# 8.10 Propagation Velocities

When a wave is used to carry a message through a medium or along a transmission line, information is encoded into the wave’s amplitude, frequency, or phase. A simple example is shown in Fig. 8-25, where a high-frequency sinusoidal wave of frequency f is amplitude-modulated by a low-frequency Gaussian pulse. The waveform in Fig. 8-25(b) is the result of multiplying the Gaussian pulse shape in Fig. 8-25(a) by the carrier waveform.

# Figure 8-25 The amplitude-modulated high-frequency waveform in (b) is the product of the Gaussian-shaped pulse with the sinusoidal high-frequency carrier in (a).

By Fourier analysis, the waveform in Fig. 8-25(b) is equivalent to the superposition of a group of sinusoidal waves with specific amplitudes and frequencies. Exact equivalence may require a large (or infinite) number of frequency components, but in practice, it is often possible to represent the modulated waveform to a fairly high degree of fidelity with a wave group that extends over a relatively narrow bandwidth surrounding the high-frequency carrier f. The velocity that the envelope—or equivalently the wave group—travels through the medium is called the group velocity u g. As such, u g is the velocity of the energy carried by the wave group and of the information encoded in it. Depending on whether or not the propagation medium is dispersive, u g may or may not be equal to the phase velocity u p. In Section 2-1.1, we described a dispersive transmission line as one “on which the phase velocity is not a constant as a function of frequency,” and as a result the shape of a pulse transmitted through it gets progressively distorted as it moves down the line. A rectangular waveguide constitutes a dispersive transmission line because the phase velocity of a TE or TM mode propagating through it is a strong function of frequency [per Eq. (8.108)], particularly at frequencies close to the cutoff frequency f m n. As we see shortly, if f≫f m n, the TE and TM modes become approximately TEM in character—not only in terms of the directional arrangement of the electric and magnetic fields but also in terms of the frequency dependence of the phase velocity.

We now examine u p and u g in more detail. The phase velocity defined as the velocity of the sinusoidal pattern of the wave is given by

u p=ω β, (8.112)

while the group velocity u g is given by

u g=1 d β/d ω. (8.113)

Even though we will not derive Eq. (8.113) in this book, it is nevertheless important that we understand its properties for TE and TM modes in a metal waveguide. Using the expression for β given by Eq. (8.107),

where—as before—u p 0 is the phase velocity in an unbounded dielectric medium. In view of Eq. (8.108) for the phase velocity u p,

Above cutoff (f>f m n), u p≥u p 0 and u g≤u p 0. As f→∞, or more precisely as (f m n/f)→0, TE and TM modes approach the TEM case for which u p=u g=u p 0.

A useful graphical tool for describing the propagation properties of a medium or transmission line is the ω-β diagram. In Fig. 8-26, the straight line starting at the origin represents the ω-β relationship for a TEM wave propagating in an unbounded medium (or on a TEM transmission line). The TEM line provides a reference to which the ω-β curves of the TE/TM modes can be compared. At a given location on the ω-β line or curve, the ratio of the value of ω to that of β defines u p=ω/β, whereas it is the slope d ω/d β of the curve at that point that defines the group velocity u g. For the TEM line, the ratio and the slope have identical values (hence, u p=u g), and the line starts at ω=0. In contrast, the curve for each of the indicated TE/TM modes starts at a cutoff frequency specific to that mode below which the waveguide cannot support the propagation of a wave in that mode. At frequencies close to cutoff, u p and u g assume very different values; in fact, at cutoff u p=∞ and u g=0. On the other end of the frequency spectrum—at frequencies much higher than f m n—the ω-β curves of the TE/TM modes approach the TEM line. We should note that for TE and TM modes, u p may easily exceed the speed of light, but u g will not. Since it is u g that represents the actual transport of energy, Einstein’s assertion that there is an upper bound on the speed of physical phenomena is not violated.

# Figure 8-26 ω-β diagram for TE and TM modes in a hollow rectangular waveguide. The straight line pertains to propagation in an unbounded medium or on a TEM transmission line.

Figure 8 26 Full Alternative Text

So far, we have described the fields in the guide, but we have yet to interpret them in terms of plane waves that zigzag along the guide through successive reflections. To do just that, consider the simple case of a TE 10 mode. For m=1 and n=0, the only nonzero component of the electric field given by Eq. (8.110) is E˜y, so

E˜y=−j ω μ k c 2 π a H 0 sin π x a e−j β z. (8.116)

Using the identity sin θ=(e j θ−e−j θ)/2 j for any argument θ, we obtain

E˜y=ω μ π H 0 2 k c 2 a(e−j π x/a−e j π x/a)e−j β z=E 0′(e−j β(z+π x/β a)−e−j β(z−π x/β a))=E 0′(e−j β z′−e−j β z′′), (8.117)

where we have consolidated the quantities multiplying the two exponential terms into the constant E 0′. The first exponential term represents a wave with propagation constant β traveling in the z′ direction, where

z′=z+π x β a, (8.118a)

and the second term represents a wave traveling in the z′′ direction with

z′′=z−π x β a. (8.118b)

From the diagram shown in Fig. 8-27(a), it is evident that the z′ direction is at an angle θ′ relative to z, and the z′′ direction is at an angle θ′′=−θ′. This means that the electric field E˜y (and its associated magnetic field H˜) of the TE 10 mode is composed of two TEM waves shown in Fig. 8-27(b) with both traveling in the +z direction by zigzagging between the opposite walls of the waveguide. Along the zigzag directions (z′ and z′′), the phase velocity of the individual wave components is u p 0, but the phase velocity of the combination of the two waves along z is u p.

# Figure 8-27 The TE 10 mode can be constructed as the sum of two TEM waves.

(a) z' and z′′ propagation directions

Figure 8 27-a Full Alternative Text

(b) TEM waves

Figure 8 27-b Full Alternative Text

# Example 8-10: Zigzag Angle

For the TE 10 mode, express the zigzag angle θ′ in terms of the ratio (f/f 10), and then evaluate it at f=f 10 and for f≫f 10.

Solution: From Fig. 8-27,

θ 10′=tan−1 π β 10 a,

where the subscript 10 has been added as a reminder that the expression applies to the TE 10 mode specifically. For m=1 and n=0, Eq. (8.106) reduces to f 10=u p 0/2 a. After replacing β with the expression given by Eq. (8.107) and replacing a with u p 0/2 f 10, we obtain

θ′=tan−1 1(f/f 10)2−1.

At f=f 10, θ′=90°, which means that the wave bounces back and forth at normal incidence between the two side walls of the waveguide, making no progress in the z direction. At the other end of the frequency spectrum, when f≫f 10, θ′ approaches 0 and the wave becomes TEM-like as it travels straight down the guide.

Concept Question 8-12: For TE waves, the dominant mode is TE 10, but for TM the dominant mode is TM 11. Why is it not TM 10?

Concept Question 8-13: Why is it acceptable for u p to exceed the speed of light c, but not so for u g?