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  • [flute plays Crash Course theme]

  • That’s a familiar tune!

  • How do instruments, like this guitar, create music?

  • Weve talked about the science of sound, and some of the properties of sound waves.

  • But when we talk about sound waves in the context of music, there are all kinds of fascinating properties and weird rules to talk about.

  • I’m talking about the music that comes from string, wind, and brass instruments.

  • String instruments create sound when their strings vibrate in the air.

  • And in order to understand how these instruments work, you have to realize that making music is not just an art.

  • It’s ALSO a science.

  • [Theme Music]

  • Sound, youll recall, is a wave: a ‘longitudinalwave.

  • This means that the medium that the wave travels through oscillates -- or moves back and forth

  • -- in the same direction that the wave is moving.

  • But string, wind, and brass instruments use a special kind of wavetheyrestanding waves.'

  • A standing wave is a wave that looks like it isn't moving.

  • Itsamplitudemay change, but it isn't traveling anywhere.

  • Standing waves are the result of two other things waves do, both of which weve talked about before: reflection and interference.

  • Reflection is what happens when a wave reaches the end of a path, and then moves back along the same path.

  • That’s what happens when you send a pulse down a fixed rope -- it reaches the end, and then comes right back.

  • When we send a continuous wave down the rope, that’s when interference comes into play.

  • The wave reaches the end of the rope and is reflected, but there are more peaks on the way.

  • As the peaks pass each other, they interfere with one another, changing their sizes.

  • Usually, you end up with crests and troughs that are different sizes and various distances apart.

  • But at certain frequencies, the reflected waves interfere in such a way that you end up

  • with a wave that seems to stay perfectly still, with only its amplitude changing.

  • That’s a standing wave, and it can happen both in strings and in the air in pipes.

  • And that's what makes music: Standing waves with different frequencies correspond to different musical notes.

  • Now, in order to understand how standing waves operate, you should get to know their anatomy.

  • The points of a standing wave that don’t oscillate are called nodes,

  • and the points at the maximum height of the peaks are antinodes.

  • And here’s something cool.

  • If you look at a string on a stringed instrument, you can actually see where the nodes and antinodes are.

  • The standing wave creates peaks along the string, and between those peaks, there are points that just stay still.

  • So the peaks are the antinodes, and the points that don’t oscillate are nodes.

  • And if one or both of the string’s ends are fixed, then each fixed end is a node too, because it’s stuck in place.

  • Now, in a pipe, the standing waves are made of air molecules moving back and forth.

  • But the areas where molecules oscillate the most

  • (including those near any open ends of the pipe) form the peaks, and therefore the antinodes.

  • And between those peaks, as well as at any closed ends of the pipe, are areas where molecules don’t move at all; those are the nodes.

  • Generally, musicians make their music using the frequencies of these standing waves.

  • But the nature of these waves depends a lot on what the ends of the string or pipe look like.

  • Remember how a wave traveling down a rope gets reflected differently, depending on whether the end of the rope is fixed or loose?

  • A fixed end will invert the wave -- turning crests into troughs, and vice versa --

  • while a loose end will just reflect it without inverting it.

  • The same thing holds true for air in a pipe: a closed end will invert the wave, while an open end won’t.

  • So the properties of a standing wave will be a little different,

  • depending on whether it’s made with a string with two fixed ends, or a pipe with two ends open,

  • or a string or a pipe with one end fixed, and the other open.

  • A string with two fixed ends -- like in a piano -- is probably the simplest way to understand standing waves.

  • Because, we know that no matter what, the wave made by a fixed string will have at least two nodes -- one at each end.

  • And in its most basic form, it would have just one antinode, in the middle.

  • So the wave is basically a peak that moves from being a crest to a trough and vice versa

  • like some kind of one-dimensional jump rope.

  • This most basic kind of standing wave is known as the fundamental -- or the 1st harmonic.

  • It’s the simplest possible standing wave you can have, with the fewest nodes and antinodes.

  • There are other, more complex standing waves that you can have, too.

  • These are known as overtones.

  • Overtones build on the fundamental, incrementally: each overtone adds a node and an antinode.

  • So each of these overtones is related to the fundamental wave -- and all of the overtones are related to each other.

  • Together, the fundamental wave and the overtones make up what are known as harmonics.

  • The fundamental is the 1st harmonic, and the overtones are higher-numbered harmonics.

  • With each node-and-antinode pair that’s added to the standing wave,

  • the number of the harmonic goes up: 2nd harmonic, 3rd harmonic, and so on.

  • Now, physicists sometimes express harmonics in terms of wavelength.

  • For example, for a string with two fixed ends, youll notice that the fundamental covers exactly half a wavelength.

  • A full wavelength of the wave would span two peaks: a crest and a trough,

  • but the fundamental spans exactly one peak, which is half the wavelength.

  • So, for the fundamental of a string with two fixed ends, the length of the string is equal to half a wavelength.

  • The second-simplest standing wave you can have on a string with two fixed ends has 3 nodes --

  • one at each end, and one in the middle -- plus 2 antinodes in between the nodes.

  • It’s called the 2nd harmonic, and the string holds exactly one wavelength.

  • You can probably guess what the 3rd harmonic looks like: it has 4 nodes and 3 antinodes,

  • and the string holds 1.5 -- or, 3/2 -- wavelengths.

  • You may have started to notice a pattern: For a standing wave on a given length of string,

  • the number of wavelengths that fit on the string is equal to the number of the harmonic, divided by 2.

  • So, now we have an equation that relates the wavelength of a standing wave to the number of the harmonic and the length of the string.

