TABLE OF CONTENTS For a propagating mode we obtain from (7.1a) for a wave propagating in the positive z-direction the electric field intensity
| (7.2) |  |
The velocity by which a plane of constant phase is propagating is called the phase velocity. We obtain the phase velocity by setting the exponential term in (7.2) constant
| (7.3) |  |
A harmonic electromagnetic wave exhibits a phase velocity, which in general depends on frequency. The frequency dependence of the phase velocity may be caused by the geometric properties of the waveguides as well as by the frequency dependence of the permittivity and permeability of the material filling the waveguide. A wave packet as depicted in Figure 7.2 may be considered a superposition of harmonic waves. The electric field of such a wave packet may be described by
| (7.4) |  |
We assume the phase coefficient ?( ?) to be frequency-dependent. Considering a narrowband wave packet, the phase term ?t - ?( ?) z may be expanded in a certain frequency interval around the center frequency ? 0 into a Taylor series,
| (7.5) |  |
After inserting (7.5) into (7.4), we obtain
| (7.6) |  |
The exponential term describes a harmonic wave propagating in the z-direction with an angular frequency ? 0. This harmonic wave propagates according to (7.3) with a phase velocity c. The integral in (7.6) describes the envelope of the wave. Setting t- (d ?/d ?) z constant we obtain the velocity
