case, this paper provides a useful addition to existing published evidence regarding coupled systems and is a valid extension of the work in Staneva et al, 2016. I would therefore recommend this paper for publication, but with some additions/corrections related to the points below. —- Specific Comments Section 2.1 Whilst this information may well be published in the authors’ previous papers, it would be useful to those reading this paper in isolation if some extra details on the update frequencies of atmosphere and river forcing data were provided. Authors: More information about the model setup, including a description of the open boundary forcing, atmospheric forcing and river runoff, has been included in Section 2.1. Additional references were also added. Section 2.2 This is an extreme case in shallow water, so please could the source term parameterizations for bottom friction and depth induced breaking dissipation that were used in the wave model be stated? Authors: Additional information about the parameterizations used in our model setup, including more references, was provided in Section 2.2. Section 2.3 I found the statements that "<u> is the sum of the Eulerian current and the Stokes drift" and "Thus the divergence of the radiation stress is the only (to second order) force related to waves in the momentum equations." somewhat contradictory. In the equations, Mellor (2011) has been followed correctly and I see the basic point about radiation stress being the difference between coupled and uncoupled systems, so just wondering if the authors can review the text in this section for clarity.
    [Show full text]
  • On the Transport of Energy in Water Waves
    On The Transport Of Energy in Water Waves Marshall P. Tulin Professor Emeritus, UCSB [email protected] Abstract Theory is developed and utilized for the calculation of the separate transport of kinetic, gravity potential, and surface tension energies within sinusoidal surface waves in water of arbitrary depth. In Section 1 it is shown that each of these three types of energy constituting the wave travel at different speeds, and that the group velocity, cg, is the energy weighted average of these speeds, depth and time averaged in the case of the kinetic energy. It is shown that the time averaged kinetic energy travels at every depth horizontally either with (deep water), or faster than the wave itself, and that the propagation of a sinusoidal wave is made possible by the vertical transport of kinetic energy to the free surface, where it provides the oscillating balance in surface energy just necessary to allow the propagation of the wave. The propagation speed along the surface of the gravity potential energy is null, while the surface tension energy travels forward along the wave surface everywhere at twice the wave velocity, c. The flux of kinetic energy when viewed traveling with a wave provides a pattern of steady flux lines which originate and end on the free surface after making vertical excursions into the wave, even to the bottom, and these are calculated. The pictures produced in this way provide immediate insight into the basic mechanisms of wave motion. In Section 2 the modulated gravity wave is considered in deep water and the balance of terms involved in the propagation of the energy in the wave group is determined; it is shown again that vertical transport of kinetic energy to the surface is fundamental in allowing the propagation of the modulation, and in determining the well known speed of the modulation envelope, cg.
    [Show full text]
  • NONLINEAR DISPERSIVE WAVE PROBLEMS Thesis by Jon
    NONLINEAR DISPERSIVE WAVE PROBLEMS Thesis by Jon Christian Luke In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1966 (Submitted April 4, 1966) -ii­ ACKNOWLEDGMENTS The author expresses his thanks to Professor G. B. Whitham, who suggested the problems treated in this dissertation, and has made available various preliminary calculations, preprints of papers, and lectur~ notes. In addition, Professor Whitham has given freely of his time for useful and enlightening discussions and has offered timely advice, encourageme.nt, and patient super­ vision. Others who have assisted the author by suggestions or con­ structive criticism are Mr. G. W. Bluman and Mr. R. Seliger of the California Institute of Technology and Professor J. Moser of New York University. The author gratefully acknowledges the National Science Foundation Graduate Fellowship that made possible his graduate education. J. C. L. -iii­ ABSTRACT The nonlinear partial differential equations for dispersive waves have special solutions representing uniform wavetrains. An expansion procedure is developed for slowly varying wavetrains, in which full nonl~nearity is retained but in which the scale of the nonuniformity introduces a small parameter. The first order results agree with the results that Whitham obtained by averaging methods. The perturbation method provides a detailed description and deeper understanding, as well as a consistent development to higher approximations. This method for treating partial differ­ ential equations is analogous to the "multiple time scale" meth­ ods for ordinary differential equations in nonlinear vibration theory. It may also be regarded as a generalization of geomet­ r i cal optics to nonlinear problems.