  • Once you get a handle on wavelength, you can figure out the aspect of the wave that musicians care about most -- the frequency.

  • Weve already established that a wave’s wavelength, times its frequency, is equal to its velocity,

  • which will be the same for each harmonic, because a wave’s velocity only depends on the medium it’s traveling through.

  • So a standing wave’s frequency will be equal to its velocity divided by its wavelength.

  • For the fundamental with two fixed ends, we already know that the wavelength is twice the string's length.

  • So the frequency of that fundamental standing wave -- known as the fundamental frequency

  • -- is equal to the velocity, divided by twice the length of the string.

  • We write it as f, with a subscript of 1.

  • Now what about the frequency of the second harmonic -- the standing wave with 3 nodes and 2 antinodes?

  • It will be equal to the velocity, divided by the length of the string.

  • Which is twice the fundamental frequency.

  • And the frequency of the third harmonic, with its 4 nodes and 3 antinodes,

  • will be equal to three times the fundamental frequency.

  • So, were starting to see another pattern here:

  • The frequency of a standing wave with two fixed ends will just be equal to the number of the harmonic, times the fundamental frequency.

  • In fact, that’s one way to define harmonics:

  • The number of a harmonic is equal to the number you multiply by the fundamental frequency, to get the harmonic’s frequency.

  • This math is what makes musical instruments work.

  • When you press down a key on a piano, you make a hammer strike a string, creating standing waves in that string.

  • Every string in the piano is tuned so that its fundamental frequency --

  • which depends on the string’s mass, length, and tension -- corresponds to a given note.

  • Middle C, for example, is 261.6 Hz.

  • Guitars are also tuned so that the fundamental frequencies of their strings, correspond to set notes.

  • And when you press down on the strings in certain places,

  • you change the length of the active part of string so that its fundamental frequency corresponds to a different note.

  • So, for a standing wave with two fixed ends, we can relate wavelength, frequency, velocity,

  • the length of the string, and the number of the harmonic.

  • And we can do the exact same thing for a standing wave with two loose ends -- in an open pipe, for example, like in a flute.

  • A standing wave in a pipe with two open ends is kind of the opposite, of the wave with two fixed ends:

  • Instead of having a node at each end, it has an antinode at each end.

  • So the fundamental standing wave for a pipe with two open ends will have two antinodes,

  • and one node in the middle of the wave.

  • Then, the 2nd harmonic will have three antinodes and two nodes, and so on.

  • But each harmonic still covers the same number of wavelengths.

  • Remember how the fundamental wave for a string with two fixed ends covered half of a wavelength?

  • The fundamental wave for a pipe with two open ends also covers half of a wavelength.

  • That half is just in a different section of the wave.

  • And just like a string with two fixed ends, the second harmonic for a pipe with two open ends also covers a full wavelength.

  • It’s just that, in the case of the pipe, the wave starts and ends with a peak instead of a node.

  • So the equations for wavelength and frequency for a standing wave with two open ends

  • will be the same as they were for a standing wave with two fixed ends.

  • So, weve covered guitars and pianos and flutes!

  • But a pipe with one closed end and one open end works a little differently.

  • These kinds of pipes are used in instruments like pan flutes, where you blow across the top of a closed pipe to make music.

  • Here, standing waves need a separate set of equations, for a couple of reasons:

  • First, the closed end of the pipe will be a node, because the air molecules aren’t oscillating there.

  • And the open end will be an antinode, because that’s where there’s a peak in the oscillations.

  • Which means that the simplest wave you can make in this pipe will stretch from one node, to one peak.

  • But that’s only a span of a quarter of a wavelength in the pipe.

  • Before, with both a string fixed at both ends, and an open pipe, the fundamental spanned half a wavelength.

  • The fact that a pan-flute pipe only covers a quarter of a wavelength changes things.

  • Because, remember: the frequency of each harmonic is equal to the number of the harmonic, times the fundamental frequency.

  • But for a pipe that’s closed on one end, you can’t double the fundamental frequency,

  • or quadruple it -- or multiply it by any even number.

  • Because it would result in a wave that would need a node on both ends, or a peak on both ends.

  • Which isn't possible.

  • So, a pipe that’s closed on one end can’t have even-numbered harmonics.

  • All of this helps explain why musical instruments sound different, even when theyre playing the same note.

  • When you play a note, youre creating the fundamental wave, plus some of the other harmonics -- the overtones.

  • And for each instrument, different harmonics will have different amplitudes -- and therefore sound louder.

  • But because of the physics of standing waves, instruments that have pipes with one closed end

  • won't create the even-numbered harmonics at all.

  • That’s why a C on the flute sounds so different from a C on, say, the bassoon!

  • Today, you learned about standing waves, and how theyre made up of nodes and antinodes.

  • We discussed harmonics, and how to find the frequency of a standing wave on a string with

  • two fixed ends, a pipe with two open ends, and a pipe with one closed end.

  • Finally, we explained why a pipe with one closed end can’t have even-numbered harmonics.

  • Crash Course Physics is produced in association with PBS Digital Studios.

  • You can head over to their channel and check out a playlist of the latest episodes from

  • shows like First Person, PBS Game/Show, and The Good Stuff.

  • This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio

  • with the help of these amazing people and our equally amazing graphics team is Thought Cafe.

[flute plays Crash Course theme]

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音楽の物理学。クラッシュコース物理学#19 (The Physics of Music: Crash Course Physics #19)

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    kingon に公開 2021 年 01 月 14 日
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