# Real image of a sawtooth standing wave in a musical string?

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• pkc111
In summary, the conversation discusses the theory of standing waves and harmonics in stringed instruments, as well as the use of Fourier analysis to understand the motion of vibrating strings. The participants also share videos and explain the difference between standing and traveling waves, and how different factors can affect the shape of a vibrating string. They also mention the possibility of using data collection software to analyze the sound and motion of a plucked guitar string.
pkc111
TL;DR Summary
Anyone got a video or animated gif of a standing sawtooth or square wave in a musical string?
Hi there
I am teaching resonance and standing waves in stringed instruments at the moment at high school.
The theory states that a number of standing waves simultaneously (harmonics) exist in a naturally vibrating musical string, but with varying amplitudes, the 1 st harmonic being loudest.
From my research on Fourier theory these standing waves should result in a standing sawtooth or square wave in say, a guitar string (lets assume plucked in the centre). I want to be able to show the students a real vibrating instrument wave slowed down so they can see clearly this composite wave...but alas.. so far no luck.
Any help would be appreciated.
Thanks

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Orthoceras
These don't look like standing waves..they look like traveling waves? The crest of a standing wave would reflect along the midline, not move position along the string ?? And the saw tooth shape in the plucked string video (first) looks like it was created by the initial position of the plucker's hand and not as result of the sinusoidal waves adding together.

Klystron, DaveE and sophiecentaur

Yes, in reality the kink travels along the string, and the kink is reflected at the ends. This is called Helmholtz motion. At any point in time the shape of the string can be decomposed in sinusoidal components by Fourier analysis, but I am not sure this would be helpful to understand the Helmholtz motion.

pkc111 said:
These don't look like standing waves..they look like traveling waves? The crest of a standing wave would reflect along the midline, not move position along the string ?? And the saw tooth shape in the plucked string video (first) looks like it was created by the initial position of the plucker's hand and not as result of the sinusoidal waves adding together.
The initial shape of the string defines the relative amplitudes and phases of the various modes at the start.
The first video makes the point that the higher order modes soon decay so you are not left with the possibility of a 'sawtooth' shape. i.e. it's more complicated than at first sight.
The situation of 'standing waves' never really arises except when all that's left is energy in the fundamental mode.

pkc111 said:
From my research on Fourier theory these standing waves should result in a standing sawtooth or square wave in say, a guitar string (lets assume plucked in the centre).

That's not my understanding. You can approximate, to any accuracy desired, any periodic function with a linear combination of any other periodic function. For example, you can add up a series of sine waves to produce a square wave, but you can also produce any other shape wave. A plucked guitar string, for example, can be observed with a strobe light, and it's not a square wave or a sawtooth wave. Not even close.

If you have access to data collection software you should be able to do your own fast Fourier transform of the sound coming from a plucked guitar string. You could then follow that recipe to construct the wave form. It wan't be a square wave or a sawtooth wave. Not even close. This would be a good guided exercise for your students.

Klystron
Mister T said:
A plucked guitar string, for example, can be observed with a strobe light, and it's not a square wave or a sawtooth wave. Not even close.
It's hardly surprising. The velocities for different frequencies may not be the same and the end effect will affect the relative phases. The sawtooth shape will only exist when it is actually created. Dispersion will destroy it (plus, as mentioned earlier, the different rates of decay).

Thank you for your reples. It still makes no sense to me..Is the theory wrong that harmonics (standing waves) can coexist in a uniform plucked string? Has anybody seen a string undergoing a sawtooth or square standing wave oscillation ...for any period of time.?Do they have a link to the vid ?

pkc111 said:
It still makes no sense to me..Is the theory wrong that harmonics (standing waves) can coexist in a uniform plucked string?
The harmonics co-exist, it's just that they don't have to add up to a square wave or a sawtooth wave. As I said, you can view the motion of a plucked guitar string with a strobe light. Look here. At about 2:40 into the video you can see a very low-tech demonstration for your students. Note the various shapes of the wave medium. None come close to sawtooth or square.

Thanks. Thats fine I get that they don't add up to a sawtooth or square wave. But that doesn't really help. I need to see a case where they do add to one in a string (even briefly). I can't even picture what a sawtooth standing wave looks like...does the peak stay at the same end? Any animations?
By the way the vid you referred Mister T shows forced oscillations imposed at a particular frequency on a string, its not free oscillation of a string as in a guitar.

pkc111 said:
I can't even picture what a sawtooth standing wave looks like...does the peak stay at the same end?
Looks like the teeth on a hand saw. Sort of.