    [Show full text]
  • Method of Studying Modulation Effects of Wind and Swell Waves on Tidal and Seiche Oscillations
    Journal of Marine Science and Engineering Article Method of Studying Modulation Effects of Wind and Swell Waves on Tidal and Seiche Oscillations Grigory Ivanovich Dolgikh 1,2,* and Sergey Sergeevich Budrin 1,2,* 1 V.I. Il’ichev Pacific Oceanological Institute, Far Eastern Branch Russian Academy of Sciences, 690041 Vladivostok, Russia 2 Institute for Scientific Research of Aerospace Monitoring “AEROCOSMOS”, 105064 Moscow, Russia * Correspondence: [email protected] (G.I.D.); [email protected] (S.S.B.) Abstract: This paper describes a method for identifying modulation effects caused by the interaction of waves with different frequencies based on regression analysis. We present examples of its applica- tion on experimental data obtained using high-precision laser interference instruments. Using this method, we illustrate and describe the nonlinearity of the change in the period of wind waves that are associated with wave processes of lower frequencies—12- and 24-h tides and seiches. Based on data analysis, we present several basic types of modulation that are characteristic of the interaction of wind and swell waves on seiche oscillations, with the help of which we can explain some peculiarities of change in the process spectrum of these waves. Keywords: wind waves; swell; tides; seiches; remote probing; space monitoring; nonlinearity; modulation 1. Introduction Citation: Dolgikh, G.I.; Budrin, S.S. The phenomenon of modulation of short-period waves on long waves is currently Method of Studying Modulation widely used in the field of non-contact methods for sea surface monitoring. These processes Effects of Wind and Swell Waves on are mainly investigated during space monitoring by means of analyzing optical [1,2] and Tidal and Seiche Oscillations.
    [Show full text]
  • Quantification of Stokes Drift As a Mechanism for Surface Oil Advection in the Gulf of Mexico During the Deepwater Horizon Oil Spill Matthew Clark
    Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2015 Quantification of Stokes Drift as a Mechanism for Surface Oil Advection in the Gulf of Mexico during the Deepwater Horizon Oil Spill Matthew Clark Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES QUANTIFICATION OF STOKES DRIFT AS A MECHANISM FOR SURFACE OIL ADVECTION IN THE GULF OF MEXICO DURING THE DEEPWATER HORIZON OIL SPILL By MATTHEW CLARK A Thesis submitted to the Department of Earth, Ocean, and Atmospheric Science in partial fulfillment of the requirements for the degree of Master of Science 2015 Matthew Clark defended this thesis on June 30, 2015. The members of the supervisory committee were: Mark Bourassa Professor Co-Directing Thesis Eric Chassignet Professor Co-Directing Thesis Robert Hart Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the thesis has been approved in accordance with university requirements. ii ACKNOWLEDGMENTS Thanks to everyone who helped with this, especially Dr. Mark Bourassa, who gave me far more time and patience than I deserved. Thanks to COAPS in general for giving me the opportunity and direction for this. Thanks to Deep-C and GoMRI for most of the funding and NOAA for the rest. Thanks to my parents who did not like the idea of me moving so far away but also did not discourage me from doing so. iii TABLE OF CONTENTS List of Tables ...................................................................................................................................v List of Figures ...............................................................................................................................
    [Show full text]
  • Three-Wave Backward Optical Solitons Carlos Montes, Pierre Aschieri, Antonio Picozzi, C
    Three-Wave Backward Optical Solitons Carlos Montes, Pierre Aschieri, Antonio Picozzi, C. Durniak, Majid Taki To cite this version: Carlos Montes, Pierre Aschieri, Antonio Picozzi, C. Durniak, Majid Taki. Three-Wave Backward Optical Solitons. Without Bounds: A Scientific Canvas of Nonlinearity and Complex Dynamics, 2013, pp.405-434. 10.1007/978-3-642-34070-3_34. hal-00835664 HAL Id: hal-00835664 https://hal.archives-ouvertes.fr/hal-00835664 Submitted on 19 Jun 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Three-Wave Backward Optical Solitons 1, 1 2 3 3 Carlos Montes ∗, Pierre Aschieri , Antonio Picozzi , Céline Durniak , and Majid Taki 1CNRS, Laboratoire de Physique de la Matière Condensée, Université de Nice - Sophia Antipolis, F-06108 Nice Cedex 2, France 2CNRS, Institut Carnot de Bourgogne, F-21078 Dijon Cedex, France 3Laboratoire de Physique des Lasers, Atomes et Molecules, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d’Asq Cedex, France ∗Corresponding author: [email protected] Abstract Three-wave solitons backward propagating with respect to a pump wave are generated in nonlinear optical media through stimulated Brillouin scattering (SBS) in optical fibers or through the non-degenerate three-wave interaction in quadratic (χ(2)) nonlinear media.
    [Show full text]