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## Class 11 Physics (India)

### Course: Class 11 Physics (India)>Unit 19

Lesson 4: Standing waves

# 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 waveWaves 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.
NodePositions on a standing wave where the wave stays in a fixed position over time because of destructive interference.
AntinodePositions on a standing wave where the wave vibrates with maximum amplitude.
Fundamental frequencyLowest frequency of a standing wave that has the fewest number of nodes and antinodes.
HarmonicA 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.
Figure 1: A wave pulse reflected from a free end returns right side up. A wave pulse reflecting from a fixed end is 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, equals, start fraction, lambda, divided by, 2, end fraction, and there are two nodes and one antinode.
Figure 2: For the fundamental frequency of a standing wave between two fixed ends, the wavelength is double the length of the string.
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.
Figure 3: For the second harmonic of a standing wave between two fixed ends, the wavelength is the length of the string and its frequency is twice the fundamental frequency.
Figure 4: For the third harmonic of a standing wave between two fixed ends, the wavelength is two-thirds the length of the string and its frequency is triple the fundamental frequency.

## 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.