Exercise 7.2.2: Which modes does a source excite?

$f_{co,1}=c/2b=5\mathrm{GHz}$, $f_{co,2}=10\mathrm{GHz}$, $f_{co,3}=15\mathrm{GHz}$. At $12\mathrm{GHz}$: TEM, TE${}_1$, TM${}_1$, TE${}_2$, TM${}_2$. With $\lambda =2.5\mathrm{cm}$, $s_1=\sqrt{1-(5/12{)}^2}=0.909$, $s_2=\sqrt{1-(10/12{)}^2}=0.553$:

Overlap $\propto \sin (n\pi /2)$: $1$ for $n=1$, $0$ for $n=2$. Only TE${}_1$ (the source sits on the node of TE${}_2$).

$\sin (\pi /4)=0.707$, $\sin (\pi /2)=1$: TE${}_1$ and TE${}_2$.

TEM: ${\int }_0^{b}dy=b\ne 0$. TM${}_n$: ${\int }_0^b\cos (n\pi y/b)dy=0$ for every $n$. Only the TEM wave.

TEM: $b/2\ne 0$. TM${}_n$: ${\int }_0^{b/2}\cos (n\pi y/b)dy=\frac{b}{n\pi }\sin \frac{n\pi }{2}$: nonzero for $n=1$, zero for $n=2$. TEM and TM${}_1$.

Every source also drives evanescent modes ($n\ge 3$) that it does not cancel by symmetry, for instance TE${}_3$ in (a) ($\sin (3\pi /2)=-1$) and TM${}_3$ in (d). They decay with ${\alpha }_3=(3\pi /b)\sqrt{1-(12/15{)}^2}=188.5Np/m$, $1637dB/m$, and are gone within a few millimetres.