(Equation 3.25 )

Figure 3.16. Snapshots of a traveling wave at two instants of time, t

Figure 3.17. The variation of E at one point in space as a function of time.

f = 1/T (Equation 3.26 )

(Equation 3.27 )

where T is the period of the function and t(a) A given periodic function [ f ( t ) ] versus time (t).

(b) The square of the function [ f

(Equation 3.28 )

where g(Equation 3.29 )

In free space, v is equivalent to the speed of light (c). In a dielectric material the velocity of the wave is slower than that of free space.

A spherical wave. The wave fronts are spherical surfaces. The wave propagates radially outward in all directions.

- The wave fronts are spheres.
**E**,**B**, and the direction of propagation (**k**) are all mutually perpendicular.- E/H = (called the wave impedance). For free space, E/H = 377 ohms. For the sinusoidal steady-state fields, the wave impedance, , is a complex number that includes losses in the medium in which the wave is traveling.
- Both
**E**and**H**vary as 1/r, where r is the distance from the source. - Velocity of propagation is given by v = 1 / .The velocity is less and the wavelength is shorter for a wave propagating in matter than for one propagating in free space. For sinusoidal steady-state fields, the phase velocity is the real part of the complex number 1 / . The imaginary part describes attenuation of the wave caused by losses in the medium.

- The wave fronts are planes.
**E**,**H**, and the direction of propagation (**k**) are all mutually perpendicular.- E/H = (called the wave impedance). For free space, E/H = 377 ohms. For the sinusoidal steady-state fields, the wave impedance, , is a complex number that includes losses in the medium in which the wave is traveling.
**E**and**H**are constant in any plane perpendicular to**k**.- Velocity of propagation is given by v = 1/ The velocity is less and the wavelength is shorter for a wave propagating in matter than for one propagating in free space. For sinusoidal steady-state fields, the phase velocity is the real part of the complex number 1/ . The imaginary part describes attenuation of the wave caused by losses in the medium.

Figure 3.20 shows a planewave. **E** and **H** could have any directions in the plane as long as they are perpendicular to each other. Far away from its source, a spherical wave can be considered to be approximately a planewave in a limited region of space, because the curvature of the spherical wavefronts is so small that they appear to be almost planar. The source for a true planewave would be a planar source, infinite in extent.

A planewave.

electric circuit theory (Kirchhoff's laws) | |

microwave theory or electromagnetic-field theory | |

optics or ray theory |

d = 2 L^{2} / (Equation 3.30)

where

The boundary between the near-field and far-field regions is not sharp because the near fields gradually become less important as the distance from the source increases.

Cross-sectional views of the electric- and magnetic-field lines in the TEM mode for coaxial cable and twin lead.

Schematic diagrams of two-conductor transmission lines.

Total waves, incident plus reflected.

(b) Total voltage as a function of position for various times through a full cycle, and the envelope of the standing wave.

(c) Total current as a function of position at various times through a full cycle, and the envelope of the standing wave.

Top half of the envelope resulting from an incident and reflected voltage wave.

The values of S range from unity to infinity. For the standing wave shown in Figure 3.23(b) ,

S = . A wave pattern is called a standing wave only when nodes exist, so the minimum value of the sinusoid is zero.

where is the magnitude of the *reflection
coefficient*--the ratio of the reflected wave's magnitude to
the incident wave's magnitude. For a terminated transmission
line (the load impedance is equal to the characteristic
impedance), the reflection coefficient is zero and the
standing-wave ratio is unity.

Field variation of the TE

(a) as would be seen looking down the waveguide and

(b) as seen looking at the side of the waveguide. The solid lines are electric field and the dotted lines are magnetic field.

where

The cutoff frequency for the TE

The relative cutoff frequencies for a few modes are shown in Figure 3.26. Both m and n cannot be zero for any mode, because that would require *all *the fields to be zero. For the same reason, neither m nor n can be zero for the TM modes.

Some relative cutoff frequencies for a waveguide with b = a/2, normalized to that of the TE

**Go to Chapter 3.3**

**Return to Table of Contents.**

Last modified: June 24, 1997

© October 1986, USAF School of Aerospace Medicine, Aerospace Medical Division (AFSC), Brooks Air Force Base, TX 78235-5301