Rogue Waves. Alex Andrade Mentor: Dr. Ildar Gabitov. Physical Mechanisms of the Rogue Wave Phenomenon Christian Kharif, Efim Pelinovsky
|
|
- Mae Rich
- 6 years ago
- Views:
Transcription
1 Rogue Waves Alex Andrade Mentor: Dr. Ildar Gabitov Physical Mechanisms of the Rogue Wave Phenomenon Christian Kharif, Efim Pelinovsky
2 Rogue Waves in History Rogue Waves have been part of marine folklore for centuries. Seafarers speak of walls of water, or of holes in the sea, which appears without warning in otherwise benign conditions.
3 Significant Wave Height Hs Significant Wave Height, Hs Is the average wave height (through to crest) of the one-third largest waves. It is commonly used as a measure of the height of ocean waves.
4 A Rogue Wave is not a Tsunami Tsunamis are a specific type of wave not caused by geological effects. In deep water, tsunamis are not visible because they are small in height and very long in wavelength. They may grow to devastating proportions at the coast due to reduced water depth.
5 Then, what is a Rogue Wave? Also called Freak or Giant Waves, Rogue Waves are waves whose height, Hf is more than twice the significant wave height, Hs: AI = H f H s > 2 AI= Abnormality Index Rogue Wave in the North Sea AI = 3.19, Hf = m Hf Hs
6 Why is important its study? a) Norwegian dreamer, 2005 c) Sinking of tanker Prestige in 2002 b) Norwegian tanker Wilstar, 1974 d) Sinking of the World Glory tanker in 1968.
7 Recognition of the phenomenon Kharif et.al., 2009 (b) The New Year Wave AI = 2.24, Hf = 26 m (c) A hole in the sea AI = 2.46, Hf = 9.3 m (d) A freak group AI = 2.23, Hf = 13.71
8 Possible physical mechanisms of Rogue Wave Generation 1. Linear mechanisms 1. Geometrical or Spatial Focusing 2. Wave-current Interaction 3. Focusing Due to Dispersion 2. Nonlinear mechanisms 1. Weakly nonlinear rogue wave packets in deep and intermediate depths
9 The Water Wave problem The water wave problem reduces to solve the system of equations: The difficulty in solving water wave problems arises from the nonlinearity of kinematic and dynamic boundary conditions. Where: =Laplace Operator; Φ=velocity potential; η=water surface elevation; g= gravity; h= Water depth, Z= position in the vertical axis.
10 Linear Mechanisms Linear theory is constructed on the assumptions: 1. ka<<1 (Wave steepness; an important measure in deep water) 2. a/h<<1 (Important measure in shallow water). With these assumptions, the nonlinear terms can be neglected and the corresponding system of equations to be solved is linear: Where: =Laplace Operator; Φ=velocity potential; η=water surface elevation; g= gravity; h= Water depth, Z= position in the vertical axis.
11 Geometrical Focusing of Water Waves Coast shape or seabed directs several small waves to meet in phase. Their crest heights combine to create a freak wave. The result is spatial variations of the kinematic and dynamic variables of the problem. Coast of Finnmark, Norway. 1976
12 Geometrical Focusing of Water Waves If the water depth, h=h(x), the shallow water wave is described by the ordinary differential equation: g d dx h(x) dη dx + ω 2 gh(x)k 2 η =0 in the vicinity of caustics, it has the form of the Airy equation d 2 η dx 2 k2 L xη =0 And its solution is described by the Airy function η(x) =const Ai( xk2/3 L 1/3 ) Where: g= gravity; h= Water depth; η=water surface elevation, x=distance, ω=wave frequency, k=wave number; Ai()=Airy function.
13 Extreme waves often occur in areas where waves propagate into a strong opposing current. The first theoretical models of the freak wave phenomenon considered wave current interaction. Wave-Current Interaction Generalizing the Airy function used for the Geometrical Focusing of Water Waves: η(x) =const Ai ( 8 U/ x Ω(k ) ) 1 3 k (x x 0 ) exp(ik ωt) Where: U= velocity of the current; Ω=Wave frequency; η=water surface elevation, x0=position of the blocking point, ω=wave frequency, k*=wave number at the blocking point; Ai()=Airy function; t=time.
14 Dispersion enhancement of transient wave groups Waves with similar frequency will group together and separate from other wave groups. This process of self-sorting is called dispersion. Trains of waves traveling in the same direction but at different speeds pass through one another. When they synchronize, they combine to form large waves.
