1. A rectangular, air filled  waveguide has dimension 2 cm x 1 cm. For what range of frequencies, there is a “single mode” operation in the guide?.

The cutoff frequency for (m,n) mode is . Substituting values, the cutoff for   (1,0) mode is 7.5 GHz and for (2,0) is 15 GHz. The frequency for (0,1) is also 15 GHz. All other modes have higher cut off. Thus in order that only one mode propagates, the operating frequency should be in the range  GHz. In this range, only TE₁₀ mode operates. Recall that there is no TM mode with either of the indices being zero.

2. The cutoff frequency for a TE10 mode in an air filled waveguide is 1.875 GHz. What would be the cutoff frequency of this mode if the guide were to be filled with a dielectric of permittivity ?

The cutoff frequency is proportional to . Thus the frequency would be reduced by a factor of 3 making it 625 MHz.

3. In an air filled waveguide of dimension 2 cm x 2 cm, the x component of the electric field is given by

Identify the propagating mode, determine  the frequency of operation  and  find  and .

Since one of the standing wave factor is missing, one of the mode indices is zero. Thus it is a TE mode. The electric field of TEmn mode is given by  (with a=b)

Thus the mode is TE₀₂ mode.  From the relation . Given

the operating frequency is 15.74 GHz.

Since  . Further, by definition for TE mode

Here we have . The ratio

so that