Group Velocity

10 Key excerpts on "Group Velocity"

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  Classical Dynamics of Particles and Systems

  • Jerry B. Marion(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  This can be accomplished only if it is possible to start and stop a wave train and thereby impress a signal on the wave. This constitutes forming a wave packet. As a consequence of this fact, it is the Group Velocity, not the phase velocity, that corresponds to the velocity at which a signal 15.6 FOURIER INTEGRAL REPRESENTATION OF WAVE PACKETS 499 may be transmitted.*t We shall make this point clear when, in Section 15.7, we consider the propagation of energy along a lattice or loaded string. 15.6 Fourier Integral Representation of Wave Packets It was implicit in the discussion of the preceding section that we were considering the description of the wave function for a wave packet for which the amplitude distribution A(k) was given. We may wish, on the other hand, to obtain the distribution function A(k) which describes a given wave function. This problem is similar to that of calculating the Fourier series which represents a given function. For that case, we found [see Eq. (14.147)] for — π < x < π, f(x) = -£ + ]T (a r cosrx + b r sinrx) (15.62) 2 r = 1 where [see Eqs. (14.150)]: l f +rr a r = -f(x) cos rx dx, r = 0,1,2,... (15.63a) 1 Γ +π b=- f(x) sin rxdx, r = 1,2,3,... (15.63b) rcJ-π For our present purposes it will prove more convenient to express f(x) in a complex exponential series, 00 f(x)= £ c r e~ ir -π<χ<π (15.64) r = — oo * The Group Velocity corresponds to the signal velocity only in media that are nondisper-sive (in which case the phase, group, and signal velocities are all equal) and in media of normal dispersion (in which case the phase velocity exceeds the group and signal velocities). In media with anomalous dispersion, the Group Velocity may exceed the signal velocity (and, in fact, may even become negative or infinite!). We need only note here that a medium in which the wave number k is complex exhibits attenuation, and the dispersion is said to be anomalous. If k is real, there is no attenuation, and the dispersion is normal.

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  Electromagnetic Optics of Thin-Film Coatings

  Light Scattering, Giant Field Enhancement, and Planar Microcavities

  • Claude Amra, Michel Lequime, Myriam Zerrad(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  (3.42) For a large number of so-called quasi-monochromatic beams, the line width Δω is small compared with the central frequency ω 0 . If we assume that, over this spectral range, variations in the derivative ∂ω ∂k can be neglected, then the expression for the Group Velocity becomes v g = ∂ω ∂k     ω 0 . (3.43) Consequence It is of interest to compare this Group Velocity with the phase velocity defined in the harmonic regime, where the angular frequency is assumed to be ω 0 . Let us expand on the expression for v g : 1 v g = ∂k ∂ω = ∂ ∂ω  nω c  = n c + ω c ∂n ∂ω ⇒ v g = c n + ω ∂n ∂ω . (3.44) We see that, in the absence of dispersion, the phase and group velocities are identical: ∂n ∂ω = 0 ⇒ v g = c n = v . (3.45) On the other hand, this is no longer true for dispersive media, a prop- erty used to slow light down in cases in which the dispersion is very high 3.4 Propagation of a Gaussian Pulse 53 (this is the concept of slow light ). Bearing in mind the property of n(ω) al- ready mentioned, namely that the derivative ∂n ∂ω is positive in a transparent window, note that the Group Velocity is always less than the phase velocity: ∂n ∂ω > 0 ⇒ v g < c n = v . (3.46) We conclude by recalling that the Group Velocity is associated with the transmission of information (detecting the passage of a pulse): this is not the case for phase velocity. 3.4 Propagation of a Gaussian Pulse 3.4.1 Introduction In many experiments or applications such as (in particular) long distance communication using optical fiber (long-haul fiber optic network ), it is im- portant that the pulses of light should propagate with the least amount of distortion. Hence, after having established the notion of Group Velocity as applied to the centroid of the pulse, we now turn our attention to the temporal shape of the beam that accompanies this centroid as it propagates. Consider the special case of a pulse whose temporal envelope is a Gaussian profile, i.e., A 0 (t) = A 0 e −( t τ ) 2 .