No, the peak is assymetric (see black line below) and only goes from node to node in a Fourier model analysis of a string fixed at both ends. It is just the sum of the overtones drawn from superposition principle. See the coloured overtones below.
Are there any vids of a string briefly undergoing standing wave motion like this, or even animations of this shape behaving as a standing wave?

My visualisation problem of this shape standing wave is this:
If a shape like the black line were to be one extreme position of a standing wave in a string then I assume the peak (at the left in this case) would reflect to the bottom left position at the other extreme momentarily later, resulting in a high amplitude movement at the left and lower amplitude movement in the right half of the string.
So, if this were the case, in reality if I plucked a string in the middle it would vibrate with higher amplitude at one end of the string than the other? This doesn't feel right..and at which end would it vibrate most always? and what makes that the preferred end?
That why I actually need to see it really happening to be convinced it happens at all, or at least an animated model of it to see that I have the assumptions right.

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Nyquist says that to to see the detail you wish to see you need to strobe at twice the frequency of the highest harmonic. For this to be stationary requires everything to be linear (I think). Obviously a square wave is impossible and it seems a triangle wave is unlikely. The fact that you can induce the octave and the third with your finger is pretty impressive and that is easy to see.
Also do you know the demonstration using Chladny plates and a rosin bow??This is for 2D oscillations of a membrane...really good lecture stuff

Klystron
Nice to know hutchpd, but its not what I am asking

pkc111 said:
Nice to know hutchpd, but its not what I am asking
I was trying to tell you that such movies will not exist because of the nonlinearities in the real world.
The pleasure is mutual.

pkc111 said:
So, if this were the case, in reality if I plucked a string in the middle it would vibrate with higher amplitude at one end of the string than the other? This doesn't feel right.

Initially there is a traveling wave, no standing wave. I think your question is related to asking at what point in time exactly the traveling pulse transforms into a standing wave, using the standard model of a string. It is nice to examine this on the Phet interactive webpage 'wave on a string'. Set the damping to 'none' and watch the pulse traveling forever at a constant speed v. Theoretically, if someone would decompose the pulse in Fourier components (Phet doesn't), he would find out that all components travel at the same constant speed v, including the fundamental component, which has wavelength λ=2L. Next, repeat the simulation at nonzero damping, and notice that the amplitude is rapidly decreasing, but the speed remains the same. The pulse wave never transforms into a stationary standing wave! So why is a guitar or a piano different than this model, which nonstandard feature causes the development of the standing wave? I don't know, but I guess the fixed ends aren't really immobile, and maybe this gradually slows down the fundamental wave.

The frequencies of the different modes of a string are NOT harmonically related. This is why they are correctly called Overtones. The practicalities of real instruments (even stringed) account for the ‘colour’ of the sounds.
The Modes are determined by the wavelengths (not frequency) and the details of how waves are reflected by the ends of the strings - giving different effective lengths for all modes.
Talking in simple terms about Fourier Analysis of the waveform of the sound is missing a very important point.
PS The shape of the wave in the diagram at the bottom of the above post cannot represent a ‘plucked’ string. To get that shape would require the string to be released from that actual initial shape.

Klystron
Ok so now I'm really confused.
My assumptions are:
1 A large series of sinusoidal harmonics (standing waves) exist in a string which is resonating and fixed at both ends.
2. Plucking a guitar string produces string resonance.
3. The amplitude of the harmonics decreases with order.
4. The law of superposition holds so that the string takes the shape of the sum of the displacements of the component harmonic standing waves.
5. This shape of the sum of the harmonic displacements is a square wave as given by Fourier theory.

Could someone please tell me which of these assumptions is wrong?

Thank you

pkc111 said:
5. This shape of the sum of the harmonic displacements is a square wave as given by Fourier theory.
Where do you get that from? In purely mathematical Fourier theory, the sum could add up to any waveform.

anorlunda said:
Where do you get that from? In purely mathematical Fourier theory, the sum could add up to any waveform.
Very true ..Thank you...That is where I have been going wrong.

Wrong assumption is that the high orders are harmonics.
The high order modes do not last long enough to produce a sawtooth, even if the phases were intact.
Etc.
Any analysis of this situation has to be appropriate.

sophiecentaur said:
PS The shape of the wave in the diagram at the bottom of the above post cannot represent a ‘plucked’ string. To get that shape would require the string to be released from that actual initial shape.

