Home exercise 7.3.2: Frequency from a slotted line

${\lambda }_g=2\times 2.40=4.80\mathrm{cm}$.

With $\tau =2\pi /{\lambda }_g$, $k=2\pi /\lambda$ and $\pi /a=2\pi /{\lambda }_c$, ${\lambda }_c=2a$:

${\lambda }_c=4.572\mathrm{cm}$, so $1/{\lambda }^2=1/4.80^2+1/4.572^2=0.09124\mathrm{c}{\mathrm{m}}^{-2}$, $\lambda =3.311\mathrm{cm}$ and $f=c/\lambda =9.06\mathrm{GHz}$.

$v_p=c{\lambda }_g/\lambda =3\times 10^8\times 4.80/3.311=4.35\times 10^{8}m/s$ and $v_g=c\lambda /{\lambda }_g=2.07\times 10^{8}m/s$.

At $10\mathrm{GHz}$ $\lambda =3\mathrm{cm}$, $1/{\lambda }_g^2=1/9-1/4.572^2=0.06327\mathrm{c}{\mathrm{m}}^{-2}$ and ${\lambda }_g=3.98\mathrm{cm}$: the minima are $1.99\mathrm{cm}$ apart.

$v_p{v}_g=4.35\times 2.07\times 10^{16}=9.0\times 10^{16}=c^2$, and ${\lambda }_g>\lambda$ as it must be for a waveguide mode.