Problem 7.2.3: TM_1 parameters and the zig-zag

Air: $v=c$, $\eta ={\eta }_0=120\pi \Omega$. $b=0.015\mathrm{m}$, $f=15\mathrm{GHz}$, $\lambda =2\mathrm{cm}$, $k=100\pi =314.2rad/m$.

$f_{co,1}=\frac{3\times 10^8}{2(0.015)}=10\mathrm{GHz}$, $f_{co,2}=20\mathrm{GHz}$. Propagating: TEM, TM${}_1$ and TE${}_1$.

$f_{co,1}/f=2/3$, $s=\sqrt{1-4/9}=\sqrt{5}/3=0.745$:

The two plane waves have $|k|=k$ and the transverse component $h_1=\pi /b$, so $\sin \theta =h_1/k=f_{co,1}/f=2/3$ and $\theta =41.8^{\circ }$ (check: $\cos \theta =0.745=s$).

A burst travels at the group velocity:

The zig-zag path is longer than the axis by $1/\cos \theta =1.342$, which is exactly the ratio $4.47/3.33$.

$v_p{v}_g=4.02\times 2.24\times 10^{16}=9.0\times 10^{16}=c^2$ and $Z_{TM}{Z}_{TE}=281.0\times 505.8=1.42\times 10^5={\eta }_0^2$.