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  Quantum Mechanics For Applied Physics And Engineering

  • Albert T. Jr. Fromhold(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  As a second check, let us apply this relation to electromagnetic waves propagating in free §7] WAVE-PACKET SOLUTIONS 63 space, for which the dispersion relation is (&) = ck. We obtain immediately u g = c, which is in complete accord with our previous conclusions in §2 that a packet of electromagnetic waves moves with a Group Velocity equal to the universal phase velocity c. 7.5 Group Velocity in Three Dimensions The above relation for the Group Velocity is quite generally valid for one-dimensional systems; it is therefore worthwhile to extend it to the three-dimensional case. The Group Velocity must in this case be a vector quantity v g , with components given by the derivative of co(k) with respect to k x , k y9 and fe z , respectively. Thus v g = [F k co(k)] k = ko (Group Velocity of a wave packet). (1.186) The relation $ = ήω between total energy and frequency is valid for any dispersion relation co(k), so an alternate expression for the Group Velocity is Vg = h~V k (k) We should ask ourselves what effects are introduced if higher derivatives in the Taylor series expansion for ( ;) are important. For free particles, ω = hk 2 /2m, so that use of only the first two terms in the Taylor series approximation (1.180) results in the neglect of the second term (/2)(^/dk 2 ) k = ko (k - k 0 ) 2 = h{k - k 0 ) 2 /2m (1.188) with respect to the first term (Ao/dk) k = ko (k- k 0 ) = hk 0 {k - k 0 )/m. (1.189) The ratio of the second term to the first term is (k — k 0 )/2k 0 , and thus it can be quite small if the spread in the values of A: over the packet is small relative to the value of k at the peak of x(k) for the packet, namely k 0 . By referring to the general expression (1.182) for the one-dimensional wave packet, we see that addition of this second term leads to the exponential factor exp[— ih(k — k 0 ) 2 t/2m] in the integrand.

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  Principles of Seismology

  • Agustín Udías, Elisa Buforn(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  13 Wave dispersion. Phase and group velocities We have seen that surface waves in layered media are dispersed; that is, their velocity is a function of the frequency (or period). Thus, for an impulsive time function at the source, surface waves at some distance are formed by trains of waves, different frequencies arriving at different times. This is an important phenomenon that requires further consideration. Arrival times, amplitudes, and phases for each frequency depend, then, on the dispersion equation. In Section 5.4 we saw that, if the phase velocity is a function of the frequency, then the velocity of energy transport is not the same, but equal to the Group Velocity, or the velocity of propagation of wave groups. We will consider now, with more detail, the general problem of wave dispersion and the relation between phase and group velocities and apply it to surface waves generated by earthquakes in the Earth. The study of the dispersion curves of phase and group velocities of surface waves is used to determine the structure of the Earth ’ s crust and mantle. 13.1 Phase and group velocities The displacement of a sinusoidal wave of angular frequency ω and wave number k that propagates in the x direction is given by u ð x ; t Þ ¼ A sin ½ð kx ω t Þ þ ϕ Š ; ð 13 : 1 Þ where the phase velocity, or the velocity of propagation of each value of the phase, is c ¼ ω = k : ð 13 : 2 Þ For monochromatic waves in a homogeneous medium, c is constant and for each value of ω there is a single value of k . In this case, the velocity of energy transport is equal to the phase velocity ( Section 5.4 ); this is the case for body waves. If the phase velocity is a function of the frequency c ( ω ), then we can also write k ( ω ) and ω ( k ), and we can use as the independent variable either k or ω .

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  Quantum Mechanics I

  The Fundamentals

  • S. Rajasekar, R. Velusamy(Authors)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  Such states are called coherent states. The behaviour of localized quantum wave packet has lately become an object of much interest to forge a link between classical and quantum mechanics. We can consider the wave packets synthesized by the superposition of a great many bound states centred around a rather large mean value n ¯ of the quantum number n. Such a wave packet may begin by mimicking classical dynamics, but after a short while, quantum effects set in, causing, at first a spreading and, at much later times, a partial or total restoration of the original form at a point, which is not necessarily the same as that where the packet was initially localized. In this chapter, we discuss some of the features of wave packets. 10.2 Phase and Group Velocities Consider a plane wave u = e i (k x − ω t) of frequency ν = ω / (2 π) and wavelength λ = 2 π / k. k x − ω t is called phase of the wave. The velocity of the wave is simply the velocity of a point at which the phase is constant. Such a position is given by x = constant + (ω / k) t = constant + v p t, where v p = ω / k is known as the phase velocity of the wave. Suppose we superpose a large number of waves of the form u = e i (k x − ω t) and denote it as ψ (x, t). It is given by ψ (x, t) = 1 2 π ∫ − ∞ ∞ C (k) e i (k x − ω t) d k. (10.1a) As this equation appears as the Fourier inverse transform we write C (k) = 1 2 π ∫ − ∞ ∞ ψ (x, t) e − i (k x − ω t) d x. (10.1b) Although, t appears explicitly in the integral in Eq. (10.1b), C (k) does not depend on t provided ψ is a superposition of plane waves. In this case ∂ C (k) / ∂ t = 0 : ∫ − ∞ ∞ ∂ ψ ∂ t + i ω ψ e − i (k x − ω t) d x = 0. (10.2) The implication is ψ (x, t) can be replaced by ψ (x, 0) in Eqs. (10.1). The wave function ψ (x, t) given by Eq. (10.1a) is called a wave group or a wave packet. Its localizations in x and p x (or k) can be adjustable and can be made very narrow

