# Confusion about overtones and a slow-motion video of a plucked string

Until now, I thought I understood the basic idea behind a "timbre" of a sound: The specific combination of overtones.

For example, this waveshape

illustrates a combination of the first harmonic (the basic frequency) and the second harmonic. The combination of overtones is different from a clarinet to a guitar, for example, and somehow determines the "timbre" of the sound. In particular, a plucked string has a waveshape of the fundamental frequency and some overtones, I thought.

Now I came across the following video of a plucked string in slow-motion: YouTube link

In this video, the "wave mountain" propagates from one end of the string to the other back and forth and it looks nothing like the waveshape from above.

How can the relation between these two waves be explained? Why is the plucked guitar string not of the form from above? What am I getting wrong here?

• ‘The combination of overtones … somehow determines the "timbre" of the sound‘ No. The overtones make up part of the timbre of an instrument, and not even the most important part in some cases. Noises and transient components are very important ingredients of timbre also. Apr 11 at 11:40
• Also, you can look into how two propagating waves combine to form a stationary wave. It begins with a transient, where the fundamental modes are not really dominant yet. After a few bouncing back and forth and the end points, only the resonant wavelengths remain, fundamental and its overtones.
– Tom
Apr 11 at 17:22

At a basic level, the waveshape in the image you posted is a graph of sound pressure over time, while the actual string in the video is a graph of string displacement over its length. The two are related, but the transformation from displacement of the string to sound in air is quite complex, and we shouldn't really expect that they would visually look similar.

For starters, the string itself is only long enough to show half of a wave cycle. If you were to pick a specific point on the string and plot its vertical displacement over time, you might get a graph more similar to the first waveform (@flawr did exactly that- see below). Or, you might not. The visual shape of a waveform is very sensitive to small changes in phase.

Here is a graph made by @flawr showing the vertical displacement of the string over time:

We can see the periodic motion that we expect, a clear fundamental frequency, a square-ish wave shape, and some "fuzz" from higher harmonics. This waveform resembles what an electric guitar pickup would "hear" if it were placed at the white line near the middle of the string (if this were an electric guitar string).

• Thanks, this answers my question. In particular, when I measure the deflection of my eardrum (on the y axis) over time (the x axis), the sting in the video induces a curve similar to the picture/waveform in my question, right? Apr 11 at 17:45
• Yes, it would be a similar waveform, perhaps with some phase shift or small amplitude differences. Apr 11 at 17:52
• Ok. Yes, only similar, sure. Thanks! Apr 11 at 17:52
• For a violin, much of the sound will depend upon the force the string exerts on the bridge in the horizontal direction perpendicular to the strings. Force on attempting to push the top of the bridge toward the treble side of the instrument will tend to push the back of the violin away from the front, while force pushing toward the bass side will push the back of the violin toward the back. Apr 11 at 22:54
• To illustrate the point, I made an image of the string of the video in question by recording a column of pixels and displaying the time component as second axis: i.stack.imgur.com/coArn.jpg Apr 15 at 12:05

The Fourier transform allows us to decompose the shape of the string into a collection of sinusoids which correspond to the fundamental frequency and the (mostly) audible harmonics that give the note its color.

As you pluck a string and your finger loses contact with it, the string's shape is the sum of a bunch of harmonic movements that were frozen in time, now that you've let go, they continue their motion.

We can make a little experiment, we can model a standing wave by taking the static graph of `sin(x)` and multiplying each point by `cos(t)` where `t` represents time:

We can do this to model what a string moving as the sum of two modes might look like:

here `f(x)` represents the sum of the two motions. Notice this is like your wave in your image but moving.

We can generalize this to be a sum of as many modes as we want, let's try and construct how we would see the string moments after plucking:

Note that this behavior models almost exactly what you saw in your video minus the damping.

Feel free to experiment with the following desmos set-ups:

As per your question, the relation between the wave in your picture and the one produced by plucking is that when you pluck a string, you force it into a particular wave shape which is the sum of many harmonics and thus cannot fall into the pure shape in your picture which only uses two harmonics.

Technically the wave shape you provided is possible, assuming you can start your string in that configuration, and then "let go" of it.

The waves that we can see traveling along a resonating string are not the same as the waveform we hear when that same string is amplified.

One major thing you’re missing is that the waveform pictured in your question is not the only way to sum a fundamental and its first overtone. There are phase differences which can be introduced that are not audible but change the shape of the wave.

Also, the overtones present in a plucked string can vary greatly depending on how and where it’s plucked. So the video may have actually had fewer overtones.

The simplest fundamental answer is you can’t watch a video of a string vibrating and easily see the overtone series that we can hear. You might be able to reconstruct it if you have video of the entire string and you spend a lot of time studying it closely, but even then the frame rate of the video is likely to not capture all the overtones.

Disclaimer: I don't know the physics well, just combining what we see there with what we experience.

The string in the video is plucked closer to one end than the other, much like a guitar or many other string instruments. Think about how the video would look different if the pluck happened in the exact center of the string. There would be no "traveling" from one end of the string to the other; the entire string would move back and forth, with the peak of its amplitude staying in the center.