15 Dispersion enhancement of transient wave groups The wave amplitude satisfies the energy balance equation A 2 t and its solution is found explicitly, A(x, t) = + x (c gra 2 )=0 A 0 (x c gr t) 1+t(dc0 /d(x c gr t)) At each focal point, the wave becomes extreme, having infinite amplitude A 1 Tf t Kinematic approach assumes slow variations of the amplitude and frequency along the wave group, and this approximation is not valid in the vicinity of the focal points. Where: A=Amplitude; A0=Initial amplitude; Cgr=Velocity of the group; c0=initial velocity; x0=position of the blocking point, Tt= Focusing time.
16 Dispersion enhancement of transient wave groups Generalizations of the kinematic approach in linear theory can be done by using various expressions of the Fourier integral for the wave field near the caustics. η(x, t) = + η(k)exp(i(kx ωt))dk This integral can be calculated for smooth freak waves (initial data), for instance for a Gaussian pulse with amplitude, A0, in the long wave approximation η(x, t) = A 0 k 3 h 2 ct 2 exp 1 2h 2 ctk 2 x ct h 2 ctk 4 Ai x ct h 2 ctk 4 h 2 ct 2 This equation model the freak wave formation in a dispersive wave packet on shallow water. η(x,t)=water displacement; A0= Initial wave amplitude; k= wavenumber (spatial frequency of the wave in radians per unit distance); h= Water depth; c= Phase velocity; x= distance; t= time; Ai= Airy function
17 Dispersion enhancement
18 Nonlinear Mechanisms When wave amplitude increases beyond a certain range, the linear wave theory may become inadequate. The reason is that those higher order terms that have been neglected in the derivation become increasingly important as wave amplitude increases. The linear theory assumptions are no longer valid 1.ka<<1 (Wave steepness; an important measure in deep water) 2.a/h<<1 (Important measure in shallow water).
19 Weakly nonlinear rogue wave packets in deep i and intermediate depths Simplified nonlinear model of 2D quasi-periodic deep-water wave trains is based on the nonlinear Schrödinger equation: A t + c gr A x = ω 0 8k 2 0 where the surface elevation is given by 2 A x 2 + ω 0k0 2 2 A 2 A η(x, t) = 1 2 (A(x, t)exp(i(k 0x w 0 t)) + c.c + ) One solution to the nonlinear Schrödinger equation corresponds to the so-called algebraic breather 4(1 + 2iω 0 t) A(x, t) =A 0 exp(iω 0 t) k 2 0 x 2 +4ω 02 t 2 Where: A=Amplitude; A0=Initial amplitude; k0= Wavenumber; ω0=frequency of the carrier wave; h= Water depth; cgr= Group phase velocity; c.c=complex conjugate; (...)=weak highest armonics of the carrier wave.
20 Weakly nonlinear rogue wave packets in deep and intermediate depths (Algebraic breather graph)
21 Future Work Weakly nonlinear rogue waves in shallow water. Conclusions 1. Precise physical mechanisms causing the rogue waves phenomenon are still unknown. 2. Rogue Waves should be considered to reduce the number of ships sinked worldwide. 3. Main physical mechanisms believed to produce rogue waves were presented. 4. The dispersion enhancement mechanisms has been used to create Rogue Waves in lab conditions.
22 References 1. A. S. Christin Kharif, Efim Pelinovsky. Rogue Waves in the Ocean. Springer Berlin Heidelberg, C. Kharif and E. Pelinovsky. Physical mechanisms of the rogue wave phenomenon. European Journal of Mechanics - B/Fluids, 22(6): , Last checked: 02/18/ Last checked: 04/7/2010
Physical mechanisms of the Rogue Wave phenomenon
Physical mechanisms of the Rogue Wave phenomenon Manuel A. Andrade. Mentor: Dr. Ildar Gabitov. Math 585. 1 "We were in a storm and the tanker was running before the sea. This amazing wave came from the
More informationPhysical mechanisms of the Rogue Wave phenomenon
Physical mechanisms of the Rogue Wave phenomenon Final Report Manuel A. Andrade. Mentor: Dr. Ildar Gabitov. Math 585. 1 "We were in a storm and the tanker was running before the sea. This amazing wave
More informationExperimental study of the wind effect on the focusing of transient wave groups
Experimental study of the wind effect on the focusing of transient wave groups J.P. Giovanangeli 1), C. Kharif 1) and E. Pelinovsky 1,) 1) Institut de Recherche sur les Phénomènes Hors Equilibre, Laboratoire
More informationPhysical mechanisms of the rogue wave phenomenon
European Journal of Mechanics B/Fluids 22 (2003) 603 634 Physical mechanisms of the rogue wave phenomenon Christian Kharif a,, Efim Pelinovsky b a Institut de recherche sur les phenomenes hors equilibre
