Graduated thickness of top plates

[Alan Carruth]

Any extensive system will have a number of different modes of vibration, theoretically as many as the number of elements it's made of. Thus a line of ten coupled pendulums will have ten possible modes of vibration. A guitar top is a 'distributed' system, with theoretically an infinite number of elements. In practice, of course, many of those modes would be at very high frequencies. I'm not mathematically competent at that level, but I'm told that guitar plates tend to have resonant modes that are more or less arithmetically spaced out: one every 50 Hz or so, say. I've been able to find something on the order of a dozen modes on guitar tops up top about 1000 Hz, using Chladni patterns. Above that frequency it's difficult to drive them hard enough to get the glitter to bounce.

The guitar as a whole is a system, of course. It's possible to find around a dozen 'back' modes below 1000 Hz too, in the same way, and a similar number of internal 'air' modes using microphone probes. Of course holography can show more. And then there's the sides, and the neck...

Up around 600-800 Hz or so there are so many modes that the overlap; the peaks are closer together than the half-power band widths, so they couple very strongly. When they do each mode so coupled will shift in pitch, which may affect the way it couples with another mode. I'm told that in this 'resonance continuum' it is literally impossible to predict in advance what the actual behavior of the instrument wil be with any precision. The output in, in some sense, a vector sum of all the modes working together. You can say that if you see ten output peaks in the octave between 500-1000 Hz, there are ten strong resonances in that range, but you can't say with any assurance which one contributes the most to any particular peak, and there may be no one part of the guitar that's vibrating strongly at that exact pitch.

So, what we can control 'up front' is the stuff that happens in the lowest range; up to 500 Hz or so. This establishes the 'character' of the sound: whether it sounds like a Parlor, or a Dreadnaught, or an archtop. Apparently whether it sounds like a good Parlor or Dreadnaught or archtop depends on what happens up in the 'resonance continuum', where you have no direct control of the pitches.

What seems to matter in that range is the number of peaks per octave, and the ratio of peak height to dip depth which is determined mostly be the losses in the system. There seem to be 'best' ranges of values for both the peak number and peak-to-dip ratio. The peak number depends on how many resonances we build in, and the losses depend on the damping of the structure. This damping, in turn, seems to be limited to some extent by the damping of the material: it's hard to produce a low damping structure from high damping material. OTOH, arched plates will tend to have effectively lower damping in their higher modes because of the way the arching moves the pitches up. Thus a maple back on an archtop might have effectively the same high frequency damping as a flat rosewood back.

I didn't mean to get so deeply into this, but you can see that it gets complicated when you think about it...

[Marshall Dixon]

...Thus a maple back on an archtop might have effectively the same high frequency damping as a flat rosewood back...

Can we say that increasing stiffness decreases damping?

[Alan Carruth]

Marshall Dixon asked:
"Can we say that increasing stiffness decreases damping?"

No; actually, it's the other way around, sort of. Increasing the bending stiffness at low frequencies by arching the plate produces lower effective damping by shifting their resonant pitches upward.

First, I made a mistake in that post: post in haste, repent at leisure... It's the low-order modes that get shifted upward in an arched plate, not the higher order ones. Thew lowest pitched mode on the assembled guitar is the 'ring' type mode where the top acts as a loudspeaker. The curvature of the plate stiffens it against that sort of bending, and raises the pitch. It's similar to using a cone shape for a loudspeaker, rather then making it flat. In the speaker the cone prevents the thin paper from bending at lower pitches, so the whole thing responds as a unit up to a fairly high frequency, but at some point it can break up into smaller vibrating areas rather suddenly. On a curved arch the added stiffness raises the pitch of the lower order modes, and the cross over into more complex vibration patterns is not as sudden. As you go up in pitch the vibrating areas get smaller and smaller. At some point, when the effective curvature of each vibrating element is small, it works more like a flat plate.

The damping factor of a particular mode is found by looking at how sharply defined the peak is. You take the most active frequency, and find the frequencies on either side of that point where the activity is 3 dB lower (70.7% of the amplitude) for the same amount of driving power. At those points the energy stored in the system is 1/2 the peak energy, so this defines a 'half power band width' for the system. Higher losses produce a wider band width. In a wood plate most of the loss is internal, due to the nature of the particular piece of wood.

If you take the center frequency for the band, where it's most active, and divide that by the half power band width you get a number called the 'Q value' or 'quality factor' of the oscillator. I high Q system has low losses. If you tap on it it produces a more secure sense of pitch, since more of the energy of vibration is focused on that peak frequency: high Q materials like aluminum can ring at a well defined pitch, where foam plastic, with it's high losses, goes 'thud'. Arching a maple back plate makes it act more like a rosewood one at low pitches, since it moves the center frequency upward, but the band width doesn't change, since that mostly depends on the wood properties. The higher order modes are less effected by the arch.

I'm sorry if that caused any confusion.

[Marshall Dixon]

Thanks for that information Alan.