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  Beyond Physics

  Or the Idealisation of Mechanism

  • Sir Oliver Lodge(Author)
  • 2015(Publication Date)
  • Routledge

    (Publisher)

  time, and accordingly we get successive alternations of sound and silence, well known as beats. All these phenomena may be called generically “beat” phenomena. Sometimes the coincidences are real, so that it is a real distribution of energy, as in the case of light and sound; sometimes it is merely an optical illusion depending on the observer, as in the case of the palings. (When both sets of palings happen to be identical in width, or what corresponds with wave-length or frequency, the relative or apparent difference between them is due to their different distances from the observer.)

  A similar sort of thing can be observed in the case of water waves, or with any waves whose velocity is dependent on wave-length. In that case the coincidences determine some special feature, perhaps a bigger wave than usual, or some other mark of identification. And such coincidental waves are known as group waves; they have a velocity of their own, which is called the Group Velocity. This is slower than the true wave velocity, having indeed half its value in the case of water waves: the true waves may be seen passing through the group. I believe that on the open ocean patches or lumps of these group waves can be seen, preserving a sort of identity, and having a secondary or derived speed of their own. The theory of such group waves was worked out most clearly by the late Lord Rayleigh. They are a recognised phenomenon with many applications.

  ENERGY TRANSMISSION BY WAVES

  Two things, two different processes or functions, are involved in all wave motion: the transmission of energy, and the propagation of waveforms. This double aspect always exists, though in light and sound it is obscured because the speed of energy-transfer, in those special cases, happens to be equal to the speed of wave-propagation. In a dispersive

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  Fluid Mechanics

  • Ira M. Cohen, Pijush K. Kundu(Authors)
  • 2004(Publication Date)
  • Academic Press

    (Publisher)

  The magnitude of phase velocity is c = ω/K , and the direction of propagation is that of K . We can therefore write the phase velocity as the vector c = ω K K K , (7.18) where K /K represents the unit vector in the direction of K . 204 Gravity Waves Figure 7.3 Wave propagating in the xy -plane. The inset shows how the components c x and c y are added to give the resultant c . From Figure 7.3, it is also clear that the phase speeds (that is, the speeds of propagation of lines of constant phase) in the three Cartesian directions are c x = ω k c y = ω l c z = ω m . (7.19) The preceding shows that the components c x , c y , and c z are each larger than the resultant c = ω/K . It is clear that the components of the phase velocity vector c do not obey the rule of vector addition . The method of obtaining c from the components c x and c y is illustrated at the top of Figure 7.3. The peculiarity of such an addition rule for the phase velocity vector merely reflects the fact that phase lines appear to propagate faster along directions not coinciding with the direction of propagation, say the x and y directions in Figure 7.3. In contrast, the components of the “Group Velocity” vector c g do obey the usual vector addition rule, as we shall see later. We have assumed that the waves exist without a mean flow. If the waves are superposed on a uniform mean flow U , then the observed phase speed is c 0 = c + U . A dot product of the forementioned with the wavenumber vector K , and the use of equation (7.18), gives ω 0 = ω + U • K , (7.20) where ω 0 is the observed frequency at a fixed point, and ω is the intrinsic frequency measured by an observer moving with the mean flow. It is apparent that the frequency 4. Surface Gravity Waves 205 of a wave is Doppler shifted by an amount U • K due to the mean flow.