More informationRogue Waves. Thama Duba, Colin Please, Graeme Hocking, Kendall Born, Meghan Kennealy. 18 January /25
1/25 Rogue Waves Thama Duba, Colin Please, Graeme Hocking, Kendall Born, Meghan Kennealy 18 January 2019 2/25 What is a rogue wave Mechanisms causing rogue waves Where rogue waves have been reported Modelling
More informationOn deviations from Gaussian statistics for surface gravity waves
On deviations from Gaussian statistics for surface gravity waves M. Onorato, A. R. Osborne, and M. Serio Dip. Fisica Generale, Università di Torino, Torino - 10125 - Italy Abstract. Here we discuss some
More informationStatistical properties of mechanically generated surface gravity waves: a laboratory experiment in a 3D wave basin
Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a 3D wave basin M. Onorato 1, L. Cavaleri 2, O.Gramstad 3, P.A.E.M. Janssen 4, J. Monbaliu 5, A. R. Osborne
More informationMarine Hydrodynamics Lecture 19. Exact (nonlinear) governing equations for surface gravity waves assuming potential theory
13.021 Marine Hydrodynamics, Fall 2004 Lecture 19 Copyright c 2004 MIT - Department of Ocean Engineering, All rights reserved. Water Waves 13.021 - Marine Hydrodynamics Lecture 19 Exact (nonlinear) governing
More informationMarine Hydrodynamics Lecture 19. Exact (nonlinear) governing equations for surface gravity waves assuming potential theory
13.021 Marine Hydrodynamics Lecture 19 Copyright c 2001 MIT - Department of Ocean Engineering, All rights reserved. 13.021 - Marine Hydrodynamics Lecture 19 Water Waves Exact (nonlinear) governing equations
More informationFundamentals of Fluid Dynamics: Waves in Fluids
Fundamentals of Fluid Dynamics: Waves in Fluids Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/ tzielins/ Institute
More information13.42 LECTURE 2: REVIEW OF LINEAR WAVES
13.42 LECTURE 2: REVIEW OF LINEAR WAVES SPRING 2003 c A.H. TECHET & M.S. TRIANTAFYLLOU 1. Basic Water Waves Laplace Equation 2 φ = 0 Free surface elevation: z = η(x, t) No vertical velocity at the bottom
More information2. Theory of Small Amplitude Waves
. Theory of Small Amplitude Waves.1 General Discussion on Waves et us consider a one-dimensional (on -ais) propagating wave that retains its original shape. Assume that the wave can be epressed as a function
More informationRogue waves in large-scale fully-non-linear High-Order-Spectral simulations
Rogue waves in large-scale fully-non-linear High-Order-Spectral simulations Guillaume Ducrozet, Félicien Bonnefoy & Pierre Ferrant Laboratoire de Mécanique des Fluides - UMR CNRS 6598 École Centrale de
More informationLecture 11: Internal solitary waves in the ocean
Lecture 11: Internal solitary waves in the ocean Lecturer: Roger Grimshaw. Write-up: Yiping Ma. June 19, 2009 1 Introduction In Lecture 6, we sketched a derivation of the KdV equation applicable to internal
More informationLinear and Nonlinear Rogue Wave Statistics in the Presence of Random Currents
Linear and Nonlinear Rogue Wave Statistics in the Presence of Random Currents Lev Kaplan (Tulane University) In collaboration with Alex Dahlen and Eric Heller (Harvard) Linghang Ying and Zhouheng Zhuang
More informationDispersion relations, stability and linearization
Dispersion relations, stability and linearization 1 Dispersion relations Suppose that u(x, t) is a function with domain { < x 0}, and it satisfies a linear, constant coefficient partial differential
More informationExperiments on extreme wave generation using the Soliton on Finite Background
Experiments on extreme wave generation using the Soliton on Finite Background René H.M. Huijsmans 1, Gert Klopman 2,3, Natanael Karjanto 3, and Andonawati 4 1 Maritime Research Institute Netherlands, Wageningen,
More informationFigure 1: Surface waves
4 Surface Waves on Liquids 1 4 Surface Waves on Liquids 4.1 Introduction We consider waves on the surface of liquids, e.g. waves on the sea or a lake or a river. These can be generated by the wind, by
More informationThe role of resonant wave interactions in the evolution of extreme wave events
The role of resonant wave interactions in the evolution of extreme wave events Richard Gibson & Chris Swan Department of Civil and Environmental Engineering Imperial College London SW7 2AZ United Kingdom
More informationModel Equation, Stability and Dynamics for Wavepacket Solitary Waves
