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hugh evans wrote:
"Rather the string vibration is close enough to a resonant peak in the instrument response itself that it's inducing sympathetic vibrations, aka wolf tones."
It's actually trickier than that, if I understand what you're driving at. In cases where there is a resonance in the instrument that causes the top to move a lot at the pitch of a played note, or a partial of one, the string resonance can actually be 'split', so that it shows up at two different frequencies close to each other. In addition, the low impedance of the top at resonance can bring it close enough to to the impedance of the string to seriously reduce sustain.
All of the math that describes what vibrating strings are supposed to do starts out by assuming that the ends are 'fixed': totally immobile. If this were the case on a guitar, you would not hear it, so we know that condition is not being met.
The main way that strings drive the top is through the 'transverse' force developed as the string vibrates. There are a couple of ways to look at this, but one way to think of it is that as the string moves 'up and down' it pulls the top along with it. The string can also move 'side to side', of course, but, except for the possibility of rolling on the top of the saddle, the bridge itself won't move that way enough to make a difference: you can consider the bridge to be a 'fixed' termination for crosswise motion over most of the frequency range of the guitar. That's not true in the 'vertical' direction, particularly at the 'main top' and 'main air' pitches (Trevor's two (1,1) monopole modes). If the end of the string is moving, the stationary point that defines the vibrating length will not be at the top of the saddle, and the string will thus 'want' to make a different pitch than its supposed to, but _only_ when it's moving 'vertically'. As Trevor points out, the effect of a strong top resonance is to 'repel' the string pitch: it will be either lower or higher than the top resonant pitch, but can't match it.
Normally, of course, when you play a string, you don't set it off in a purely 'vertical' or 'horizontal' direction; it's moving at some angle. If, say, it's the open G string, and the 'main top' resonance is just above that, at say, 196 Hz, what will happen is that the 'horizontal' part of the vibration of the string will produce the correct pitch of 195.9 Hz, while the 'vertical' motion will proceed at a lower frequency. The string will 'beat' with itself at whatever the difference frequency is. If that difference approaches seven Hz or so, it will sound a lot like a fret buzz. This is not really a 'sympathetic vibration'; nothing else needs to get involved, since the string is beating with itself.
The really extreme case of this is the 'wolf' on the 'cello. There are four major ingredients to this recipe:
1) the string is bowed, so there's a constant input of energy,
2) the bowing constrains the motion of the string to a single plane, in the direction that the top of the bridge is moving the most,
3) the bridge itself is very tall, so that a small motion of the top can induce a large travel at the top of the bridge, and
4) there is a strong resonance that causes the bass bar to move a lot vertically, while the sound post is 'nailed'.
When this happens (all too frequently) the impedance of the top of the bridge at the string end for the fundamental of the note (remember that impedance is frequency dependent) becomes very nearly equal tot he characteristic impedance of the string itself. In essence, te bridge 'looks like' a continuation of the string at that frequency, so the string loses its termination, but _only_ at that frequency. There's usually a decent impedance mismatch at the octave, so the string simply shifts upward in pitch. When this occurs, the vibration of the body at the troublesome fundamental is no longer being fed, and it does out quickly, since this is usually a strong radiator of sound. Once the body vibration dies away, the top of the bridge stops moving, and the string suddenly 'sees' a proper termination, so it shifts back down to the fundamental,and the whole thing starts over.
If you could bow the open G on a guitar in the 'vertical' direction, you might well get the same sort of response, particularly on some Classicals or Flamencos with very light bridges. As it is, I've seen guitars that had a pretty exact pitch match between the top mode and a played note where there is not much of a problem. In some cases the mass of the bridge is sufficient by itself to impose a reasonable termination on the string. In other instances, the fundamental does indeed 'come and go', but it's not very noticeable in the overall transient of the sound.
There are _lots_ of other 'wolf' notes that lurk in the guitar, and some do involve vibration of the back or air as well as the top and the string. This makes the hunt challenging, but often, once you've found out the cause, it's pretty easy to fix a wolf.
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