Physics
Phase Velocity
Phase velocity refers to the speed at which the phase of a wave propagates through space. It is the rate at which the phase of a wave, such as a light wave or sound wave, changes at a specific point in space over time. Phase velocity is a key concept in wave mechanics and is distinct from group velocity, which describes the speed at which the overall shape of a wave moves.
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8 Key excerpts on "Phase Velocity"
eBook - PDFElectronic Properties of Crystalline Solids
An Introduction to Fundamentals
- Richard Bube(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
1.6 Summary 17 υ = οΙ(εμ) υ2 , where c = 3 χ 10 10 cm/sec, the velocity of light in v a c u u m . In a material with electrical conductivity, the Phase Velocity becomes complex, implying an attenuation of the wave, which corresponds to the p h e n o m e n o n of optical absorption. W e consider some of these relations further in later portions of this b o o k where they are needed. 1.6 Summary T h e properties of waves a n d of wave motion are i m p o r t a n t for the under-standing of a variety of topics relevant to the electronic processes that occur in solids. In this chapter we have briefly summarized certain basic character-istics of wave motion, and have given a few examples of classical wave systems. T h e analysis of wave problems starts with the formulation of a wave equation based on a knowledge of the system involved. This wave equation involves space a n d time derivatives of a suitable displacement. T h e general solution of this wave equation is obtained, a n d specific solutions are gener-ated by consideration of b o u n d a r y conditions or other restrictions imposed by physical reality. T h e specific solutions correspond to certain allowed frequencies that are consistent with the b o u n d a r y conditions a n d other possible restrictions. T h e major requirement imposed by b o u n d a r y conditions that terminate the displacement at the ends of the region of interest is that an integral n u m b e r of half-wavelengths must be contained within the region. T h e variation of the phase a n d g r o u p velocities of a wave with wave-n u m b e r are given by the dispersion relationship between frequency a n d wavenumber. F o r long-wavelength vibrations in a string or rod, a n d for electromagnetic waves in a n o n a b s o r b i n g medium, the velocity is constant a n d the Phase Velocity is equal to the g r o u p velocity.
eBook - PDF- Jerry B. Marion(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
(15.18), that the general solution to the wave equation produces, even in the one-dimensional case, a complicated system of exponential factors. For the purposes of further discussion, we shall restrict our attention to the particular combination Ψ(χ, 0 = Aé i(cot — kx) (15.26) This equation describes the propagation to the right (larger x) of a wave which possesses a well-defined frequency ω. Certain physical situations can be quite adequately approximated by a wave function of this type—e.g., the propogation of a monochromatic light wave in space, or the propagation of a sinusoidal wave along an infinite string (or one which is terminated in such a way that reflection is minimized). The phase φ of the wave described by Eq. (15.26) is defined to be the argument of the real part of the wave function : φ = œt — kx (15.27) If the phase remains constant in time, then the form of the wave function remains the same. Such a condition defines a Phase Velocity For velocity of propagation of the wave form. To insure φ = const, we set d
eBook - PDFOcean Waves and Oscillating Systems: Volume 8
Linear Interactions Including Wave-Energy Extraction
- Johannes Falnes, Adi Kurniawan(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
3.2 Dispersion, Phase Velocity and Group Velocity Both acoustic waves and electromagnetic waves are non-dispersive, which means that the Phase Velocity is independent of the frequency. The dispersion relation (the relationship between ω and k) is ω = ck, (3.7) where c is the constant speed of sound or light, respectively. Gravity waves on water are, in general, dispersive. As will be shown later, in Chapter 4 (Section 4.2), the relationship for waves on deep water is ω 2 = gk, (3.8) where g is the acceleration of gravity. Using this dispersion relationship, we find that the Phase Velocity is v p ≡ ω/k = g/ω = g/k. (3.9) Note that in the study of the propagation of dispersive waves (for which the Phase Velocity depends on frequency), we have to distinguish between phase 3.3 WAVE POWER AND ENERGY TRANSPORT 49 velocity and group velocity. Let us assume that the dispersion relationship may be written as F (ω, k) = 0, where F is a differentiable function of two variables. Then the group velocity is defined as v g = dω dk = − ∂ F/∂ k ∂ F/∂ω . (3.10) As shown in Problem 3.1, if several propagating waves with slightly different frequencies are superimposed on each other, the result may be interpreted as a group of waves, each of them moving with the Phase Velocity, whereas the ampli- tude of the group of individual waves are modulated by an envelope moving with the group velocity. Here let us just consider the following simpler example of two superimposed harmonic waves of angular frequency ω ± Δω and angular repetency k ± Δk, namely p(x, t) = D cos { (ω − Δω) t − (k − Δk) x} + D cos { (ω + Δω) t − (k + Δk) x} = 2D cos { (Δω) t − (Δk) x} cos (ωt − kx) , (3.11) where an elementary trigonometric identity has been used in the last step.
