For the guide of Problem \(13.14\), and at the \(32 \mathrm{GHz}\) frequency, determine the difference between the group delays of the highest-order mode (TE or TM) and the TEM mode. Assume a propagation distance of \(10 \mathrm{~cm}\).

We are given a frequency of 32 GHz, which we need to convert into Hz: \(f = 32 \times 10^9 Hz\). To find the phase velocity of a mode, we use the formula: $$ v_p = \frac{c}{\sqrt{1 - (\frac{m}{2\pi a})^2(\frac{c}{f})^2)} $$ where \(c\) is the speed of light (\(3 \times 10^8 m/s\)), \(a\) is the radius of the waveguide, \(f\) is the frequency, and \(m\) is the mode number (0 for TEM mode, maximum number for the highest-order mode). Since we're not given the radius of the waveguide (\(a\)), we will define the phase velocity ratio between the highest-order mode (let's denote \(v_{p,max})\) and the TEM mode (\(v_{p,TEM}\)): $$ \frac{v_{p,max}}{v_{p,TEM}} = \sqrt{1 - (\frac{m}{2\pi a})^2(\frac{c}{f})^2} $$

We now need to calculate the group velocity for both modes, using the formula: $$ v_g = c\sqrt{1 - (\frac{m}{2\pi a})^2(\frac{c}{f})^2} $$ We can simplify the group velocity ratio by noting that: $$ \frac{v_{g,max}}{v_{g,TEM}} = \frac{v_{p,max}}{v_{p,TEM}} $$ From step 1, we have the ratio of phase velocities, which is also the ratio of group velocities between the highest-order mode and the TEM mode.

Now we can calculate the group delay difference between the highest-order mode and the TEM mode that will be experienced over the given propagation distance (10 cm). The group delay for both modes is given by the formula: $$ \tau_g = \frac{d}{v_g} $$ where \(d\) is the propagation distance. Now, we can find the group delay difference (\(\Delta \tau_g\)) directly as: $$ \Delta \tau_g = \frac{d_{max}}{v_{g,max}} - \frac{d_{TEM}}{v_{g,TEM}} $$ However, since the distance is equal for both the highest-order mode and the TEM mode, we can simplify the difference expression to: $$ \Delta \tau_g = d \left(\frac{1}{v_{g,max}} - \frac{1}{v_{g,TEM}}\right) $$ We know from step 2 that the ratio of the group velocities is the same as the ratio of the phase velocities: $$ \frac{v_{g,max}}{v_{g,TEM}} = \frac{v_{p,max}}{v_{p,TEM}} $$ Using this relation, we can evaluate the group delay difference without knowing the actual value of the radius (\(a\)): $$ \Delta \tau_g = d \left(\frac{v_{g,TEM} - v_{g,max}}{v_{g,max} v_{g,TEM}}\right) $$ We can then plug in the given propagation distance, \(d = 10\; cm = 0.1\; m\), and the ratio of the group velocities from step 2: $$ \Delta \tau_g = 0.1 \times \left(\frac{v_{g,TEM} - v_{g,max}}{v_{g,max}v_{g,TEM}}\right) $$ Since we only need to find the difference in group delays, we don't need to compute the actual values of the group velocities. Instead, we can plug in the velocity ratios: $$ \Delta \tau_g = 0.1 \times \left(\frac{1 - \sqrt{1 - (\frac{m}{2\pi a})^2(\frac{c}{f})^2}}{\sqrt{1 - (\frac{m}{2\pi a})^2(\frac{c}{f})^2}}\right) $$ Plugging in the given values of frequency (32 GHz) and the speed of light (\(3 \times 10^8 m/s\)), the expression for \(\Delta \tau_g\) can be evaluated numerically for a specific value of mode number (\(m\)) and waveguide radius (\(a\)).