Non Linear Wave

6 Key excerpts on "Non Linear Wave"

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  eBook - PDF

  Sir James Lighthill And Modern Fluid Mechanics

  • Lokenath Debnath(Author)
  • 2008(Publication Date)
  • ICP

    (Publisher)

  Chapter 8 Linear and Nonlinear Waves in Fluids “Our present analytical methods seem unsuitable for the so-lution of the important problems arising in connection with nonlinear partial differential equations and, in fact, with virtu-ally all types of nonlinear problems in pure mathematics. The truth of this statement is particularly striking in the field of fluid dynamics .... ” John Von Neumann “Everything is vague to a degree you do not realize till you have tried to make it precise.” Bertrand Russell “I believe there is no philosophical high-road in science, with epistemological signposts. No, we are in a jungle and find out way by trial and error, building our road behind us as we pro-ceed.” Max Born 8.1 Sound Waves and Shock Waves In his 1956 famous paper “On Viscosity Effects in Sound Waves of Fi-nite Amplitude” written in honor of the seventieth birthday of Sir G. I. Taylor, Lighthill initiated the theory of nonlinear acoustics. According to the linearized theory of sound, pressure waves in gases initially at rest are propagated with constant wave velocity and constant energy. Sinusoidal acoustic waveforms are likely to be altered or attenuated by various ther-modynamically irreversible processes (or viscosity), or other diffusive effects 165 166 Sir James Lighthill and Modern Fluid Mechanics including heat conduction, relaxation effects, and nonlinear effects which emerge from the theory of plane waves of finite amplitude. In view of dominant effects of nonlinearity, not only does the velocity of sound waves change, being greater in the compression phase of a sound wave than in the expansion phase, but also the waves are convected with the fluid, which is propagating in the direction of propagation in the compression part and in the opposite direction in the expansion part. Both effects significantly dis-tort the waveform by causing the compression part to move forward faster than the expansion part.

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  Nonlinear Ocean Waves and the Inverse Scattering Transform

  • Alfred Osborne(Author)
  • 2010(Publication Date)
  • Academic Press

    (Publisher)

  International Geophysics , Vol. 97, No. (Suppl C), 2010

  ISSN: 0074-6142

  doi: 10.1016/S0074-6142(10)97001-0

  1 Brief History and Overview of Nonlinear Water Waves

  Alfred R. Osborne

  1.1 Linear and Nonlinear Fourier Analysis

  Man has long been intrigued by the study of water waves, one of the most ubiquitous of all known natural phenomena. Who has not been fascinated by the rolling and churning of the surf on a beach or the often-imposing presence of large waves at sea? How many countless times have ship captains logged the treacherous encounters with high waves in the deep ocean or later reported (if they were lucky) the damage to their ships? Man’s often strained friendship with the world’s oceans, and its waves and natural resources, has endured at least since the beginning of recorded history and perhaps even to the invention of ocean going vessels thousands of years ago. But it is only in the last 200 years that the study of water waves has been placed on a firm foundation, not only from the point of view of the physics and mathematics, but also from the perspective of experimental science and engineering.

  While water waves are one of the most common of all natural phenomena, they possess an extremely rich mathematical structure. Water waves belong to one of the most difficult areas of fluid dynamics (Batchelor, 1967 ; Lighthill, 1986 ) and wave mechanics (Whitham, 1974 ; Stoker, 1957 ; LeBlond and Mysak, 1978 ; Lighthill, 1978 ; Mei, 1983 ; Drazin and Johnson, 1989 ; Johnson, 1997 ); Craik, 2005 , namely the study of nonlinear, dispersive waves in two-space and one-time dimensions. The governing equations of motion are coupled nonlinear partial differential equations in two fields: the surface elevation , η(x, t ), and the velocity potential , φ(x, t ). Analytically, these equations are difficult to solve because of the nonlinear boundary conditions that are imposed on an unknown free surface . This set of equations is known as the Euler equations

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  Essentials of Photonics

  • Alan Rogers(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  In a semi-classical description of light propagation in dielectric media, the optical electric field drives the atomic/molecular oscillators of which the material is composed, and these oscillators become secondary radiators of the field; the primary and secondary fields then combine vectorially to form the resultant wave. The phase of this wave (being different from that of its primary) determines a velocity of light different from that of free space, and its amplitude determines a scattering/absorption coefficient for the material.

  Nonlinear behaviour occurs when the secondary oscillators are driven beyond the linear response; as a result, the oscillations become nonsinusoidal. Fourier theory dictates that, under these conditions, frequencies other than that of the primary wave will be generated (Figure 9.1 ).

  The fields necessary to do this depend upon the structure of the material, because this dictates the allowable range of sinusoidal oscillation at given frequencies. Clearly, it is easier to generate large amplitudes of oscillation when the optical frequencies are close to natural resonances, and one expects (and obtains) enhanced nonlinearity there. The electric field required to produce nonlinearity in material therefore varies widely, from ~106 Vm⁐1 up to ~1011 Vmr−1 , the latter being comparable with the atomic electric field. Even the lower of these figures, however, corresponds to an optical intensity of ~109 Wm−2 , which is only achievable practically with laser sources. It is for this reason that the study of nonlinear optics only really began with the invention of the laser, in 1960.

  The magnitude of any given nonlinear effect will depend upon the optical intensity, the optical path over which the intensity can be maintained, and the size of the coefficient that characterizes the effect.

