7.3.9 TE₁₀ at a glance, and what comes next

cutoff	$f_{co,10}=v/2a$, ${\lambda }_c=2a$
propagation	$\tau =k\sqrt{1-(f_c/f{)}^2}$, ${\lambda }_g=2\pi /\tau =\lambda /\sqrt{1-(f_c/f{)}^2}$
velocities	$v_p=v/\sqrt{1-(f_c/f{)}^2}$, $v_g=v\sqrt{1-(f_c/f{)}^2}$
impedance	$Z_{TE}=\eta /\sqrt{1-(f_c/f{)}^2}$
fields	$E_y=E_0\sin \frac{\pi x}{a}{\mathrm{e}}^{-\mathrm{j}\tau z}$, $H_x=-E_y/Z_{TE}$, $H_z=\mathrm{j}\frac{\pi E_0}{\omega \mu a}\cos \frac{\pi x}{a}{\mathrm{e}}^{-\mathrm{j}\tau z}$
power	$P=|E_0{|}^{2}ab/4Z_{TE}$
loss	${\alpha }_c=\frac{R_s}{\eta b\sqrt{1-(f_c/f{)}^2}}[1+\frac{2b}{a}(f_c/f{)}^2]$

The higher modes have $f_{co,mn}=\frac{v}{2}\sqrt{(m/a{)}^2+(n/b{)}^2}$, TE with $m$ or $n$ zero allowed, TM with $m,n\ge 1$. Section 7.4 applies the same recipe in cylindrical coordinates to a hollow circular pipe of radius $a$ (Figure 7.3.14). Bessel functions $J_m(h\rho )$ replace the sines, and the walls fix $h$ through the zeros of $J_m$ (TM) or of $J_m^{\prime }$ (TE). The dominant mode there is TE${}_{11}$, with $f_c=1.841v/2\pi a$. Section 7.5 then turns to the coaxial guide, which, having two conductors, carries a TEM wave again.