Standing Wave: Definition, Formula & Examples

A ​standing wave​ is a stationary wave whose pulses do not travel in one direction or the other. It is typically the result of the superposition of a wave moving in one direction with its reflection moving in the opposite direction.

To know what the combination of waves will do to a given point in a medium at a given point in time, you simply add what they would be doing independently. This is called the ​principle of superposition​.

For example, if you were to plot the two waves on the same graph, you would simply add their individual amplitudes at each point to determine the resultant wave. Sometimes the resultant amplitude will have a larger combined magnitude at that point, and sometimes the effects of the waves will partially or completely cancel each other.

If both waves are in phase, meaning their peaks and valleys line up perfectly, they combine together to make a single wave with a maximum amplitude. This is called ​constructive interference​.

If the individual waves are exactly out of phase, meaning the peak of one lines up perfectly with the valley of the other, then they cancel each other out, creating zero amplitude. This is called ​destructive interference​.

If you attach one end of a string to a rigid object and shake the other end up and down, you send wave pulses down the string that then reflect at the end and move back, interfering with the stream of pulses in opposite directions. There are certain frequencies that you can shake the string at that will produce a standing wave.

A standing wave is formed as a result of the wave pulses moving to the right periodically constructively and destructively interfering with the wave pulses moving to the left.

​Nodes​ on a standing wave are points where the waves always destructively interfere. ​Antinodes​ on a standing wave are points that oscillate between perfect constructive interference and perfect destructive interference.

In order for a standing wave to form on such a string, the length of the string must be a half-integer multiple of the wavelength. The lowest-frequency standing wave pattern will have a single "almond" shape in the string. The top of the "almond" is the antinode, and the ends are the nodes.

The frequency at which this first standing wave, with two nodes and one antinode, is achieved is called the ​fundamental frequency​ or the ​first harmonic​. The wavelength of the wave that produces the fundamental standing wave is ​λ = 2L​, where ​L​ is the length of the string.

Each frequency at which the string driver oscillates that produces a standing wave beyond the fundamental frequency is called a harmonic. The second harmonic produces two antinodes, the third harmonic produces three antinodes and so on.

The frequency of the nth harmonic relates to the fundamental frequency via

\(f_n=nf_1\)

The wavelength of the nth harmonic is

\(\lambda = \frac{2L}{n}\)

where ​L​ is the length of the string.

The speed of the waves producing the standing wave can be found as the product of frequency and wavelength. For all harmonics, this value is the same:

\(v=f_n\lambda_n = nf_1\frac{2L}{n}=2Lf_1\)

For a particular string, this wave speed can also be pre-determined in terms of the tension and mass density of the string as:

\(v=\sqrt{\frac{F_T}{\mu}}\)

​_F_T​ is the tension force, and ​μ_​ is the mass per unit length of the string.

​Example 1:​ A string of length 2 m and linear mass density 7.0 g/m is held at tension 3 N. What is the fundamental frequency at which a standing wave will be produced? What is the corresponding wavelength?

​Solution:​ First we must determine the wave speed from the mass density and tension:

\(v=\sqrt{\frac{3}{.007}}=20.7\text{ m/s}\)

Use the fact that the first standing wave occurs when the wavelength is 2​L​ = 2 × (2 m) = 4 m, and the relationship between wave speed, wavelength and frequency to find the fundamental frequency:

\(v=\lambda f_1 \implies f_1=\frac{v}{\lambda}=\frac{20.7}{4}=5.2\text{ Hz}\)

The second harmonic ​_f₂​ = 2 × ​f₁​ = 2×5.2 = 10.4 Hz, which corresponds to a wavelength of 2​L_​/2 = 2 m.

The third harmonic ​_f₃​ = 3 × ​f₁​ = 3 × 5.2 = 10.4 Hz, which corresponds to a wavelength of 2​L_​/3 = 4/3 = 1.33 m

And so on.

​Example 2:​ Just like standing waves on a string, it is possible to produce a standing wave in a hollow tube using sound. With the waves on a string, we had nodes on the ends, and then additional nodes along the string, depending on the frequency. However, when a standing wave is created by having one or both ends of the string free to move, it is possible to create standing waves with one or both ends being antinodes.

Similarly, with a standing sound wave in a tube, if the tube is closed on one end and open on the other, the wave will have a node on one end and an antinode on the open end, and if the tube is open on both ends, the wave will have antinodes on both ends of the tube.

For example, a student uses a tube with one open end and one closed end to measure the speed of sound by looking for sound resonance (an increase in volume of sound indicating the presence of a standing wave) for a 540-Hz tuning fork.

The tube is designed so that the closed end is a stopper that can be slid up or down the tube in order to adjust the effective length of the tube.

The student begins with the tube length almost 0, hits the tuning fork and holds it near the open end of the tube. The student then slowly slides the stopper, causing the effective tube length to increase, until the student hears the sound increase significantly in loudness, indicating resonance, and the creation of a standing sound wave in the tube. ​This first resonance occurs when the tube length is 16.2 cm.​

Using the same tuning fork, the student further increases the length of the tube until she hears another resonance at a ​tube length of 48.1 cm​. The student does this again, and gets a third resonance at ​tube length 81.0 cm​.

Use the student's data to determine the speed of sound.

​Solution:​ The first resonance happens at first possible standing wave. This wave has one node and one antinode, making the length of the tube = 1/4λ. So 1/4λ = 0.162 m or λ = 0.648 m.

Second resonance happens at the next possible standing wave. This wave has two nodes and two antinodes, making the length of the tube = 3/4λ. So 3/4λ = 0.481 m or λ = 0.641 m.

Third resonance happens at the third possible standing wave. This wave has three nodes and three antinodes, making the length of the tube = 5/4λ. So 5/4λ = 0.810 m or λ = 0.648 m.

The average experimentally determined value of λ is then

\(\lambda = (0.648 + 0.641 + 0.648)/3 = 0.6457\text{ m}\)

The experimentally determined speed of sound is

\(v=\lambda f = = 0.6457 \times 540 = 348.7\text{ m/s}\)