# 3.13: Standing Waves

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- 6279

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 \(\PageIndex{2}\).

## Contributors and Attributions

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 Licensed with CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0. Report adoption of this book here. If you are a professor reviewing, adopting, or adapting this textbook please help us understand a little more about your use by filling out this form.