Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Dispersion

The phase velocity is related to the phase constant by $v_{p} = ω / β$ . In a lossless dielectric $β = ω ε μ$ and the phase velocity is a constant. In some media, however, $β$ is not a linear function of $ω$ , and different frequencies travel with different velocities: the plasma, Eq. (6.2.40), and the good conductor, Eq. (6.2.25), are two examples. A signal that carries information always occupies a band of frequencies. If its component frequencies travel with different velocities, the signal is distorted as it travels. This phenomenon is called dispersion, and a medium in which it occurs is a dispersive medium.

The behaviour of a medium is best displayed in the dispersion diagram, $ω$ against $β$ (Figure 6.2.14). A lossless dielectric is a straight line through the origin. In a plasma, $ω^{2} = ω_{p}^{2} + c^{2} β^{2}$ is a hyperbola that starts at the cutoff $ω_{p}$ and approaches the light line $ω = c β$ . The phase velocity at an operating point is the slope of the secant from the origin, $ω / β$ . The slope of the tangent, $d ω / d β$ , is the group velocity defined below.

The dispersion diagram of a collisionless plasma, \omega^{2}=\omega_p^{2}+c^{2}\beta^{2}, with the secant and the tangent at the operating point c\beta/\omega_p=1.2. Here v_p=1.30\,c and v_g=0.77\,c. A lossless dielectric would be a straight line through the origin, with v_g=v_p. — **Figure 6.2.14.** The dispersion diagram of a collisionless plasma, $ω^{2} = ω_{p}^{2} + c^{2} β^{2}$ , with the secant and the tangent at the operating point $c β / ω_{p} = 1.2$ . Here $v_{p} = 1.30 c$ and $v_{g} = 0.77 c$ . A lossless dielectric would be a straight line through the origin, with $v_{g} = v_{p}$ .

The wave packet

An electromagnetic wave carrying information usually consists of a small spread of frequencies, the side bands, loaded on a high carrier frequency. Together they form a wave packet, a group of frequencies, and the velocity of this group is called the group velocity. The simplest packet is made of two waves of equal amplitude at the angular frequencies $ω_{0} \pm Δ ω$ , with the phase constants $β_{0} \pm Δ β$ and $Δ ω ≪ ω_{0}$ . With $cos A + cos B = 2 cos \frac{A - B}{2} cos \frac{A + B}{2}$ ,

E (z, t) = E_{0} cos [(ω_{0} + Δ ω) t - (β_{0} + Δ β) z] + E_{0} cos [(ω_{0} - Δ ω) t - (β_{0} - Δ β) z] = slow envelope 2 E_{0} cos (Δ ω t - Δ β z) \times fast carrier cos (ω_{0} t - β_{0} z) . (6.2.48)

A point of constant carrier phase, $ω_{0} t - β_{0} z =$ const., moves with $ω_{0} / β_{0} = v_{p}$ . A point of constant envelope phase, $Δ ω t - Δ β z =$ const., moves with $Δ ω /Δ β$ , and in the limit $Δ ω \to 0$ this becomes a derivative.

Group velocity

v_{g} = \frac{1}{d β / d ω} = \frac{d ω}{d β}, while v_{p} = \frac{ω}{β} . (6.2.49)

In plain words

The crests of the carrier move with $v_{p}$ , the envelope that carries the message moves with $v_{g}$ . They are equal only when $β$ is proportional to $ω$ .

**Figure 6.2.15.** The two-tone wave packet of Eq. (6.2.48) at one instant: a fast carrier inside a slow envelope (dashed). In a dispersive medium the two move with different speeds.

Group velocity in a plasma

For a plasma, $β = \frac{ω}{c} 1 - ω_{p}^{2} / ω^{2} = \frac{1}{c} ω^{2} - ω_{p}^{2}$ , so

\frac{d β}{d ω} = \frac{1}{c} \frac{ω}{ω ^{2} - ω _{p}^{2}} = \frac{1}{c 1 - ω _{p}^{2} / ω ^{2}} . (6.2.50)

Phase and group velocity in a plasma

v_{p} v_{p} v_{g} = \frac{c}{1 - ω _{p}^{2} / ω ^{2}}, v_{g} = c 1 - \frac{ω _{p}^{2}}{ω ^{2}}, = c^{2} . (6.2.51)

In plain words

In a plasma the crests run faster than light and the envelope slower, by the same factor, and the product of the two speeds is always $c^{2}$ .

It seems that the phase velocity exceeds the speed of light, but this does not violate special relativity. A single-frequency plane wave is an idealisation: it has no beginning and no end and carries no message. Real signals are sums of many frequencies travelling as a group, and once single-frequency plane waves are understood, the group of them that carries the information can be understood too. Energy and information travel with $v_{g}$ , and in the plasma $v_{g} < c$ always (Figure 6.2.16). Near the cutoff $v_{g} \to 0$ and a pulse crawls. An HF pulse crossing the ionosphere is therefore delayed (the group delay of Problem 6.2.3), while its carrier phase advances. Dual-frequency GPS receivers remove the resulting range error through its $1/ f^{2}$ dependence (Exercise 6.2.8).

Phase and group velocity in a collisionless plasma. The phase velocity exceeds c and diverges at cutoff, the group velocity stays below c and vanishes there. Far above f_p both approach c. — **Figure 6.2.16.** Phase and group velocity in a collisionless plasma. The phase velocity exceeds $c$ and diverges at cutoff, the group velocity stays below $c$ and vanishes there. Far above $f_{p}$ both approach $c$ .

Move $f$ below towards $f_{p}$ and watch the envelope fall behind while the crests race ahead. The layer is that of Problem 6.2.3, $200 km$ thick with $f_{p} = 6 MHz$ , crossed by two tones $5 kHz$ either side of $f_{0} = 10 MHz$ .

InteractiveLoading simulation…

Common mistake · Faster than light

Concluding from $v_{p} > c$ that something travels faster than light, or that $v_{p} = c^{2} / v_{g}$ breaks relativity. Nothing physical moves with a crest: in a packet the crests are born at its back, run through it and die at its front (Problem 6.2.3). Energy and information move with the envelope, at $v_{g} < c$ .

Quick check

In a plasma $v_{p} = 1.25 c$ at some frequency. What is $v_{g}$ , and what is $f / f_{p}$ ?

$v_{g} = c^{2} / v_{p} = 0.8 c$ . From $1 - f_{p}^{2} / f^{2} = 0.8$ , $f_{p} / f = 0.6$ and $f / f_{p} = 1.67$ , exactly the layer of Problem 6.2.3.

A good conductor is dispersive too. There $β = ω μ σ /2$ grows only as $ω$ , and Home exercise 6.2.3 shows that $v_{g} = 2 v_{p}$ : the group is faster than the phase. A medium with $v_{g} < v_{p}$ , such as the plasma, is called normally dispersive, one with $v_{g} > v_{p}$ anomalously dispersive.

Table 6.2.3. The applications of this section.

Application	Physics used	Worked out in
Talking to a submarine	$σ / ω ε ≫ 1$ , $δ = 1/ π f μ σ$ , $8.686 d / δ$ dB	Problems 6.2.1, 6.2.4, Worked example 6.2.2
Ionospheric HF skywave	$ε_{p} = ε_{0} (1 - f_{p}^{2} / f^{2})$ , $f_{MUF} = f_{o} F_{2} sec θ_{i}$	Problems 6.2.2, 6.2.3, Home exercise 6.2.2

6.2.7 Group velocity

Dispersion

The wave packet

Group velocity in a plasma