p. 1/1 Model Equation, Stability and Dynamics for Wavepacket Solitary Waves Paul Milewski Mathematics, UW-Madison Collaborator: Ben Akers, PhD student p. 2/1 Summary Localized solitary waves exist in the
More informationTsunamis and ocean waves
Department of Mathematics & Statistics AAAS Annual Meeting St. Louis Missouri February 19, 2006 Introduction Tsunami waves are generated relatively often, from various sources Serious tsunamis (serious
More informationIntroduction to Marine Hydrodynamics
1896 190 1987 006 Introduction to Marine Hydrodynamics (NA35) Department of Naval Architecture and Ocean Engineering School of Naval Architecture, Ocean & Civil Engineering Shanghai Jiao Tong University
More informationLecture 1: Introduction to Linear and Non-Linear Waves
Lecture 1: Introduction to Linear and Non-Linear Waves Lecturer: Harvey Segur. Write-up: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More informationA short tutorial on optical rogue waves
A short tutorial on optical rogue waves John M Dudley Institut FEMTO-ST CNRS-Université de Franche-Comté Besançon, France Experiments in collaboration with the group of Guy Millot Institut Carnot de Bourgogne
More informationExtreme waves, modulational instability and second order theory: wave flume experiments on irregular waves
European Journal of Mechanics B/Fluids 25 (2006) 586 601 Extreme waves, modulational instability and second order theory: wave flume experiments on irregular waves M. Onorato a,, A.R. Osborne a,m.serio
More informationNumerical Simulation of Water Waves Modulational Instability under the Effects of Wind s Stress and Gravity Force Relaxation
Open Journal of Marine Science, 016, 6, 93-10 Published Online January 016 in SciRes. http://www.scirp.org/journal/ojms http://dx.doi.org/10.436/ojms.016.61009 Numerical Simulation of Water Waves Modulational
More informationChapter 4 Water Waves
2.20 Marine Hdrodnamics, Fall 2018 Lecture 15 Copright c 2018 MIT - Department of Mechanical Engineering, All rights reserved. Chapter 4 Water Waves 2.20 - Marine Hdrodnamics Lecture 15 4.1 Exact (Nonlinear)
More informationThree Dimensional Simulations of Tsunami Generation and Propagation
Chapter 1 Earth Science Three Dimensional Simulations of Tsunami Generation and Propagation Project Representative Takashi Furumura Authors Tatsuhiko Saito Takashi Furumura Earthquake Research Institute,
More informationOccurrence of Freak Waves from Envelope Equations in Random Ocean Wave Simulations
Occurrence of Freak Waves from Envelope Equations in Random Ocean Wave Simulations Miguel Onorato, Alfred R. Osborne, Marina Serio, and Tomaso Damiani Universitá di Torino, Via P. Giuria, - 025, Torino,
More informationNonlinear mechanism of tsunami wave generation by atmospheric disturbances
Natural Hazards and Earth System Sciences (001) 1: 43 50 c European Geophysical Society 001 Natural Hazards and Earth System Sciences Nonlinear mechanism of tsunami wave generation by atmospheric disturbances
More informationSAMPLE CHAPTERS UNESCO EOLSS WAVES IN THE OCEANS. Wolfgang Fennel Institut für Ostseeforschung Warnemünde (IOW) an der Universität Rostock,Germany
WAVES IN THE OCEANS Wolfgang Fennel Institut für Ostseeforschung Warnemünde (IOW) an der Universität Rostock,Germany Keywords: Wind waves, dispersion, internal waves, inertial oscillations, inertial waves,
More informationNonlinear Wave Theory
Nonlinear Wave Theory Weakly Nonlinear Wave Theory (WNWT): Stokes Expansion, aka Mode Coupling Method (MCM) 1) Only applied in deep or intermediate depth water ) When truncated at a relatively high order,
More informationModeling and predicting rogue waves in deep water
Modeling and predicting rogue waves in deep water C M Schober University of Central Florida, Orlando, Florida - USA Abstract We investigate rogue waves in the framework of the nonlinear Schrödinger (NLS)
More informationLecture 7: Oceanographic Applications.
Lecture 7: Oceanographic Applications. Lecturer: Harvey Segur. Write-up: Daisuke Takagi June 18, 2009 1 Introduction Nonlinear waves can be studied by a number of models, which include the Korteweg de
More informationFreak Waves in the Ocean Victor S. Lvov, Weizmann Institute, Israel
Freak Waves in the Ocean Victor S. Lvov, Weizmann Institute, Israel Ships are disappearing all over the world s oceans at a rate of about one every week. These drownings often happen in mysterious circumstances.
