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# Consider a rectangular wave guide with dimensions $2.28\mathrm{~cm}\times1.01\mathrm{~cm}$. What TE modes will propagate in this wave guide, if the driving frequency is $1.70\times10^{10}\mathrm{~Hz}$? Suppose you wanted to excite only one TE mode; what range of frequencies could you use? What are the corresponding wavelenghts (in open space)?

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For the driving frequency to excite a mode, it needs to be larger than the mode frequency:

$\begin{gather*} \omega>c\pi\sqrt{(m/a)^2+(n/b)^2}=\omega_{mn}\qquad/:2\pi\\ f>\dfrac{c}{2}\sqrt{(m/a)^2+(n/b)^2}=f_{mn}\\ \end{gather*}$

For $a=2.21\:cm$ and $b=1.01\:cm$ we have the mode frequencies:

$\begin{gather*} f_{10}=6.58\cdot10^9\:Hz\tag{m=1,\:n=0}\\ f_{20}=1.32\cdot10^{10}\:Hz\tag{m=2,\:n=0}\\ f_{11}=1.62\cdot10^{10}\:Hz\tag{m=1,\:n=1}\\ f_{01}=1.49\cdot10^{10}\:Hz\tag{m=0,\:n=1}\\ \end{gather*}$

All of the higher combinations have frequencies higher than the driving one, and so do not get excited.

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