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George Wilson wrote:
"The string has only a certain amount of energy when it vibrates,to impart a movement in the bridge. It is elementary physics that the grater the mass of a body is,even in outer space,the greater the moment of inertia is to set it in motion. "
What's the job of the bridge? IMO, it's to tell the string how long it is, so it will know what pitch to make. In theory, a string only works properly, making a nice harmonic series of partials and so on, when both ends are fixed.
This is a problem: if the bridge doesn't move you won't get any sound out of the guitar, because it's the movement of the top that makes the sound and the bridge is glued to the top. So, as so often happens with the guitar, we're loking at some sort of compromise; we want the bridge/top system to be stationary enough so that the string doesn't act up, and mobile enough to produce some sound.
So how do you get the bridge to stay still, or at least, sorta still? Well, you can add mass, or you can add stiffness. Adding either raises the 'impedance' of the bridge. Mechanical impedance is simply the ratio of force over velocity at a given frequency; anything that makes the bridge harder to move adds to the impedance. You can add stiffness, or mass, or damping, and get higher impedance, but each one will have a different effect on the frequency response.
'Resonance', BTW, is just the frequency where you get the most motion for a given input of power, so it's the frequency where the impedance is at a minimum.
We all know that adding mass in a given system drops the resonant frequency. Adding mass doesn't change the impedance much at low frequencies, but raises it more as you go higher. Using a heavier bridge on a given top cuts down on the treble response in general, so the balance shifts toward the bass.
Adding stiffness has the opposite effect; making it hard to move the top at low frequencies, but not adding as much impedance at higher pitches. The resonant frequency of the system goes up, and the guitar gets more treble balanced.
Technically, it's said that 'mass reactance' and 'stiffness reactance' are equal, and opposite in sign, at resonance, so that they cancel out. As much energy is stored in stiffness when the system is fully deflected (and not moving) as is stored in inertia when the system is not deflected, but moving at it's maximum velocity. The only thing that's left to overcome is the 'resistive reactance': you need to replace any energy that's lost to friction, sound production, or what have you. Adding resistance doesn't change the resonant frequency, but it does cut down the amplitude at resonance, without having much effect off resonance. The output curve get's 'flatter', and, maybe, less 'interesting'.
Strings have impedance too. The 'characteristic impedance' of a string is proportional to the square root of (mass per unit length times tension). The actual impedance of a given string depends in a more complicated way on the length as well. In the case of a string you can think of it as a measure of how hard the string can push on the bridge, with the frequency response stuff working pretty much the same way. The impedance of a string is high off resonance, and low at it's resonant pitches.
Any time you hook two objects together that have different impedances some energy is reflected at the boundary. When you hook together things that have the same impedance, all of the energy can cross the boundary: in fact, as far as the energy is concerned, there _is_ no boundary. So the job of the bridge can be redefined as that of presenting an impedance mismatch to the strring that will keep enough energy in it to allow it to vibrate with a stable frequency.
The 'cello 'wolf note' is a good example of what happens when there's not enough impedance mismatch at the bridge. In that case, a strong resonance of the body causes the top of the bridge to move so much at that frequency that the string doesn't 'see' it as a stationary end. All of the energy at that frequency leaks out into the body of the 'cello, and the string is no longer maintaining that partial, the fundamental of the note. The pitch of the string jumps up an octave (those frequencies 'see' a decent impedance mismatch), and the body resonance, which is no longer being driven by the string dies out. When it does, and the top of the bridge stops moving, the string suddenly discovers that it can make it's fundamental again, and the whole thing starts over. This happens several times every second, causing the 'bleating' or 'growling' sound characteristic of a 'wolf tone'.
I have been doing some tests on a classical guitar that has it's 'main top' resonant mode at 195 Hz, just below the pitch of the open G string. That note is loud, and lacks sustain: a classic 'guitar wolf'. If we were bowing the string, it might do just what the 'cello does. One thing it does do is 'split' the pitch of the string. If you pluck the note and analyse the spectrum, it has two peaks; one at 194 Hz and another at 199. You hear the pitch of G at 195.9 Hz, and so does the electronic tuner, but it comes and goes in strength a few times per second. When I used an electronic gizmo to drive the D string capoed up to that pitch it wants to go at 195.6 Hz in the 'vertical' direction (relative to the soundboard) and 194.1 'horizontally'. So there's another outcome of too close an impedance match.
Adding either stiffness or mass to that bridge are would help solve the problem; give the note more sustain and elmininate the 'beating'. With the higher impedance it might show up in a weaker form at some other frequency, and whether that new 'wolf' pitch was higher or lower than the original would depend on which thing, stiffness or mass, I added to solve the problem. So neither mass nor stiffness by itself is 'evil', it all depends on what the system looks like, and what you want it to do.
Boy, this post has gotten long! I'll just close by saying that violins and banjos are somewhat different problems, and leave it to the reader's imagination to say why.
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