More informationRogue Waves: Refraction of Gaussian Seas and Rare Event Statistics
Rogue Waves: Refraction of Gaussian Seas and Rare Event Statistics Eric J. Heller (Harvard University) Lev Kaplan (Tulane University) Aug. 15, 2006 Cuernavaca: Quantum Chaos (RMT) 1/27 Talk outline: Introduction:
More informationModulational instability in the presence of damping
Perspectives on Soliton Physics February 17, 2007 Modulational instability in the presence of damping Harvey Segur University of Colorado Joint work with: J. Hammack, D. Henderson, J. Carter, W. Craig,
More informationarxiv: v1 [nlin.cd] 21 Mar 2012
Approximate rogue wave solutions of the forced and damped Nonlinear Schrödinger equation for water waves arxiv:1203.4735v1 [nlin.cd] 21 Mar 2012 Miguel Onorato and Davide Proment Dipartimento di Fisica,
More informationEvolution of a nonlinear wave field along a tank: experiments and numerical simulations based on the spatial Zakharov equation
J. Fluid Mech. (), vol. 7, pp. 7 9. Printed in the United Kingdom c Cambridge University Press 7 Evolution of a nonlinear wave field along a tank: experiments and numerical simulations based on the spatial
More informationLong-time solutions of the Ostrovsky equation
Long-time solutions of the Ostrovsky equation Roger Grimshaw Centre for Nonlinear Mathematics and Applications, Department of Mathematical Sciences, Loughborough University, U.K. Karl Helfrich Woods Hole
More informationOCN660 - Ocean Waves. Course Purpose & Outline. Doug Luther. OCN660 - Syllabus. Instructor: x65875
OCN660 - Ocean Waves Course Purpose & Outline Instructor: Doug Luther dluther@hawaii.edu x65875 This introductory course has two objectives: to survey the principal types of linear ocean waves; and, to
More informationarxiv: v1 [physics.flu-dyn] 2 Sep 2016
Predictability of the Appearance of Anomalous Waves at Sufficiently Small Benjamin-Feir Indices V. P. Ruban Landau Institute for Theoretical Physics RAS, Moscow, Russia (Dated: October, 8) arxiv:9.v [physics.flu-dyn]
More informationA note on the shoaling of acoustic gravity waves
A note on the shoaling of acoustic gravity waves USAMA KADRI MICHAEL STIASSNIE Technion Israel Institute of Technology Faculty of Civil Environmental Engineering Technion 32, Haifa ISRAEL usamakadri@gmailcom
More informationTheoretical Seismology. Astrophysics and Cosmology and Earth and Environmental Physics. FTAN Analysis. Fabio ROMANELLI
Theoretical Seismology Astrophysics and Cosmology and Earth and Environmental Physics FTAN Analysis Fabio ROMANELLI Department of Mathematics & Geosciences University of Trieste romanel@units.it 1 FTAN
More informationFAST COMMUNICATION THREE-DIMENSIONAL LOCALIZED SOLITARY GRAVITY-CAPILLARY WAVES
COMM. MATH. SCI. Vol. 3, No., pp. 89 99 c 5 International Press FAST COMMUNICATION THREE-DIMENSIONAL LOCALIZED SOLITARY GRAVITY-CAPILLARY WAVES PAUL A. MILEWSKI Abstract. In a weakly nonlinear model equation
More informationv t + fu = 1 p y w t = 1 p z g u x + v y + w
1 For each of the waves that we will be talking about we need to know the governing equators for the waves. The linear equations of motion are used for many types of waves, ignoring the advective terms,
More informationAn Optimal Dimension of Submerged Parallel Bars as a Wave Reflector
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 32(1) (2009), 55 62 An Optimal Dimension of Submerged Parallel Bars as a Wave Reflector
More informationNumerical simulations of modulated waves in a higher-order Dysthe equation
Numerical simulations of modulated waves in a higher-order equation Alexey Slunyaev,) and Efim Pelinovsky -5) ) Institute of Applied Physics, Nizhny Novgorod, Russia ) Institute of Applied Physics, Nizhny
More informationChapter 9. Barotropic Instability. 9.1 Linearized governing equations
Chapter 9 Barotropic Instability The ossby wave is the building block of low ossby number geophysical fluid dynamics. In this chapter we learn how ossby waves can interact with each other to produce a
More informationThe effect of a background shear current on large amplitude internal solitary waves
The effect of a background shear current on large amplitude internal solitary waves Wooyoung Choi Dept. of Mathematical Sciences New Jersey Institute of Technology CAMS Report 0506-4, Fall 005/Spring 006
More informationWave Phenomena Physics 15c. Lecture 11 Dispersion
Wave Phenomena Physics 15c Lecture 11 Dispersion What We Did Last Time Defined Fourier transform f (t) = F(ω)e iωt dω F(ω) = 1 2π f(t) and F(w) represent a function in time and frequency domains Analyzed
More informationConcepts from linear theory Extra Lecture
