Waves on a string fixed at both ends

[cavis]

Hi all,
I've got a question about waves and standing waves on a string fixed at both ends. I understand why only certain discrete wavelengths / frequencies are allowed to generate standing waves on a string such as a guitar string. My question pertains to understand what happens when a guitar string is plucked.

At the instant the string is plucked, the instantaneous shape of the string does not resemble any of the standing wave modes, but I understand that the pluck essentially sets the string vibrating with several modes at once... the fundamental and the higher harmonics. I guess I'm trying to visualize the shape of the string when plucked as being the sum of a Fourier series made up by adding up essentially an infinite number of standing wave frequencies.

My questions are:

1) Is my description of exciting a number of different standing wave frequencies at once correct?

2) Can frequencies other than the set standing wave frequencies be generated by the initial pluck? Do these other frequencies rapidly die out (which is why they seem to be seldom mentioned at all in descriptions in introductory texts), or are they never excited at all? If they're never excited at all, why is that the case?

I guess the main thing I'm struggling with is how a seemingly arbitrary string shape generated by the pluck can be represented just by the standing wave patterns.

Cheers,

Chris.

 

[Simon Bridge]

Hi all,
I've got a question about waves and standing waves on a string fixed at both ends. I understand why only certain discrete wavelengths / frequencies are allowed to generate standing waves on a string such as a guitar string. My question pertains to understand what happens when a guitar string is plucked.

At the instant the string is plucked, the instantaneous shape of the string does not resemble any of the standing wave modes, but I understand that the pluck essentially sets the string vibrating with several modes at once... the fundamental and the higher harmonics. I guess I'm trying to visualize the shape of the string when plucked as being the sum of a Fourier series made up by adding up essentially an infinite number of standing wave frequencies.

My questions are:

1) Is my description of exciting a number of different standing wave frequencies at once correct?

Yes.

2) Can frequencies other than the set standing wave frequencies be generated by the initial pluck?

If you pluck a guitar string from the middle, the shape of the string is a triangle. The triangle shape can be represented as a sum over harmonics.

Do these other frequencies rapidly die out (which is why they seem to be seldom mentioned at all in descriptions in introductory texts), or are they never excited at all? If they're never excited at all, why is that the case?

The extra harmonics typically die out - this is mainly due to damping, but the sub-harmonics in a note are what gives different instruments their characteristic sound. Some sub-harmonics are reinforced by the geometry of the instrument - compare an (acoustic) Bass guitar with a regular guitar and with a Ukelele.

Perhaps a more general idea is needed?

1. If you distort a spring at one end, creating a pulse, you know the pulse travels to the other end, inverts, then travels back (fixed ends) right?

2. So, if you distort the far end instead, the pulse travels to the near-end, inverts, then goes back. So far so good.

3. But what happens if you were to make a pulse exactly in the center of the spring?
Which way does it go?

4. The answer is that 1 & 2 are incorrect. In each case the pulse splits in half - each half going in opposite directions. If you look closely at the initial two cases you'll see that each pulse is being followed by an inverted version.

5. when you pluck a string, you make a big triangle pulse stretching the whole length of the string. You should be able to split that into two equal sized pulses going in opposite directions, inverting a the ends, and then reinforcing appropriately to make a funny looking standing wave ... do it.

With no losses, the standing wave starts out as an upright triangle - the top gets flat - the flat part spreads but the slope parts keeps it's shape - until it is totally flat and starts building an inverted triangle.

With losses, some parts of the string lose energy faster than others - the shape that is sustained will be sinusoidal ... in the math, we see the Fourier transform lose the higher frequency terms.

 

[cavis]

Thank you for the response, Simon. It helps a lot. May I ask another quick question?
A classic demo involving waves on a string is to attach a vibrator to one end of the string whose other end is a fixed end. The vibrator oscillates up and down with a very small amplitude (so this end of the string is effectively a node also) and can excite large standing waves in the string if the vibrator frequency is the same as a harmonic frequency for the string.

What I'm curious about is the case where the vibrator frequency is not equal to a standing wave frequency... here the string does not vibrate up and down in a nice standing wave pattern. But, if I understand Fourier's theorem correctly, the sinusoidal waves that are being generated on the string by the vibrator can still be visualized as a sum of the harmonic standing waves on the string since any periodic function on the string should be representable as the sum of the harmonics even though the frequency of the sinusoidal waves being generated is not itself a harmonic frequency. Is this correct?

I also understand that when we hear the sound of a string being plucked our ear interprets the sound's pitch as essentially the fundamental frequency plus a bunch of harmonics which determine the tone. If we were to consider the case describe above with a string deliberately excited at a non-harmonic frequency, would we still perceive the pitch of the sound emanating from the string as being the fundamental for the string?

Thanks again for your response!

Chris.

 

[sophiecentaur]

The extra harmonics typically die out - this is mainly due to damping, but the sub-harmonics in a note are what gives different instruments their characteristic sound.

Even for a lossless string, the only modes that can be exited will be the fundamental and overtones. Energy just wouldn't be coupled into any other frequencies due to the boundary conditions.

 

[Simon Bridge]

@sophiecentaur: yeh - I wouldn't want anyone to think I meant that any frequencies not consistent with the boundary conditions could be excited. I mean only that the fundamental + overtones so set form a basis and any waveform on the string can be represented as a linear superposition of them. The components that are not otherwise reinforced die out - mainly due to damping. I found a demo:

... I tried to find one with a less lossy string, plucked dead center, but no luck.
However, it does illustrate a lot of what I was talking about and can be used as a clarification.

 

[Simon Bridge]

A classic demo involving waves on a string is to attach a vibrator to one end of the string whose other end is a fixed end. The vibrator oscillates up and down with a very small amplitude (so this end of the string is effectively a node also) and can excite large standing waves in the string if the vibrator frequency is the same as a harmonic frequency for the string.

If you drive a string like that, you can get a node quite close to the multivibrator - the effective length of the string is to that point (as far as your-level calculations are concerned.)

Start at resonance for the fundamental - you get a high amplitude standing wave.
As you increase the frequency of the multivibrator the amplitude of this wave decreases - after a while there is very little excitation at all... keep increasing the frequency and the next harmonic appears and increases in amplitude.

There are other "natural modes" due to the physical realities of the experiment. See discussion: http://www.colorado.edu/physics/phys4830/phys4830_fa01/lab/n1106.htm for examples.

You can also see low-amplitude traveling waves at the in-between frequencies. In demos, the amplitude of the multivibrator is usually chosen to be small so that the traveling waves just make the string look a bit blurry to the eye.

There are a bunch of video demos on youtube that show all this - go look.

I also understand that when we hear the sound of a string being plucked our ear interprets the sound's pitch as essentially the fundamental frequency plus a bunch of harmonics which determine the tone. If we were to consider the case describe above with a string deliberately excited at a non-harmonic frequency, would we still perceive the pitch of the sound emanating from the string as being the fundamental for the string?

You can do the experiment and listen to it and see.

When you drive the string you normally hear the driving frequency as a hum.
These are really the sorts of things you just have to experience to "get" so next time you have the opportunity - take it.