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  Waves in Layered Media

  • Leonid Brekhovskikh(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Consequently, for given values of r and t, these frequencies will be predominant. The relation between t and r, corresponding to the point of stationary phase for a given o>, determines the propagation velocity of a group of waves composed of mutually interfering sinusoidal waves with frequencies close to ω, that is, the so-called Group Velocity U. The points of stationary phase are (the subscript I on k x is omitted temporarily) άφ_ ί Γ ^(ω) = ( ) > da> dco P R O P A G A T I O N O F A S O U N D P U L S E I N A L I Q U I D L A Y E R 393 The last equation can also be written in the form u = c^ = d(vk) = V + Jc dv dk dk die' (31.12) where ν is the phase velocity. It is thus clear that the Group Velocity will always be smaller than the phase velocity, if the latter decreases with increasing frequency, as is the case here. The phase and group velocities of the first normal mode are shown in Fig. 137 as functions of 0.1 0.2 0.5 1.0 2.0 5.0 10 20 50 100 v -h = fh Fig. 137. The phase and group velocities of the first normal mode as functions of frequency for some concrete cases. frequency, for several definite cases. The density of the bottom (assumed homogeneous) was chosen equal to 2. The velocity of sound in the bottom c x varies from 1.05c to 3c, where c is the velocity of sound in the water. It is clear from the Figure that in all cases, the phase velocity passes through a minimum, and furthermore, that when U < c, two frequencies correspond to one value of the velocity. It will be shown below that these characteristics of the Group Velocity curves 394 W A V E P R O P A G A T I O N I N L A Y E R S are of great significance in the study of the propagation of a pulse in a layer of water. 3. Qualitative picture of pulse propagation Let us consider the qualitative picture of the propagation of a pulse in a layer, and the change of its form as it propagates. For the present, we will take only the first normal mode into account.

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  Principles of Photonics

  • Jia-Ming Liu(Author)
  • 2016(Publication Date)
  • Cambridge University Press

    (Publisher)

  The phase velocity and Group Velocity of the mode are, respectively, v p β ¼ ω β ¼ c n β , (3.177) and v g β ¼ dω dβ ¼ c N β : (3.178) As an example of the contributions of the waveguiding effect to the dispersion parameters, Fig. 3.25 shows n β , N β , and D β of the fundamental mode of a circular optical fiber in comparison to the parameters of its core and cladding materials. 3.6.2 Modal Dispersion The frequency dependence of the propagation constant β of a mode discussed above is the total intramode dispersion that includes material and waveguide contributions for the mode. Differ- ent normal modes of an anisotropic medium or an optical structure have different propagation constants at a given optical frequency. Such differences lead to modal dispersion among different modes, which is intermode dispersion. For plane waves or Gaussian modes propagating in a homogeneous anisotropic medium, modal dispersion exists due to different propagation constants for normal modes of different polarizations, such as k x , k y , and k z of the linearly birefringent principal normal modes of polarization given in (2.15), k þ and k  of the circularly birefringent principal normal modes of polarization given in (2.21), or k o and k e of the ordinary and extraordinary waves in (3.57). Such modal dispersion causes polarization dispersion. Figure 3.25 (a) Effective index of refraction and group index and (b) group-velocity dispersion of the fundamental mode as a function of wavelength. The solid curves show the effective parameters of the mode with both material and waveguide contributions. The dashed curves show only the material contribution to the core and cladding regions, labeled 1 and 2, respectively. 3.6 Phase Velocity, Group Velocity, and Dispersion 127

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  Wave Propagation in Materials and Structures

  • Srinivasan Gopalakrishnan(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  In wave propagation problems, two parameters are very important, namely, the wavenumber and the speeds of the propagation. This chapter provides a general methodology to compute these quantities for a material system. There are many types of waves that can be generated in structure. Wavenum- ber expression reveals the type of waves that are generated. Hence, in wave propagation problems, two relations are very important, namely, spectrum relations, which is a plot of the wavenumber with the frequency, and dis- persion relations, which is a plot of wave velocity with the frequency. These relations reveal the characteristics of different waves that are generated in a given material system. This chapter is organized as follows. First, the concept of phase and group speeds and their behavior in different mediums are explained. The expressions for evaluating them are derived. Next, some of the commonly used wave prop- agation terminologies are explained. This is followed by a subsection, wherein the spectral analysis of motion is explained for a general one-dimensional second-order material system. If the governing equation of the material system is derived using higher-order theories, computation of wavenumbers becomes extremely difficult. In such situations, we need to obtain them numerically. In the last part of this chapter, this issue is addressed, wherein general methods of numerically computing wavenumbers and their corresponding wave ampli- tudes is given. Introduction to Wave Propagation 109 5.1 CONCEPT OF WAVENUMBER, GROUP SPEEDS, AND PHASE SPEEDS A wave propagating in a medium can be represented as u(x, t) = e i(kx-ωt). (5.1) Eqn. (5.1) is an alternate way of representing a wave, which is given in Chapter 1 (Eqn. (1.1). In the above equation, k is the wavenumber, which specifies the behavior of the wave. The exponent in Eqn. (5.1) i(kx - ωt) is called the phase of the wave.

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