Concepts from linear theory Extra Lecture Ship waves from WW II battleships and a toy boat. Kelvin s (1887) method of stationary phase predicts both. Concepts from linear theory A. Linearize the nonlinear
More informationExperimental and numerical study of spatial and temporal evolution of nonlinear wave groups
Nonlin. Processes Geophys., 15, 91 942, 28 www.nonlin-processes-geophys.net/15/91/28/ Author(s) 28. This work is distributed under the Creative Commons Attribution. License. Nonlinear Processes in Geophysics
More informationMaking Waves in Vector Calculus
Making Waves in Vector Calculus J. B. Thoo Yuba College 2014 MAA MathFest, Portland, OR This presentation was produced
More informationDerivation of the General Propagation Equation
Derivation of the General Propagation Equation Phys 477/577: Ultrafast and Nonlinear Optics, F. Ö. Ilday, Bilkent University February 25, 26 1 1 Derivation of the Wave Equation from Maxwell s Equations
More informationFreak waves over nonuniform depth with different slopes. Shirin Fallahi Master s Thesis, Spring 2016
Freak waves over nonuniform depth with different slopes Shirin Fallahi Master s Thesis, Spring 206 Cover design by Martin Helsø The front page depicts a section of the root system of the exceptional Lie
More informationExperimental observation of dark solitons on water surface
Experimental observation of dark solitons on water surface A. Chabchoub 1,, O. Kimmoun, H. Branger 3, N. Hoffmann 1, D. Proment, M. Onorato,5, and N. Akhmediev 6 1 Mechanics and Ocean Engineering, Hamburg
More informationRational homoclinic solution and rogue wave solution for the coupled long-wave short-wave system
PRAMANA c Indian Academy of Sciences Vol. 86 No. journal of March 6 physics pp. 7 77 Rational homoclinic solution and rogue wave solution for the coupled long-wave short-wave system WEI CHEN HANLIN CHEN
More informationGoals of this Chapter
Waves in the Atmosphere and Oceans Restoring Force Conservation of potential temperature in the presence of positive static stability internal gravity waves Conservation of potential vorticity in the presence
More informationTHE PHYSICS OF WAVES CHAPTER 1. Problem 1.1 Show that Ψ(x, t) = (x vt) 2. is a traveling wave.
CHAPTER 1 THE PHYSICS OF WAVES Problem 1.1 Show that Ψ(x, t) = (x vt) is a traveling wave. Show thatψ(x, t) is a wave by substitutioninto Equation 1.1. Proceed as in Example 1.1. On line version uses Ψ(x,
More informationEuropean Journal of Mechanics B/Fluids. A modified High-Order Spectral method for wavemaker modeling in a numerical wave tank
European Journal of Mechanics B/Fluids 34 (01) 19 34 Contents lists available at SciVerse ScienceDirect European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu A modified
More informationBreather propagation in shallow water. 1 Introduction. 2 Mathematical model
Breather propagation in shallow water O. Kimmoun 1, H.C. Hsu 2, N. Homann 3,4, A. Chabchoub 5, M.S. Li 2 & Y.Y. Chen 2 1 Aix-Marseille University, CNRS, Centrale Marseille, IRPHE, Marseille, France 2 Tainan
More informationReview of Fundamental Equations Supplementary notes on Section 1.2 and 1.3
Review of Fundamental Equations Supplementary notes on Section. and.3 Introduction of the velocity potential: irrotational motion: ω = u = identity in the vector analysis: ϕ u = ϕ Basic conservation principles:
More informationSwinburne Research Bank
Powered by TCPDF (www.tcpdf.org) Swinburne Research Bank http://researchbank.swinburne.edu.au Author: Chalikov, Dmitry; Babanin, Alexander V. Title: Comparison of linear and nonlinear extreme wave statistics
More informationarxiv: v1 [physics.flu-dyn] 20 Apr 2016
Tracking breather dynamics in irregular sea state conditions A. Chabchoub,, Department of Ocean Technology Policy and Environment, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa,
More informationSuperposition of electromagnetic waves
Superposition of electromagnetic waves February 9, So far we have looked at properties of monochromatic plane waves. A more complete picture is found by looking at superpositions of many frequencies. Many
More informationARTICLE IN PRESS. JID:EJMFLU AID:2246 /FLA [m3sc+; v 1.64; Prn:1/08/2006; 14:22] P.1 (1-27)
JID:EJMFLU AID:2246 /FLA [m3sc+; v 1.64; Prn:1/08/2006; 14:22] P.1 (1-27) 3 European Journal of Mechanics B/Fluids ( ) 3 Evolution of wide-spectrum unidirectional wave groups in a tank: an experimental
More informationWave-Body Interaction Theory (Theory of Ship Waves) Lecture Notes for Graduate Course
Wave-Body Interaction Theory (Theory of Ship Waves) Lecture Notes for Graduate Course ( April 018 ) Lab of Seakeeping & Floating-Body Dynamics in Waves Dept of Naval Architecture & Ocean Engineering Osaka