  In bulk media, the magnitude of any nonlinearity is limited by diffraction effects. For a beam of power P watts and wavelength λ focused to a spot of radius r , the intensity, P/πr 2 , can be maintained (to within a factor of ~2) over a distance

  ~r2 /λ

  (Rayleigh distance), beyond which diffraction will rapidly reduce it. Hence, the product of intensity and distance is ~P/πλ , independent of r , and of propagation length (Figure 9.2

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  Waves and Oscillations in Plasmas

  • Hans L. Pecseli(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  4   Weakly Nonlinear Waves  

  In real life, we will of course never encounter a wave phenomenon which can truly be called linear (with the exception of the classical description of electromagnetic waves in vacuum), although a description based on a linear formalism as in the foregoing section can be very good, in some cases even extremely good. To describe the lowest order nonlinear corrections to the basic linear wave analysis, it is an advantage to classify wave-types according to their dispersion relation. The important observation will be that many nonlinear wave phenomena can be “classified” into either Korteweg-deVries types or nonlinear Schrödinger types, as explained in detail later. These two are the most relevant ones for plasma physics, but there are also others. The basic classification is rooted in the linear dispersion relation, where we here distinguish two cases. Already from the outset, the reader should be warned that the term “simple waves” will in the following Section 4.1.1 have a slightly different meaning compared to its standard usage. The terminology used in the following is standard in the literature.

  1) We have weakly dispersive linear waves, where the phase and group velocities are similar, though not necessarily identical (one example being acoustic or sound waves). By “weakly dispersive” we understand waves where to the lowest approximation all wavelengths (i.e., wave-numbers) propagate with the velocity, phase and group velocities being identical; see Fig. 4.1a ). We have a linear dispersion relation of the form

  ω 2

  =

  C s 2

  k 2

  , or equivalently ω = ±C s k . Corrections then appear as higher order corrections to this dispersion relation. Such corrections can originate from linear as well as nonlinear effects.

  2) We can have strongly dispersive linear waves, where the phase and group velocities are significantly different (one example being Langmuir waves); see Fig. 4.1b ). For these waves we have a local dispersion relation ω = ω0 + u g (k − k 0 ), with ω0 ≡ ω(k 0 ) and u g ≡ d ω/dk ≠ ω0 /k 0 being the group velocity, obtained for k = k 0

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  Electrodynamics of Continuous Media

  • L D Landau, J. S. Bell, M. J. Kearsley, L. P. Pitaevskii, E.M. Lifshitz, J. B. Sykes(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER XIII

  NON-LINEAR OPTICS

  Publisher Summary

  This chapter discusses nonlinear optics. The most important feature of a nonlinear medium is the generation in it of vibrations with new frequencies. If the medium is nondissipative, the frequency transformation process is subject to certain very general relationships, in addition to the obvious condition that the total energy of the vibrations at all frequencies must be conserved. The chapter also discusses optical effects arising from the nonlinear variation of the field at the primary wave frequency. The use of the nonlinear relation presupposes that there is only slight nonlinearity: The higher-order terms must be small in comparison with the terms in electric field induction. The third-order nonlinear effects include the influence of radiation with frequency on the propagation of a wave with a different frequency in the same medium.

  §107 Frequency transformation in non-linear media

  The theory of electromagnetic wave propagation in dielectric media described in the preceding chapters is based on the assumption of a linear relation between the electric field induction D and intensity E . This approximation is sufficiently accurate if (as is true in practice) E is much less than typical intra-atomic fields. Even then, however, the small nonlinear corrections to D (E ) cause qualitatively new effects and may therefore be important.

  The most important feature of a non-linear medium is the generation in it of vibrations with new frequencies. For example, if a monochromatic wave with frequency ω1 is incident on such a medium, then, as it is propagated in the medium, waves with frequencies mω 1 (m being an integer) are generated; if there is initially a set of monochromatic signals with frequencies ω1 and ω2 , the combination frequencies mω 1 + nω 2

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  Modern Optics and Photonics of Nano- and Microsystems

  • Yu. N. Kulchin(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  1

  Fundamentals of Nonlinear Optics

  Introduction

  When in ordinary life we talk about the propagation of light in matter, it is initially assumed that the characteristics of the medium do not change under its influence. Indeed, the electric fields in atoms and molecules, which are the main elements determining the structure and optical properties of matter, are very large. In this connection, the light sources that existed in the pre‐laser era could provide electric fields in the light electromagnetic wave many orders of magnitude smaller than the intra‐atomic ones. Therefore, the influence of the electromagnetic field of the lightwave on the properties of the medium turned out to be negligibly small and did not manifest itself under real conditions. Since in this case the response of the medium to the external optical action is proportional to the electric field strength in the wave, the theoretical description of the phenomena arising in the interaction of light with matter is usually called linear optics.

  The discovery of optical quantum generators (lasers) made it possible to generate optical fields with a strength comparable to that of an intra‐atomic field. The effect of such optical radiation on the medium leads to a change in its optical properties. This means that the wave in the medium experiences both self‐action and can influence the propagation of other waves in the medium. Naturally, this leads to the emergence of new phenomena that have not been observed before and requires the development of new theoretical approaches. As a result, a new field of science arose, which was called the nonlinear optics.

  The term ‘nonlinear optics’ was first introduced by the Russian scientist S.I. Vavilov in 1925, who managed to observe a decrease in the absorption of light by uranium glasses when optical radiation passes through them with high intensity. In the early 60s of the same century, after the creation of lasers, nonlinear optical phenomena became not only observable, but also turned into a serious tool for studying matter, and became the basis for the creation of completely new laser devices. Thus, the subject of nonlinear optics are the processes of interaction of light with matter, the nature of which depends on its intensity. Such processes include phenomena of resonant medium clarification, two‐photon or multiphoton light absorption, optical breakdown of the medium, generation of optical harmonics, ‘straightening’ of light, stimulated light scattering, self‐ focusing of light beams, self‐modulation of pulses, and a number of other effects manifested in fields of laser radiation. The classification of all these nonlinear optical effects is presented in detail in [1 –5

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