Exercise 7.2.4: Two modes beating along the guide

$f_{co,1}=c/2b=12\mathrm{GHz}$ and $f_{co,2}=24\mathrm{GHz}$: TEM, TM${}_1$ and TE${}_1$ propagate (the probe does not excite TE${}_1$, which has no $E_y$). $k=2\pi f/c=418.9rad/m$, $s=\sqrt{1-0.6^2}=0.8$ and ${\tau }_1=0.8k=335.1rad/m$.

On the plates $E_y(0,z)=E_1({\mathrm{e}}^{-\mathrm{j}kz}+{\mathrm{e}}^{-\mathrm{j}{\tau }_{1}z})$ and $E_y(b,z)=E_1({\mathrm{e}}^{-\mathrm{j}kz}-{\mathrm{e}}^{-\mathrm{j}{\tau }_{1}z})$, so

The field sits on the bottom plate at $z=0$, $7.5$ and $15\mathrm{cm}$ and on the top plate at $z=3.75$ and $11.25\mathrm{cm}$.

TEM: $P^{\prime }=b{E}_1^2/(2{\eta }_0)$. TM${}_1$ with peak transverse field $E_1$: $P^{\prime }=b{E}_1^2/(4Z_{TM})$ with $Z_{TM}=0.8{\eta }_0$. The ratio is $P_{TM}^{\prime }/P_{TEM}^{\prime }={\eta }_0/(2Z_{TM})=1/(2s)=0.625$.

At $22\mathrm{GHz}$: $\lambda =1.364\mathrm{cm}$, $s=\sqrt{1-(12/22{)}^2}=0.838$, $L_B=1.364/0.162=8.42\mathrm{cm}$.

$|E_y(0,z){|}^2+|E_y(b,z){|}^2=4E_1^2$ for every $z$: the field moves from plate to plate, but the two modes are orthogonal and carry their powers independently, so the total power does not change along the guide.