Chapter 21 - Superposition - Exercises and Problems - Page 623: 46

\begin{array}{|c|c|} \hline m& f_m\rm(GHz)\\ \hline 7& 10.5\\ \hline 8& 12\\ \hline 9& 13.5\\ \hline 10& 15\\ \hline 11& 16.5\\ \hline 12& 18\\ \hline 13& 19.5\\ \hline \end{array}

From the given figure, we can see that the waves inside the microwave are reflected back and forth since there are two reflectors on both sides (both ends). This is similar to a standing wave in a string that is tied from both ends. Hence the frequency is given by $$f_m=\dfrac{mc}{2L}$$ where $c$ is the speed of the light which is also the speed of the electromagnetic waves. Plugging the known; $$f_m=\dfrac{m(3\times 10^8)}{2(10\times 10^{-2})}=\boxed{1.5\times 10^9\;m}$$ So, for 10 GHz frequency, $m$ is then $$m=\dfrac{10\times 10^9}{1.5\times 10^9}=6.67=\bf 7$$ And for 20 GHz frequency, $m$ is then $$m=\dfrac{20\times 10^9}{1.5\times 10^9}=13.3=\bf 13$$ So, we have about 7 different possible frequencies from $m=7$ to $m=13$, as we see in the table below. \begin{array}{|c|c|} \hline m& f_m\rm(GHz)\\ \hline 7& 10.5\\ \hline 8& 12\\ \hline 9& 13.5\\ \hline 10& 15\\ \hline 11& 16.5\\ \hline 12& 18\\ \hline 13& 19.5\\ \hline \end{array}