The Phet animation resembles a pulse on a hammered string, like in a piano (video 3 in post #2). The propagation of a pulse on a hammered string may be equally informative as a plucked string.

sophiecentaur
Hammering a string can impress a pulse shape that's not related to the modes of the string (as when it's plucked) so that's another reason why the Attack sounds the way it does.

This may be a bit tangential from the OP.

And yet...

pkc111 said:
Summary: Anyone got a video or animated gif of a standing sawtooth or square wave in a musical string?

Hi there
I am teaching resonance and standing waves in stringed instruments at the moment at high school.
The theory states that a number of standing waves simultaneously (harmonics) exist in a naturally vibrating musical string, but with varying amplitudes, the 1 st harmonic being loudest.
From my research on Fourier theory these standing waves should result in a standing sawtooth or square wave in say, a guitar string (lets assume plucked in the centre). I want to be able to show the students a real vibrating instrument wave slowed down so they can see clearly this composite wave...but alas.. so far no luck.
Any help would be appreciated.
Thanks
This may work...

Swamp Thing said:
This may be a bit tangential from the OP.
It pretty relevant, I should say and the demo is a good one. The only point I would make is that the frequencies of those modes do not have to be harmonically related. I don't understand how so many PF contributors ignore the difference between Harmonics (time) and Modes (space) in these discussions. Anyone who has heard many instrumental sounds should appreciate this, even if the Physics is a struggle.

Remember that 'Hammond Organ Sound'? That was all to do with the fact that the sound was built up with Harmonics and not Overtones. It was not like other instruments and it sounded that way.

lloydiarth said:
This may work...

Obviously the apparent wavelength on the guitar string is not real. It is a camera artefact, the rolling shutter effect. It is beautifully explained and demonstrated with a high speed camera by SmarterEveryDay.

Orthoceras said:
...the rolling shutter effect. It is beautifully explained and demonstrated with a high speed camera by SmarterEveryDay.

If you want to really geek out on this subject:

BTW, how do you show the video link with a thumbnail and the "Click to expand..." ?

Swamp Thing said:
BTW, how do you show the video link with a thumbnail and the "Click to expand..." ?

It is unfathomable. I quoted a post which contained the full size video, and the forum automagically turned it into a thumbnail.

Orthoceras said:
the rolling shutter effect
A simple rolling shutter would produce different effects for a horizontal and a vertical string (the shapes of moving images get stretched or compressed in one direction. More generally I think it's just an artefact of sub-sampling. It would be interesting to see comparative pictures

In the video link I posted in post #26
Swamp Thing said:
-Video-
if you watch the video from 04:14, the sine wave is flipping sign about twice a second, i.e. ##\pm sin(x)##

This can't be a camera (sampling) artefact, I think?

If not... well, normally one would expect the envelope of the string to be a surface of rotation, at least when it's in steady state. But here you can see that the string seems to be constrained to, say, the upper half wave on the left and to the lower half wave on the right. This state lasts for about half a second, and then it flips to being the other way round.

Could it be that the LEDs are cleverly synchronized with the motor to achieve this effect, perhaps changing phase by 180 degrees every half second? This might work if each set of LEDs illuminates one half of the string... ?

Here is a capture from the video. One objection to the LED theory is that there is enough ambient light to see Tim's face, so even when the LED is off, we should be able to see the string, at least faintly, in the positive left hand side.

The visible region is from around ##-0.2 A sin(x)## to ##-A sin(x)##

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Swamp Thing said:
if you watch the video from 04:14, the sine wave is flipping sign about twice a second, i.e.
The polarisation of the string is rotating (as in the model with the string). That's normal for a vibrating string as there's nothing to keep it in one plane.

Swamp Thing said:
.. the sine wave is flipping sign about twice a second ..
Could it be that the LEDs are cleverly synchronized with the motor to achieve this effect, perhaps changing phase by 180 degrees every half second? This might work if each set of LEDs illuminates one half of the string?

The device does have a built-in LED strobe light. Other youtube videos of this 3D Standing Wave Machine mention it has adjustments for stroboscopic effects (flash speed, pattern select, and pulse control). Unfortunately, the description does not specify whether some of these stroboscopic effects are cleverly synchronized with the motor.

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