Lowest Mode of Rectangular Wave Guide

Consider a rectangular wave guide, infinitely long in the direction, with a width (x direction) of a and a height (y direction) of b (a > b) (see Figure 1.1). The walls are perfect conductors.

Figure 1.1

a) What are the boundary conditions on the components of B and E at the walls?

b) Write the wave equation which describes the E and B fields of the lowest mode.

Hint: The lowest mode has the electric field in the y direction only.

c) For the lowest mode that can propagate, find the phase velocity and the group velocity.

d) The possible modes of propagation separate naturally into two classes. What are these two classes and how do they differ physically?

SOLUTION

a) Because the walls are perfectly conducting, we have for E and B the boundary conditions

where n is normal to the wall, or in terms of E_z (B_z) (z is the direction of wave propagation)

(1)

b) Starting from the sourceless Maxwell equations in vacuum

(2)

(3)

(4)

(5)

and substituting E (r, t) = E (r) exp (-ωt) (same for B), we obtain

(6)

(7)

(8)

(9)

The field dependence on may be written in the form f(z) = exp [i (kz – ωt)], where k is the wave vector for the wave transmitted in the z direction. Using the fact that the electric field of the lowest mode is in the y direction only, we have, from (6)–( 7),

(10)

(11)

(12)

(13)

From (12), B_z = B (x) exp (ikz), and substituting (10) into (13), we obtain

(14)

(15)

where

(16)

Using  and B_x from (14), we get a differential equation for B_z

(17)

or

(18)

The solution of this equation satisfying the boundary conditions

is B_z = B₀ cos γx with γ = π/a. So the field in the wave guide in this mode from (14)– (15)

(19)

(20)

(21)

c) The dispersion relation for the lowest mode is found from (16):

(22)

The phase velocity v is

(23)

The group velocity u is

(24)

d) The waves propagating in the wave guides can be divided into two classes: TE (transverse electric, E_z = 0) as is the case in this problem, and TM (transverse magnetic, B_z = 0).