More informationMechanical Waves. 3: Mechanical Waves (Chapter 16) Waves: Space and Time
3: Mechanical Waves (Chapter 6) Phys3, A Dr. Robert MacDonald Mechanical Waves A mechanical wave is a travelling disturbance in a medium (like water, string, earth, Slinky, etc). Move some part of the
More informationAcoustic mode beam effects of nonlinear internal gravity waves in shallow water
Acoustic mode beam effects of nonlinear internal gravity waves in shallow water Timothy Duda Ying Tsong Lin James F. Lynch Applied Ocean Physics & Engineering Department Woods Hole Oceanographic Institution
More informationSalmon: Introduction to ocean waves
9 The shallow-water equations. Tsunamis. Our study of waves approaching the beach had stopped at the point of wave breaking. At the point of wave breaking, the linear theory underlying Propositions #1
More informationChapter 3. Shallow Water Equations and the Ocean. 3.1 Derivation of shallow water equations
Chapter 3 Shallow Water Equations and the Ocean Over most of the globe the ocean has a rather distinctive vertical structure, with an upper layer ranging from 20 m to 200 m in thickness, consisting of
More informationLecture 18: Wave-Mean Flow Interaction, Part I
Lecture 18: Wave-Mean Flow Interaction, Part I Lecturer: Roger Grimshaw. Write-up: Hiroki Yamamoto June 5, 009 1 Introduction Nonlinearity in water waves can lead to wave breaking. We can observe easily
More informationThe Evolution of Large-Amplitude Internal Gravity Wavepackets
The Evolution of Large-Amplitude Internal Gravity Wavepackets Sutherland, Bruce R. and Brown, Geoffrey L. University of Alberta Environmental and Industrial Fluid Dynamics Laboratory Edmonton, Alberta,
More informationarxiv: v3 [physics.flu-dyn] 16 Nov 2018
Maximum temporal amplitude and designs of experiments for generation of extreme waves Marwan 1,2, Andonowati 1, and N. Karjanto 3 1 Department of Mathematics and Center for Mathematical Modelling and Simulation
More informationFlows Induced by 1D, 2D and 3D Internal Gravity Wavepackets
Abstract Flows Induced by 1D, 2D and 3D Internal Gravity Wavepackets Bruce R. Sutherland 1 and Ton S. van den Bremer 2 1 Departments of Physics and of Earth & Atmospheric Sciences, University of Alberta
More informationSpatial evolution of an initially narrow-banded wave train
DOI 10.1007/s40722-017-0094-6 RESEARCH ARTICLE Spatial evolution of an initially narrow-banded wave train Lev Shemer 1 Anna Chernyshova 1 Received: 28 February 2017 / Accepted: 26 July 2017 Springer International
More informationFission of a weakly nonlinear interfacial solitary wave at a step
Fission of a weakly nonlinear interfacial solitary wave at a step Roger Grimshaw ), Efim Pelinovsky ), and Tatiana Talipova ) ) Department of Mathematical Sciences, Loughborough University, Loughborough,
More informationTHE OCCURRENCE PROBABILITIES OF ROGUE WAVES IN DIFFERENT NONLINEAR STAGES
THE OCCURRENCE PROBABILITIES OF ROGUE WAVES IN DIFFERENT NONLINEAR STAGES Aifeng Tao 1,2, Keren Qi 1,2, Jinhai Zheng 1,2, Ji Peng 1,2, Yuqing Wu 1,2 The occurrence probabilities of Rogue Waves in different
More informationChapter 16 Waves in One Dimension
Chapter 16 Waves in One Dimension Slide 16-1 Reading Quiz 16.05 f = c Slide 16-2 Reading Quiz 16.06 Slide 16-3 Reading Quiz 16.07 Heavier portion looks like a fixed end, pulse is inverted on reflection.
More informationUNIVERSITY OF SOUTHAMPTON. Answer all questions in Section A and two and only two questions in. Section B.
UNIVERSITY OF SOUTHAMPTON PHYS2023W1 SEMESTER 1 EXAMINATION 2009/10 WAVE PHYSICS Duration: 120 MINS Answer all questions in Section A and two and only two questions in Section B. Section A carries 1/3
More informationAttenuation and dispersion
Attenuation and dispersion Mechanisms: Absorption (anelastic); Scattering (elastic). P- and S-wave, bulk and shear attenuation Mathematical descriptions Measurement Frequency dependence Velocity dispersion,
More informationElectromagnetic fields and waves
Electromagnetic fields and waves Maxwell s rainbow Outline Maxwell s equations Plane waves Pulses and group velocity Polarization of light Transmission and reflection at an interface Macroscopic Maxwell
More informationLecture 9: Waves in Classical Physics
PHYS419 Lecture 9 Waves in Classical Physics 1 Lecture 9: Waves in Classical Physics If I say the word wave in no particular context, the image which most probably springs to your mind is one of a roughly
More informationDavydov Soliton Collisions
Davydov Soliton Collisions Benkui Tan Department of Geophysics Peking University Beijing 100871 People s Republic of China Telephone: 86-10-62755041 email:dqgchw@ibmstone.pku.edu.cn John P. Boyd Dept.
