# Chapter 7 Plane-Wave Propagation

# Objectives

Upon learning the material presented in this chapter, you should be able to:

1. Describe mathematically the electric and magnetic fields of TEM waves.

2. Describe the polarization properties of an EM wave.

3. Relate the propagation parameters of a wave to the constitutive parameters of the medium.

4. Characterize the flow of current in conductors and use it to calculate the resistance of a coaxial cable.

5. Calculate the rate of power carried by an EM wave in both lossless and lossy media.

# Unbounded EM Waves

It was established in Chapter 6 that a time-varying electric field produces a magnetic field and, conversely, a time-varying magnetic field produces an electric field. This cyclic pattern often results in electromagnetic (EM) waves propagating through free space and in material media. When a wave propagates through a homogeneous medium without interacting with obstacles or material interfaces, it is said to be unbounded. Light waves emitted by the sun and radio transmissions by antennas are good examples. Unbounded waves may propagate in both lossless and lossy media. Waves propagating in a lossless medium (e.g., air and perfect dielectrics) are similar to those on a lossless transmission line in that they do not attenuate. When propagating in a lossy medium (material with nonzero conductivity, such as water), part of the power carried by an EM wave gets converted into heat. A wave produced by a localized source, such as an antenna, expands outwardly in the form of a spherical wave, as depicted in Fig. 7-1(a). Even though an antenna may radiate more energy along some directions than along others, the spherical wave travels at the same speed in all directions. To an observer very far away from the source, however, the wavefront of the spherical wave appears approximately planar, as if it were part of a uniform plane wave with identical properties at all points in the plane tangent to the wavefront (Fig. 7-1(b)). Plane waves are easily described using a Cartesian coordinate system, which is mathematically easier to work with than the spherical coordinate system needed to describe spherical waves.

# Figure 7-1 Waves radiated by an EM source, such as a light bulb or an antenna, have spherical wavefronts, as in (a); to a distant observer, however, the wavefront across the observer’s aperture appears approximately planar, as in (b).

(a) Spherical wave

(b) Plane-wave approximation

Figure 7 1 Full Alternative Text

When a wave propagates along a material structure, it is said to be guided. The Earth’s surface and ionosphere constitute parallel boundaries of a natural structure capable of guiding short-wave radio transmissions in the HF band* (3 to 30 MHz); indeed, the ionosphere is a good reflector at these frequencies, thereby allowing the waves to zigzag between the two boundaries (Fig. 7-2). When we discussed wave propagation on a transmission line in Chapter 2, we dealt with voltages and currents. For a transmission-line circuit such as that shown in Fig. 7-3, the ac voltage source excites an incident wave that travels down the coaxial line toward the load, and unless the load is matched to the line, part (or all) of the incident wave is reflected back toward the generator. At any point on the line, the instantaneous total voltage v(z,t) is the sum of the incident and reflected waves—both of which vary sinusoidally with time. Associated with the voltage difference between the inner and outer conductors of the coaxial line is a radial electric field E(z,t) that exists in the dielectric material between the conductors, and since v(z,t) varies sinusoidally with time, so does E(z,t). Furthermore, the current flowing through the inner conductor induces an azimuthal magnetic field H(z,t) in the dielectric material surrounding it. These coupled fields, E(z,t) and H(z,t), constitute an electromagnetic wave. Thus, we can model wave propagation on a transmission line either in terms of the voltages across the line and the currents in its conductors or in terms of the electric and magnetic fields in the dielectric medium between the conductors.

# Figure 7-2 The atmospheric layer bounded by the ionosphere at the top and the Earth’s surface at the bottom forms a guiding structure for the propagation of radio waves in the HF band.

# Figure 7-3 A guided electromagnetic wave traveling in a coaxial transmission line consists of time-varying electric and magnetic fields in the dielectric medium between the inner and outer conductors.

Figure 7 3 Full Alternative Text

In this chapter, we focus our attention on wave propagation in unbounded media. Unbounded waves have many practical applications in science and engineering. We consider both lossless and lossy media. Even though strictly speaking uniform plane waves cannot exist, we study them in this chapter to develop a physical understanding of wave propagation in lossless and lossy media. In Chapter 8, we examine how both planar and spherical waves are reflected by and transmitted through boundaries between dissimilar media. The proc esses of radiation and reception of waves by antennas are treated in Chapter 9.