Now think (or try, if you have the means) what impact this has on the tone. Pluck a guitar string in the normal area, over the sound hole. Pluck it in the exact center. Pluck it very close to one end. You'll hear the same fundamental frequency each time—if this is, say, a high E string, then they are all Es. But the center pluck gives the "richest," "roundest," "simplest" sound, while the end plucks give "twangier," "brighter" sounds.

Why? At this point it starts getting beyond me, but my guess is that that the off-center pluck causes more higher partials. The center pluck is the closest to a sine wave. And as an off-center pluck establishes equilibrium, it gets more similar to the center pluck. What we might describe as a "twang" is us hearing a change over time in the waveform, from many higher partials at first to fewer. Meanwhile, we get the same fundamental pitch each time because of the actual mass and tension of the string; we might pluck it off center, but it still whips back and forth at the same rate as it would on-center. (This is all my guess; others who know more about the physics, please chime in if I'm wrong!)

Now, to Todd's initial point, many other things contribute to timbre besides the actual motion of the string. Play a nylon string and a steel string, both tuned to E4, and they will have the same fundamental but different timbre (because elasticity and mass). Play the same-pitched strings on a classical guitar, a dreadnought acoustic, an unplugged electric, a mandola, and a sitar, and they will all sound different... because once the vibration leaves the string it's passing through all the apparatus of body and air.

Now I realize that none of this directly addresses your central question:

How can the relation between these two waves be explained? Why is the plucked guitar string not of the form from above?

Or, to put it even more strongly: Let's say you pluck the exact center of a string. Why would we get anything other than a perfect sine wave? If what we get shows "extra" harmonics on an oscilloscope, why don't we see similar complications in the string's motion?

At this point again it gets beyond me, but I'm going to guess: Because what's recorded on the oscilloscope are sound waves, but the motion of the string is not. It's physical waves creating sound waves. Some materials get you closer to a sine wave—a nylon string, for instance, and as the "pluck" dies away and you examine the sustained pitch, these sounds get closer to sine. Now, for all I know, the difference between a steel and a nylon string would be visible—maybe a bit of extra wiggle or wobble in the overall back-and-forth vibration? Because of something about elasticity maybe?—but I imagine you'd need a much, much higher zoom and resolution than in your video. Because the big back-and-forth vibrations just give us the fundamental. Meanwhile, the placement of your pluck, and the traveling of that "kink" or "corner" up and down the string, does matter to the overtones produced. You can see it here:

or have it somewhat lay-level explained here:

But how that traveling translates into a complication of the sound wave probably has more to do with the effect of the string on the air, and we can't "see" all the partials.

• The center pluck will be closer to a square wave than a sine, in that it will only have odd harmonics. This is why the sound resembles that of a clarinet. The even harmonics are not excited by action at the center of the string because the center of the string is a point of zero amplitude for those harmonics. By contrast, it is the point of maximum amplitude for all of the odd harmonics. Similarly, if you drive the string at the 1/3 point, you should suppress the third harmonic. This is why pianos typically strike the string at the 1/7 point (or so I understand; I haven't measured). Apr 11 at 19:23
• Technically, the center pluck is closer to triangle wave than square, which is in turn closer to sine than square wave.
– ojs
Apr 12 at 10:55

In addition to what Edward said:

the waveshape in the image you posted is a graph of sound pressure over time, while the actual string in the video is a graph of string displacement over its length

there is a second crucial difference: the illustration in your question is probably meant to represent the quasi-steady-state sound after the short-lived excitations (transients) from the plucking have died down. Otherwise it would look more chaotic. What you see at the beginning of the slow-motion video is the transients. As the narrator says at 0:43, "after several cycles, the pronounced kink has disappeared, and the motion looks much more like the entire string simply bounces up and down," and that's the part that you should compare to the illustration.

Also, the illustration is not very accurate. It's really just a plot of sin x + sin 2x, which would be the sound of an instrument with first and second harmonics of the same amplitude and no other harmonics. The motion of a guitar string, even in its quasi-steady state, is much more complicated.

In this video, the "wave mountain" propagates from one end of the string to the other back and forth and it looks nothing like the waveshape from above.

There is Fourier analysis, and there are boundary conditions. A string vibrates in a superposition of modes, but the phase and relative size of those modes is still to be chosen.

Plucking a string puts the string into a starting configuration (more or less a triangular shape) that determines the relative phases and amplitudes at the moment of plucking the string, and the different modes then progress mostly independently. Higher modes tend to be significantly faster damped, and there is inharmonicity that makes the mode frequencies not perfect multiples of the same fundamental frequency. Both of these factors mean that the starting configuration of the string does not reoccur in recognizable shape for long if at all. Instead the higher partial vibrations die out fast and leave lower, more stable vibration modes as the dominant sound source.

Also the momentary shape of the string and the sound pressure over time are both "wavy" but are two completely different things. There will be a way of guiding string tension or inertial momentum to a pickup or a resonance body radiating sound, and that creates a connection between one or two string fixtures and the generated sound. But the string shape as such does not translate into sound.