Standing Waves

A standing wave, also called a stationary wave, is a wave that effectively stands in place. Standing waves don't oscillate through space.

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Standing waves are caused by the reflection and inversion of waves reflecting from a fixed end in a medium. Basically, if you take a wave and let it oscillate until it reaches a tight end, it will evidently reflect and invert. This change in the waveform will go back the way it came but if the source creating waves is generating more waves, the reflected wave will hit another incident wave and they'll superpose. This superposition is the premise behind standing waveforms: they're waves made up of a reflected and inverted wave going back and superposing with another wave like itself.

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Standing waves can occur in other ways but this is the easiest way to do it because both waves should have equal speeds and frequencies to truly superpose properly and create the standing wave effect.

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The parts of a standing wave where there's minimum absolute amplitude are called nodes, which don't move.

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The parts of a standing wave where there's maximum absolute amplitude are called antinodes. Basically, they're the extrema of a standing wave.

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Standing waves on a string are unique in that the string has a finite length, so we can define the length of a wave, or the wavelength, through the length of the medium. However, before doing so, we must first discuss harmonics.

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Harmonics arise from the fact that if you keep the nodes in place, you can create many different overtones of a different wave so long as the nodes are all in the same places. These overtones' frequencies must be in integer multiples of fundamental frequency of the wave. The harmonic number of a wave is defined through n, which is an integer.

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In terms of classifying overtones and harmonics, the second harmonic of a wave is its first overtone and the third harmonic of a wave is its second overtone. This applies universally as the 13th harmonic of a wave is its 12th overtone.

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If I have the 2nd harmonic of a given standing wave on a guitar string, that means I'm playing a sound with a frequency double the fundamental frequency on that guitar. If I play a higher nth level frequency, the frequency of the waves I create increase and since the phase velocity wouldn't be affected, the wavelength goes down. However, the stringed instrument can support longer waves if the string is longer itself, intuitively giving the equation

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where lambda is the wavelength of the standing wave, L is the length of the medium, and n is the harmonic.

So, if we take a musical instrument like a guitar, for example, we'll notice that its medium has a rigidly defined length. The strings of the guitar have rigid endpoints which will cause standing waves when plucked. These types of standing waves are string waves.

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The first thing to know about a standing wave on a string is that its velocity depends primarily on 2 factors: the tension of the string and the linear mass density of the string.

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If you've played a musical instrument, this might make a lot of sense, since you've likely noticed higher pitches(frequencies) in tuned strings. However, if you haven't, it can still be well-explained.

If you increase the tension, you're basically increasing the force keeping the waveforms on the string moving forward. Thus, if you increase the tension, you'd increase the wave speed.

For the linear mass density(kg/m), which is not the same as three-dimension density(kg/m3), this should also make sense with some intuition behind the forces at play on a string. If you increase the linear mass density, you get a greater mass at a given length so the strings moving up and down creating the transverse string wave will have greater inertia. This greater inertia will cause them to have lower speeds of propagation.

Standing waves on a string are unique in that the string has a finite length, so we can define the length of a wave, or the wavelength, through the length of the medium. However, before doing so, we must first discuss harmonics.

​

Harmonics arise from the fact that if you keep the nodes in place, you can create many different overtones of a different wave so long as the nodes are all in the same places. These overtones' frequencies must be in integer multiples of the fundamental frequency of the wave. The harmonic number of a wave is defined through n, which is an integer.

​

In terms of classifying overtones and harmonics, the second harmonic of a wave is its first overtone and the third harmonic of a wave is its second overtone. This applies universally as the 13th harmonic of a wave is its 12th overtone.

​

If I have the 2nd harmonic of a given standing wave on a guitar string, that means I'm playing a sound with a frequency double the fundamental frequency on that guitar. If I play a higher nth level frequency, the frequency of the waves I create increase and since the phase velocity wouldn't be affected, the wavelength goes down. However, the stringed instrument can support longer waves if the string is longer itself, intuitively giving the equation

Standing waves can also be produced in pipes, especially those in trumpets, flutes, and most woodwind instruments.

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An open-ended pipe is exactly what it sounds like. It's open on both sides as you blow through one end and will hear sound come out the other end. Open-ended pipes start and end at anti-nodes so they can support any integral number of harmonic frequencies. This should make sense, because air, the medium the wave ultimately goes through outside the pipe ends, can move freely about. This makes open-ended pipes very similar to strings when it comes to waves.

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For a standing wave in an open air column or pipe,

just like a string.

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Now, if you notice, the waves you produce when you play something like the clarinet aren't actually from open-ended air columns.This is because when you play a musical instrument like this, you're blowing into one end of the "pipe" with your mouth so there's only one open end. Pipes like these are called closed-end pipes, pipes with one end closed and the other one open. Waves going through closed end pipes will start at an anti-node but they must reach the endpoint of the pipe(or you could call it "the wall") of the pipe with a node. This is because the air at the closed end isn't free to move and is being reflected.

So, what's the difference between a closed end pipe and an open-ended pipe?

Well, it has to do with the fact that the closed end of the closed end pipe must be a node. This means that a closed end pipe only supports odd harmonics. For a standing wave on a string and in an open-ended pipe, you could generate frequencies that are any integer number multiplied by the fundamental. However, since you must meet different boundary conditions for a closed end pipe, its properties are expressed as this:

Basically, you replace the 2 in earlier equations with a 4 but now you must remember, n can only be odd.