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# Standing waves review

Review key terms and skills related to standing waves including how to find standing wave harmonics.

## Key terms

Term (symbol) | Meaning |
---|---|

Standing wave | Waves which appear to be vibrating vertically without traveling horizontally. Created from waves with identical frequency and amplitude interfering with one another while traveling in opposite directions. |

Node | Positions on a standing wave where the wave stays in a fixed position over time because of destructive interference. |

Antinode | Positions on a standing wave where the wave vibrates with maximum amplitude. |

Fundamental frequency | Lowest frequency of a standing wave that has the fewest number of nodes and antinodes. |

Harmonic | A standing wave that is a positive integer multiple of the fundamental frequency. |

## Standing wave harmonics

A wave that travels down a rope gets reflected at the rope’s end. If the end of the rope is free, then the wave returns right side up. If the end of the rope is fixed, then the wave will be inverted.

For a rope with two fixed ends, another wave travelling down the rope will interfere with the reflected wave. At certain frequencies, this produces standing waves where the nodes and antinodes stay at the same places over time. For all standing wave frequencies, the nodes and antinodes alternate with equal spacing.

The lowest frequency (which corresponds with the longest wavelength) that will produce a standing wave has one “bump” (see Figure 2) along the string length $L$ . This standing wave is called the fundamental frequency, with $L={\displaystyle \frac{\lambda}{2}}$ , and there are two nodes and one antinode.

Each successive harmonic has an additional node and antinode. For the second harmonic, there are two “bumps”, for the third, there are three, and so on. Examples of the second and third harmonics are shown below. A string has an infinite number of resonant frequencies.

## Common mistakes and misconceptions

The length of the standing wave depends on the length of the string. The endpoints will always be nodes, and the first harmonic’s wavelength is double the length of the string, no matter how long the string is.

## Learn more

For deeper explanations of standing waves, see our video about standing waves on strings.

To check your understanding and work toward mastering these concepts, check out our exercises:

## Want to join the conversation?

- Is antinode the same as amplitude(3 votes)
`Antinodes`

are not the same thing as the`amplitude`

, but they are very closely related. The`amplitude`

is the`distance`

from the`rest position`

of the wave to the`antinode`

. The`antinode`

is just a point along the wave that has the greatest maximum velocity in the`y-direction`

.(25 votes)

- does a standing wave have a period?(5 votes)
- Yes. All waves have a time period. A wave is the propagation of energy. So although the wave might appear to be stationary, there is energy being propagated, at a constant rate (for standing waves).(4 votes)

- I dont undestand how the antinodes create constructive, the heights are opposite to each other which means that they will substract(5 votes)
- Why are there missing harmonics for a standing wave on a string with loose(free) ends?(2 votes)
- is the antinode the same as the crest and trough?(2 votes)
- Are antinodes the same as crests in waves?(2 votes)
- why does a node form one half wavelength in from a fixed end when a standing wave is formed by reflection?(2 votes)
- At what frequency allows you to get standing waves with a loose end?(1 vote)
- At the loose end, there will be a node, since the loose end will be free to oscillate. That means that you have an antinode in the fixed end and a node in the loose end. If we call the length of the string L, all possible standing waves are the ones with wavelengths 4L, 2/3L, 2/5 L, 2/7L etc. (try to draw it yourself!). The frequency of the n'th harmonic can now be found using the relation frequency_n=speed of sound/wavelength_n.

Hope it helps!(2 votes)

- How would lambda_n=(2L)/n be derived algebraically?(1 vote)
- This formula is simply derived from the fact that each bump is equal to half a wavelength. There's no serious hard math behind it.(2 votes)

- How to calculate frequency of a stand wave when you know nodes, velocity and length(1 vote)