Page 486 - Engineering Electromagnetics, 8th Edition

468 ENGINEERING ELECTROMAGNETICS

Figure 13.14 The components of the
upward wavevector are κ m and β m , the
transverse and axial phase constants. To
form the downward wavevector, k d , the
direction of κ m is reversed.
x component, κ m ,would be reversed. In Section 12.4, we measured the phase shift
per unit distance along the x and z directions by the components, k x and k z , which
varied continuously as the direction of k changed. In our discussion of waveguides,
we introduce a different notation, where κ m and β m are used for k x and k z . The
subscript m is an integer indicating the mode number. This provides a subtle hint that
β m and κ m will assume only certain discrete values that correspond to certain allowed
directions of k u and k d , such that our coincident phase front requirement is satisfied. 4
From the geometry we see that for any value of m,

2
β m = k − κ 2 m (35)
Use of the symbol β m for the z components of k u and k d is appropriate because β m
will ultimately be the phase constant for the mth waveguide mode, measuring phase
shift per distance down the guide; it is also used to determine the phase velocity of
the mode, ω/β m , and the group velocity, dω/dβ m .
Throughout our discussion, we will assume that the medium within the guide is
lossless and nonmagnetic, so that

ω ωn
r

k = ω µ 0 = = (36)
c c

4 Subscripts (m) are not shown on k u and k d but are understood. Changing m does not affect the
magnitudes of these vectors, only their directions.