# Waves on a string

1. Apr 25, 2014

### leeone

How does where you pluck a guitar string affect the pitch of the note? Does it or does it not? Can you explain this in terms of harmonics? Does where you pluck it simply change the harmonic content and not the frequency?

Thanks.

Also how is resonance and standing waves on a string and harmonics related?

2. Apr 25, 2014

### Pythagorean

Changing the harmonics does change the frequencies, just not the dominant frequencies. The modes are determined from the length of the string (which you can shorten by fretting) so an A played on the open string is going to have fuller harmonics than the same A played on a fretted string as more modes will be audibly available.

Where you pluck it changes the relative amplitudes of the harmonics, emphasizing harmonics that are closer to integer multiples of the length between pluck region and the bridge and nut on either side, but still constrained by the length of ringing string.

Pickup location matters too. Neck pickups have a lower tone because they have more string on either side of them, whereas bridge pickups have higher tones.

3. Apr 26, 2014

### leeone

So what does this have to do with resonance? Is the string more "tuned" to resonate with the fundamental frequency (i.e the dominant frequency?) Is it called the fundamental frequency because it is the frequency which will resonate the most independent of the where you pluck it? I am still confused why the fundamental frequency is named so...and why is it the dominant one?

Also

What determines "n" in the equation? Is that what are you getting at in this quote?

f = n(v/2L) where V = √T/μ)....Does this take into account the material of the string? Linear mass density could be the same for two different materials no?

4. Apr 26, 2014

### Pythagorean

Yes, the fundamental frequency is the dominant frequency. It's determined by the length of string. You have to think about more than just a single frequency though with a plucked string. The sound that comes off such a string has several components, like the "attack" and the "decay" (read more here at the bottom), and what you hear as the long-term shape of the note is essentially the decay... that is, the fundamental dominating after you've plucked it and let the system relax to its resonant dynamics. But the attack is going to have a different harmonic content depending on where you pluck it, and even the decay will likely contain some subtle lasting harmonics specific to where you plucked it.

So, it's the higher harmonics that you'll be affecting. They're already there no matter how you pluck it, you just accentuate certain harmonics (give them a higher amplitude) based on where you pluck, since, as you noted the string has mass, the strength of the wave is weaker the farther you get from where you plucked it, so when you pluck on the mode underlying a particular harmonic (or set of harmonics), you're making that set ring louder.

I'm not sure what assumptions the vibrating string model makes about elasticity, as it's not explicit, but since the string is allowed to "change length" as it vibrates there must be some elasticity, and that must affect wave dynamics.

5. Apr 26, 2014

### leeone

I understand that there are multiple harmonics present on a string...but why is n=1 the dominant frequency? I guess I am looking for a more fundamental answer...would it be wrong to compare it to say a hydrogen atom that occupies the lowest energy state with the greatest probability? Is it the dominant frequency because that is the most energetically favorable mode? (Then again I know the energy of the string is determined by the amplitude...).

I am not sure if my question is making sense but you are definitely helping me so thanks for the replies.

6. Apr 26, 2014

### leeone

Also I was researching and I stumbled across this

"The lowest frequency is called the fundamental frequency, and the pitch it produces is used to name the note, but the fundamental frequency is not always the dominant frequency. The dominant frequency is the frequency that is most heard, and it is always a multiple of the fundamental frequency. For example, the dominant frequency for the transverse flute is double the fundamental frequency."

http://en.wikipedia.org/wiki/Timbre under harmonics....so the fundamental frequency is not necessarily the dominant frequency?

Also I am still trying to figure out if resonance is what is actually occurring when you pluck a string....Can you say that a standing wave is a result of the wave resonating with itself?

7. Apr 26, 2014

### Pythagorean

I don't know about hydrogen atoms, but with acoustics, the lower the frequency, generally the larger the thing that created it, so the more energy it has, and it can penetrate through more barriers than higher frequencies. It will go through walls, as you may know, more readily than other audible frequencies. When you pluck a string, it rings as whole and in halves and in thirds, and fourths, etc (which is where the harmonic series comes from). But the largest amplitude will come from the longest length on the strength, the string ringing as whole, which is the lowest note (with the exception of plucking harmonics that I mention below). Think about the relative hieight of the standing wave of the whole string vs. the one at 1/7 the string:

http://en.wikipedia.org/wiki/Harmonic_series_(music)

No, in general it is not. Under normal plucking conditions, it is for a guitar. Though you can pluck harmonics out in which case you might say the fundamental is the one produce by the whole string (the lowest note) but the dominant frequency is the harmonic you plucked.

8. Apr 27, 2014

### leeone

Okay I think the idea of harmonics just clicked for me, but one last question.

Can you say that a standing wave is a result of the wave resonating with itself?

9. Apr 27, 2014

### sophiecentaur

You do not need 'resonance' to have a standing wave. Waves hitting a wall will bounce back off it and form a standing wave. The standing wave will form, whatever the wavelength of the waves. You only get a resonance when there are multiple reflections, producing a build up of energy as the standing waves at each wall (or each of several walls) when the phases of all the waves happens to fit the resonance condition.

10. Apr 27, 2014

### Pythagorean

The are not the same, but they are both related to reflective boundary conditions.

Standing waves are a result of reflective boundary conditions, as sophiecentaur indicated.

Resonance comes from the geometry of the boundary conditions. The spatial distances between reflective boundaries in the system determines what wavelengths get reinforced (and thus, resonate).

11. Apr 27, 2014

### leeone

I guess I meant can you say a harmonic is a result of resonance..

12. Apr 27, 2014

### Pythagorean

That's an interesting question about cause and effect. I would argue they are both consequences of boundary conditions and one does not cause the other.

They appear to be two aspects of the same phenomena.

13. Apr 28, 2014

### sophiecentaur

Resonance is a strange thing. The power supplied in the form of an input signal ends up being dissipated, in or out of resonance. It's just that, at a resonance, the energy can build up after the signal is first applied. The lower the losses (/higher the Q of the resonance), the longer it takes to build up and the higher the final level of energy and the narrower the frequency range of the resonance peak. This applies with 'standing wave' type resonance or simple 'mass on spring' type resonance. The only difference is that in wave resonance, there are resonances at overtone frequencies as well as the fundamental.
(I always insist on using the word Overtone because there are many standing wave resonators that have Overtones that are not at Harmonic Frequencies.)