The cutoff frequency
In solving Eq. (24.16) for $k_z$, there should really be two roots—one plus and one minus. We should write \begin{equation} \label{Eq:II:24:20} k_z=\pm\sqrt{(\omega^2/c^2)-(\pi^2/a^2)}. \end{equation}  The two signs simply mean that there can be waves which propagate with a negative phase velocity (toward $-z$), as well as waves which propagate in the positive direction in the guide. Naturally, it should be possible for waves to go in either direction. Since both types of waves can be present at the same time, there will be the possibility of standing-wave solutions. Our equation for $k_z$ also tells us that higher frequencies give larger values of $k_z$, and therefore smaller wavelengths, until in the limit of large $\omega$, $k$ becomes equal to $\omega/c$, which is the value we would expect for waves in free space. The light we “see” through a pipe still travels at the speed $c$. But now notice that if we go toward low frequencies, something strange happens. At first the wavelength gets longer and longer, but if $\omega$ gets too small the quantity inside the square root of Eq. (24.20) suddenly becomes negative. This will happen as soon as $\omega$ gets to be less than $\pi c/a$—or when $\lambda_0$ becomes greater than $2a$. In other words, when the frequency gets smaller than a certain critical frequency $\omega_c=\pi c/a$, the wave number $k_z$ (and also $\lambda_g$) becomes imaginary and we haven’t got a solution any more. Or do we?  Who said that $k_z$ has to be real? What if it does come out imaginary?  Our field equations are still satisfied. Perhaps an imaginary $k_z$ also represents a wave. Suppose $\omega$ is less than $\omega_c$; then we can write \begin{equation} \label{Eq:II:24:21} k_z=\pm ik', \end{equation}  where $k'$ is a positive real number: \begin{equation} \label{Eq:II:24:22} k'=\sqrt{(\pi^2/a^2)-(\omega^2/c^2)}. \end{equation}  If we now go back to our expression, Eq. (24.12), for $E_y$, we have \begin{equation} \label{Eq:II:24:23} E_y =E_0e^{i(\omega t\mp ik'z)}\sin k_xx, \end{equation}  which we can write as \begin{equation} \label{Eq:II:24:24} E_y =E_0e^{\pm k'z}e^{i\omega t}\sin k_xx. \end{equation}  This expression gives an $\FLPE$-field that oscillates with time as $e^{i\omega t}$ but which varies with $z$ as $e^{\pm k'z}$. It decreases or increases with $z$ smoothly as a real exponential. In our derivation we didn’t worry about the sources that started the waves, but there must, of course, be a source someplace in the guide. The sign that goes with $k'$ must be the one that makes the field decrease with increasing distance from the source of the waves. So for frequencies below $\omega_c=\pi c/a$, waves do not propagate down the guide; the oscillating fields penetrate into the guide only a distance of the order of $1/k'$. For this reason, the frequency $\omega_c$ is called the “cutoff frequency” of the guide. Looking at Eq. (24.22), we see that for frequencies just a little below $\omega_c$, the number $k'$ is small and the fields can penetrate a long distance into the guide. But if $\omega$ is much less than $\omega_c$, the exponential coefficient $k'$ is equal to $\pi/a$ and the field dies off extremely rapidly, as shown in Fig. 24–7. The field decreases by $1/e$ in the distance $a/\pi$, or in only about one-third of the guide width. The fields penetrate very little distance from the source. We want to emphasize an interesting feature of our analysis of the guided waves—the appearance of the imaginary wave number $k_z$. Normally, if we solve an equation in physics and get an imaginary number, it doesn’t mean anything physical. For waves, however, an imaginary wave number does mean something. The wave equation is still satisfied; it only means that the solution gives exponentially decreasing fields instead of propagating waves. So in any wave problem where $k$ becomes imaginary for some frequency, it means that the form of the wave changes—the sine wave changes into an exponential.