7.2.7 The zig-zag picture: two plane waves

Write the TM${}_n$ field with $\sin u=({\mathrm{e}}^{\mathrm{j}u}-{\mathrm{e}}^{-\mathrm{j}u})/2\mathrm{j}$:

Each exponential is a uniform plane wave ${\mathrm{e}}^{-\mathrm{j}k\cdot r}$ (Section 6.1) with the wave vector $k={\tau }_n{a}_z\mp h_n{a}_y$, whose length is $\sqrt{{\tau }_n^2+h_n^2}=k$. The two waves travel at the speed $v$ of the filling, one obliquely upwards and one downwards, and the plates reflect them into each other with the reflection coefficient $-1$ of a perfect conductor (Section 6.4). Their direction makes the angle $\theta$ with the axis (Figure 7.2.8), where

The same holds for TE${}_n$ with $\cos (h_{n}y)$. The TEM mode is the case $\theta =0$: a single plane wave along the axis.

What the picture explains

• Cutoff: $f\to f_{co,n}$ gives $\theta \to 90^{\circ }$. The waves bounce straight up and down and make no progress ($v_g\to 0$).
• Mode condition: going up and down once, the transverse phase $2h_{n}b$ plus two reflections ($\Gamma =-1$, a phase of $\pi$ each) must repeat itself, $2h_{n}b+2\pi =2\pi m$, so $h_{n}b=n\pi$. This is transverse resonance.
• Velocities: energy runs along the rays at $v$, so its progress along the axis is $v_g=v\cos \theta$. The crests of each plane wave cut the axis at the spacing ${\lambda }_g=\lambda /\cos \theta$, so they move along it at $v_p=v/\cos \theta$. Both agree with Eq. (7.1.18) in Section 7.1, since $\cos \theta =\sqrt{1-(f_{co,n}/f{)}^2}$.
• Far above cutoff: $\theta \to 0$, nearly a plane wave along $z$.

The tool below starts with TM${}_1$ at $f=1.5f_{co,1}$, $\theta =41.8^{\circ }$, in the air-filled guide of Problem 7.2.3 ($b=1.5\mathrm{cm}$, $f_{co,1}=10\mathrm{GHz}$). Lower $f$ towards $f_{co,1}$ and watch the rays steepen, ${\lambda }_g$ grow and the dot's progress along $z$, $v_g$, slow to zero.