Standing Waves

Period and frequency of standing waves

The period of the standing wave is the time needed for an antinode to complete one full cycle of vibration. This means that the antinode oscillated from the maximum amplitude above the center line to the maximum value below the center line and back. The frequency of a wave is the number of full cycles per second.

The frequency velocity and wavelength of a wave can be found using the wave equation below, where$f$is frequency in $\mathrm{Hz}$,$v$is the velocity in$\mathrm{m}/\mathrm{s}$and$\lambda$is the wavelength in$\mathrm{m}$.

$$\begin{gathered} v=f\lambda \\ velocity=frequency·wavelength \end{gathered}$$

Phase difference

The phase difference of different points on a stationary wave depends on the number of nodes between those two points.

• If the number of nodes between two points is odd, then the points are out of phase.
• If the number of nodes between the points is even then the points are in phase.

How are standing waves created?

A standing wave is a wave pattern formed by the superposition of two or more traveling waves moving in opposite directions along the same line. The waves must have the same frequency or wavelength, and amplitude. Usually, a standing wave is formed by a traveling wave that reflects off a boundary and begins moving in the opposite direction. The original wave and the reflected wave interfere and create a standing wave.

When two identical waves traveling in opposite directions interfere, they form a standing wave. Here the green and blue traveling waves combine to form the red standing wave. Wolfgang Christian and Francisco Esquembre, CC BY-SA 4.0

Now that we understand how standing waves are formed, we need to know how to work with them. To do this we need to figure out how to mathematically describe them. Consider two traveling waves that are moving in opposite directions but are otherwise equivalent:

$$\begin{gathered} y_1(x,t)=A\sin (kx-\omega t) \\ {y}_2(x,t)=A\sin (kx+\omega t)\begin{array}{}\end{array} \end{gathered}$$

Recall that$k=\frac{2\pi }{\lambda }$is the wave number, and$\omega =2\pi f$is the angular frequency. To get a standing wave we can simply take the superposition of these two waves.

$$\begin{gathered} y(x,t)=y_1(x,t)+y_2(x,t) \\ y(x,t)=A\sin (kx-\omega t)+A\sin (kx+\omega t) \end{gathered}$$

Now recall that the sum of angles trigonometric identity is given as

$\sin (\alpha \pm \beta )=\sin \left(\alpha \right)\cos \left(\beta \right)\pm \cos \left(\alpha \right)\sin \left(\beta \right)$

Next, we apply this identity to our equation where$\alpha =kx,$and$\beta =\omega t.$Thus

$$\begin{gathered} y(x,t)=A\left[\sin \right(kx\left)\cos \right(\omega t)-\cos (kx\left)\sin \right(\omega t\left)\right]+A\left[\sin \right(kx\left)\cos \right(\omega t)+\cos (kx\left)\sin \right(\omega t\left)\right] \\ \boxed{y(x,t)=2A\sin \left(kx\right)\cos \left(\omega t\right)} \end{gathered}$$

This is our formula for a standing wave as a function of position and time.

The amplitude of a standing wave

The amplitude of a standing wave at a given time is dependent on the position you are at on the standing wave. This ranges from the maximum amplitude being a superposition of the two amplitudes of the traveling waves that make up our standing wave, or 2A, then we pass through 0 and eventually get to a minimum of -2A. You can probably see where this is going. Our Standing wave amplitude contains the sin term from our formula for a standing wave. Thus we have

$A_{standing}=2A\sin \left(kx\right)$

What are some examples of standing waves?

Below we will look at two examples: sound waves and vibrating strings.

Sound waves

Sound waves can be produced as a result of the formation of standing waves inside an air column

Sound waves can produce standing waves in air columns. This can be visualized by placing a powder inside the air column and a loudspeaker on one end which is open. The loudspeaker will produce sound waves which will be reflected once they reach the boundary. With a traveling and a reflected wave, we get standing waves at certain frequencies. The powder inside the air column will be spaced evenly indicating visually the position of nodes.

Stretched strings

Harmonics

Harmonics are different wave patterns of standing waves formed on strings with two fixed ends. The pattern of the standing wave depends on its frequency. The higher the frequency of the wave, the more harmonics appear on the wave. The simplest form of harmonic shown in the figure below is formed by the lowest frequency which is known as the first harmonic or fundamental harmonic. This consists of one loop formed by two nodes at its ends and a single antinode as shown below. The frequency of the first harmonic is dependent on the length of the string L, and the speed of the wave.

Similarly, a second harmonic or a first overtone is formed by a higher frequency and consists of three nodes and two antinodes. Finally, a third harmonic or a second overtone is formed by an even higher frequency and consists of four nodes and 3 antinodes.

Standing waves form at specific frequencies called harmonics StudySmarter Originals

By using our restriction on the wavelength and frequency, we can calculate the harmonics.

Differences between traveling and standing waves

While standing and traveling waves have some similar properties, they also have several differences. The table below summarizes the differences between a standing and traveling wave.