forming the backward wave.
Figure 3.23 Description of phase and group velocities of a forward‐moving modulated wave.
The carrier and envelope are combined to form a unified wave structure called the wave‐packet. In the case of normal dispersion, the group velocity is the energy velocity of a signal and the information travels with the group velocity [B.1, B.4, B.5, B.7, B.14, B.16]. However, in the case of anomalous dispersion, the energy velocity and group velocity are different. In this case, group velocity is not velocity of information. Moreover, the concept of group velocity applies only to a narrow‐band wave‐packet, not to the wideband signal. The controversy exists at present on the travel of information with a velocity more than the velocity of light [J.7].
Formation of Two‐Frequency Wave‐Packet
A wave‐packet is formed by a linear combination of two signals of equal magnitude with a small difference in angular frequency and phase constant. It is shown in Fig (3.24). The composite voltage wave is given by
Figure 3.24 Formation of a wave‐packet.
(3.3.9)
The carrier wave has frequency ω0 and propagation constant β0. The above expression applies to a narrow‐band signal, Δω << ω0. The envelope frequency is Δω and its propagation constant is Δβ. The carrier wave inside the envelope, shown in Fig (3.23), moves with phase velocity:
(3.3.10)
The velocity of the envelope, i.e. the group velocity, is obtained from the constant phase point on the envelope
(3.3.11)
Both the phase velocity and group velocity are represented over the (ω − β) dispersion diagram of the wave‐supporting medium. The (ω − β) diagram plays a very significant role in the wave propagation in periodic and metamaterial media. These are discussed in the chapters 18–22. Figure (3.25) shows the dispersion relation given by equation (3.3.8a). The point P on the dispersion diagram locates ( β0, ω0) . The slope ψ of a point P on the dispersion diagram, with respect to the origin, gives the phase velocity of a carrier wave, whereas the local slope ϕ at ω = ω0, i.e. a local tangent at point P, provides the group velocity:
Figure 3.25 ω − β diagram to get phase and group velocities.
(3.3.12)
At the cut‐off frequency ωp, β → 0, results in an infinite extent of the phase velocity, while the group velocity is zero, vg = 0. However, away from the cut‐off frequency, the phase velocity decreases to a limiting value
As the propagation constant increases from β = 0 at ωp to
The group velocity is less than the velocity of the EM‐wave in the free space, vg ≤ c. In summary, the plasma forms a fast‐wave normal dispersive medium that supports the forward wave having the same direction for both the phase and group velocities. Equations (3.3.8d) and (3.3.13c) for the wave propagation in a nonmagnetized plasma with μ = μ0, ε0 and v = c show that the phase and group velocities are related through the following expression:
In equation (3.3.14), the general isotropic medium supports EM‐wave propagation with velocity
A more general relation between the phase and group velocities could be obtained. In the dispersive medium, phase velocity (vp) is a function of frequency; therefore,
Figure (3.26a and b) show the dispersive behaviors of the phase velocity (vp) and propagation constant (β). The expression for the group velocity in a dispersive medium, i.e. a medium with frequency‐dependent refractive index
