# 3.13: Standing Waves

A standing wave consists of waves moving in opposite directions. These waves add to make a distinct magnitude variation as a function of distance that does not vary in time.

To see how this can happen, first consider that an incident wave $$V_0^+ e^{-j\beta z}$$, which is traveling in the $$+z$$ axis along a lossless transmission line. Associated with this wave is a reflected wave $$V_0^- e^{+j\beta z}=\Gamma V_0^+ e^{+j\beta z}$$, where $$\Gamma$$ is the voltage reflection coefficient. These waves add to make the total potential

$\begin{split} \widetilde{V}(z) & = V_0^+ e^{-j\beta z} + \Gamma V_0^+ e^{+j\beta z} \\ & = V_0^+ \left( e^{-j\beta z} + \Gamma e^{+j\beta z} \right) \end{split}$

The magnitude of $$\widetilde{V}(z)$$ is most easily found by first finding $$|\widetilde{V}(z)|^2$$, which is:

\begin{aligned}
& \tilde{V}(z) \tilde{V}^{*}(z) \\
=&\left|V_{0}^{+}\right|^{2}\left(e^{-j \beta z}+\Gamma e^{+j \beta z}\right)\left(e^{-j \beta z}+\Gamma e^{+j \beta z}\right)^{*} \\
=&\left|V_{0}^{+}\right|^{2}\left(e^{-j \beta z}+\Gamma e^{+j \beta z}\right)\left(e^{+j \beta z}+\Gamma^{*} e^{-j \beta z}\right) \\
=&\left|V_{0}^{+}\right|^{2}\left(1+|\Gamma|^{2}+\Gamma e^{+j 2 \beta z}+\Gamma^{*} e^{-j 2 \beta z}\right)
\end{aligned}

Let $$\phi$$ be the phase of $$\Gamma$$; i.e.,

$\Gamma = \left|\Gamma\right|e^{j\phi}$

Then, continuing from the previous expression:

\begin{aligned}
&\left|V_{0}^{+}\right|^{2}\left(1+|\Gamma|^{2}+|\Gamma| e^{+j(2 \beta z+\phi)}+|\Gamma| e^{-j(2 \beta z+\phi)}\right) \\
=&\left|V_{0}^{+}\right|^{2}\left(1+|\Gamma|^{2}+|\Gamma|\left[e^{+j(2 \beta z+\phi)}+e^{-j(2 \beta z+\phi)}\right]\right)
\end{aligned}

The quantity in square brackets can be reduced to a cosine function using the identity

$\cos\theta = \frac{1}{2}\left[e^{j\theta}+e^{-j\theta}\right] \nonumber$

yielding: $|V_0^+|^2 \left[ 1 + \left|\Gamma\right|^2 + 2\left|\Gamma\right| \cos\left( 2\beta z + \phi \right) \right]$

Recall that this is $$|\widetilde{V}(z)|^2$$. $$|\widetilde{V}(z)|$$ is therefore the square root of the above expression:

$\left|\widetilde{V}(z)\right| = |V_0^+| \sqrt{ 1 + \left|\Gamma\right|^2 + 2\left|\Gamma\right| \cos\left( 2\beta z + \phi \right) }$

Thus, we have found that the magnitude of the resulting total potential varies sinusoidally along the line. This is referred to as a standing wave because the variation of the magnitude of the phasor resulting from the interference between the incident and reflected waves does not vary with time.

We may perform a similar analysis of the current, leading to: $\left|\widetilde{I}(z)\right| = \frac{|V_0^+|}{Z_0} \sqrt{ 1 + \left|\Gamma\right|^2 - 2\left|\Gamma\right| \cos\left( 2\beta z + \phi \right) }$

Again we find the result is a standing wave.

Now let us consider the outcome for a few special cases.

Matched load. When the impedance of the termination of the transmission line, $$Z_L$$, is equal to the characteristic impedance of the transmission line, $$Z_0$$, $$\Gamma=0$$ and there is no reflection. In this case, the above expressions reduce to $$|\widetilde{V}(z)| = |V_0^+|$$ and $$|\widetilde{I}(z)| = |V_0^+|/Z_0$$, as expected.

Open or Short-Circuit. In this case, $$\Gamma=\pm1$$ and we find:

$\left|\widetilde{V}(z)\right| = |V_0^+| \sqrt{ 2 + 2\cos\left( 2\beta z + \phi \right) }$

$\left|\widetilde{I}(z)\right| = \frac{|V_0^+|}{Z_0} \sqrt{ 2 - 2\cos\left( 2\beta z + \phi \right) }$

where $$\phi=0$$ for an open circuit and $$\phi=\pi$$ for a short circuit. The result for an open circuit termination is shown in Figure $$\PageIndex{1}$$(a) (potential) and $$\PageIndex{1}$$(b) (current). The result for a short circuit termination is identical except the roles of potential and current are reversed. In either case, note that voltage maxima correspond to current minima, and vice versa.

Also note:

The period of the standing wave is $$\lambda/2$$; i.e., one-half of a wavelength.

This can be confirmed as follows. First, note that the frequency argument of the cosine function of the standing wave is $$2\beta z$$. This can be rewritten as $$2\pi\left(\beta/\pi\right)z$$, so the frequency of variation is $$\beta/\pi$$ and the period of the variation is $$\pi/\beta$$. Since $$\beta=2\pi/\lambda$$, we see that the period of the variation is $$\lambda/2$$. Furthermore, this is true regardless of the value of $$\Gamma$$.

Mismatched loads. A common situation is that the termination is neither perfectly-matched ($$\Gamma=0$$) nor an open/short circuit ($$\left|\Gamma\right|=1$$). Examples of the resulting standing waves are shown in Figure $$\PageIndex{2}$$. Figure $$\PageIndex{2}$$: Standing waves associated with loads exhibiting various reflection coefficients. In this figure the incident wave arrives from the right.