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Question 9.29: Consider a rectangular wave guide with dimensions 2.28 cm × ...

Consider a rectangular wave guide with dimensions 2.28 cm × 1.01 cm. What TE modes will propagate in this wave guide, if the driving frequency is 1.70 × 10^{10} Hz? Suppose you wanted to excite only one TE mode; what range of frequencies could you use? What are the corresponding wavelengths (in open space)?

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\text { Here } a=2.28 cm \text { and } b=1.01 cm , \text { so } \nu_{10}=\frac{1}{2 \pi} \omega_{10}=\frac{c}{2 a}=0.66 \times 10^{10} Hz ; \nu_{20}=2 \frac{c}{2 a}=1.32 \times 10^{10} Hz;

\nu_{30}=3 \frac{c}{2 a}=1.97 \times 10^{10} Hz ; \nu_{01}=\frac{c}{2 b}=1.49 \times 10^{10} Hz ; \nu_{02}=2 \frac{c}{2 b}=2.97 \times 10^{10} Hz ; \nu_{11}=\frac{c}{2} \sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}= 1.62 \times 10^{10} Hz . \text { Evidently just four modes occur: } 10,20,01, \text { and } 11 .

To get only one mode you must drive the waveguide at a frequency between \nu_{10} \text { and } \nu_{20} :

0.66 \times 10^{10}<\nu<1.32 \times 10^{10} Hz . \quad \lambda=\frac{c}{\nu}, \text { so } \lambda_{10}=2 a ; \lambda_{20}=a . \quad 2.28 cm <\lambda<4.56 cm .

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