eBook - PDF- R. N. Thurston, Allan D. Pierce(Authors)
- 2012(Publication Date)
- Academic Press(Publisher)
82 Emmanuel P. Papadakis activity and for various geometries of structures. It must be pointed out that both Phase Velocity and group velocity are relevant. 1.2. DEFINITION OF VELOCITY Consider a plane wave A = A 0 exp(eoi — kz) (1) of frequency / = ω/2π and amplitude A 0 traveling in the z-direction with a propagation constant k = 2π/λ = 2nf/v, where λ is wavelength and v is Phase Velocity. The variable ω is angular frequency (rad/s). Since the wave expressed is monochromatic and of infinite extent, it has a definite value for v and a specific value for k. However, k and v may be functions of ω. If this is the case, there is another velocity u known as the group velocity. This is essentially the speed of the energy centroid along z of a wave packet made up of the sum of (integral over) a spectrum of waves of many frequencies. Equation (1) would have A 0 = Α 0 (ω) and k = k(oe) to feed into the integral. When k = k(oe the medium is called dispersive. Then u φ v, and v = ν(ώ). There are ways to measure both u and v. These are covered in another book, in this series (Papadakis, 1976a) and elsewhere. The Phase Velocity is dco/dk, which reduces to u= V^T-& v df Both velocities can be measured by continuous wave methods involving the measurement of phase Φ (cycles) contained in a wave path from y 1 to y 2 as frequency is varied. The group delay t g (time for the centroid of a wave packet to go from y l to y 2 ) is άΦ/df, the slope of phase versus frequency at frequency /, while the phase delay ί φ is Φ// where Φ is the total amount of phase cranked into the wave path from zero frequency up to the frequency of interest / Methods and circuits for making these measurements were given earlier (Papadakis, 1966a; 1976a). They will not be pursued further in this chapter. 1.3. SPECIALIZATION IN THIS CHAPTER This chapter will be specialized to the accurate measurement of Phase Velocity by pulse methods.
eBook - PDF- 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 ; 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 .
eBook - PDF- 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.
- Srinivasan Gopalakrishnan(Author)
- 2022(Publication Date)
- CRC Press(Publisher)
Dispersion 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.3.1CONCEPT OF WAVENUMBER, GROUP SPEEDS AND PHASE SPEEDS
A wave propagating in a medium can be represented asu ( x , t ) =(3.1)ei ( k x − ω t )Eqn. (3.1) is an alternate way of representing 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. (3.1)i ( k x − ω t )is called the phase of the wave. If the wave moves with constant phase, then we have(3.2)= 0 , ord ( i ( k x − ω t )d t=d xd tC p=ω kIn the above equation, Cprepresents the speed of the wave that moves with constant phase and hence it is called Phase speed. From the point of view of sending information, these waves are not useful. They are the same throughout the time and the space. Some quantities must therefore be modulated, such as frequency or amplitude, in order to convey information. The resulting wave may be a perturbation that acts over a short distance, which is called a wave packet
eBook - PDF- 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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