More informationWave statistics in unimodal and bimodal seas from a second-order model
European Journal of Mechanics B/Fluids 25 (2006) 649 661 Wave statistics in unimodal and bimodal seas from a second-order model Alessandro Toffoli a,, Miguel Onorato b, Jaak Monbaliu a a Hydraulics Laboratory,
More informationSURFACE WAVE DISPERSION PRACTICAL (Keith Priestley)
SURFACE WAVE DISPERSION PRACTICAL (Keith Priestley) This practical deals with surface waves, which are usually the largest amplitude arrivals on the seismogram. The velocity at which surface waves propagate
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles? 5.6 Uncertainty Principle 5.7 Probability,
More informationInternal Wave Generation and Scattering from Rough Topography
Internal Wave Generation and Scattering from Rough Topography Kurt L. Polzin Corresponding author address: Kurt L. Polzin, MS#21 WHOI Woods Hole MA, 02543. E-mail: kpolzin@whoi.edu Abstract Several claims
More informationBasic Concepts and the Discovery of Solitons p. 1 A look at linear and nonlinear signatures p. 1 Discovery of the solitary wave p. 3 Discovery of the
Basic Concepts and the Discovery of Solitons p. 1 A look at linear and nonlinear signatures p. 1 Discovery of the solitary wave p. 3 Discovery of the soliton p. 7 The soliton concept in physics p. 11 Linear
More informationSimulations and experiments of short intense envelope
Simulations and experiments of short intense envelope solitons of surface water waves A. Slunyaev 1,), G.F. Clauss 3), M. Klein 3), M. Onorato 4) 1) Institute of Applied Physics, Nizhny Novgorod, Russia,
More informationA Low-Dimensional Model for the Maximal Amplification Factor of Bichromatic Wave Groups
PROC. ITB Eng. Science Vol. 35 B, No., 3, 39-53 39 A Low-Dimensional Model for the Maximal Amplification Factor of Bichromatic Wave Groups W. N. Tan,* & Andonowati Fakulti Sains, Universiti Teknologi Malaysia
More informationROSSBY WAVE PROPAGATION
ROSSBY WAVE PROPAGATION (PHH lecture 4) The presence of a gradient of PV (or q.-g. p.v.) allows slow wave motions generally called Rossby waves These waves arise through the Rossby restoration mechanism,
More informationSound Waves Sound Waves:
3//18 Sound Waves Sound Waves: 1 3//18 Sound Waves Linear Waves compression rarefaction Inference of Sound Wave equation: Sound Waves We look at small disturbances in a compressible medium (note: compressible
More informationThe shallow water equations Lecture 8. (photo due to Clark Little /SWNS)
The shallow water equations Lecture 8 (photo due to Clark Little /SWNS) The shallow water equations This lecture: 1) Derive the shallow water equations 2) Their mathematical structure 3) Some consequences
More informationINTERNAL SOLITARY WAVES IN THE ATMOSPHERE AND OCEAN
INTERNAL SOLITARY WAVES IN THE ATMOSPHERE AND OCEAN ROGER GRIMSHAW LOUGHBOROUGH UNIVERSITY In collaboration with: Efim Pelinovsky (Nizhny Novgorod) Tatiana Talipova (Nizhny Novgorod) Outline: 1. Observations
More informationWave Phenomena Physics 15c
Wave Phenomena Physics 5c Lecture Fourier Analysis (H&L Sections 3. 4) (Georgi Chapter ) What We Did Last ime Studied reflection of mechanical waves Similar to reflection of electromagnetic waves Mechanical
More informationPhysics, Nonlinear Time Series Analysis, Data Assimilation and Hyperfast Modeling of Nonlinear Ocean Waves
DISTRIBUTION STATEMENT A. Approved for public release; distribution is unlimited. Physics, Nonlinear Time Series Analysis, Data Assimilation and Hyperfast Modeling of Nonlinear Ocean Waves A. R. Osborne
More informationDispersive nonlinear partial differential equations
Dispersive nonlinear partial differential equations Elena Kartashova 16.02.2007 Elena Kartashova (RISC) Dispersive nonlinear PDEs 16.02.2007 1 / 25 Mathematical Classification of PDEs based on the FORM
More informationSTRUCTURE OF MATTER, VIBRATIONS AND WAVES, AND QUANTUM PHYSICS
Imperial College London BSc/MSci EXAMINATION June 2008 This paper is also taken for the relevant Examination for the Associateship STRUCTURE OF MATTER, VIBRATIONS AND WAVES, AND QUANTUM PHYSICS For 1st-Year
More information