Physics: If the node of a string is at the saddle, how does it make the top vibrate?

[Alan Carruth]

Frank S wrote:

"The kinks that you are talking about here affect the timbre but is not the source of the main sound."

What I'm talking about is shown nicely in the video posted by Cuki79. Think about the forces on the nut and saddle as the kinks travel. See also Fletcher and Rossing; "The Physics of Musical Instruments", chapter 9, section 3, where they develop the math. This fits almost exactly with measurements of the transverse and tension forces I've made.

"Using a soft release halfway between the saddle and nut eliminates this and gives the purest sine wave sound. This is usually at the 12th fret on a guitar."

As I stated, the plucking point determines the relative strengths and phase angles of the partials of the string. Plucking in the center of the string suppresses all of the even order partials, since they would all have non-moving nodes at that point. The wave form produced is a square wave with a 50% duty cycle. Most of the energy is in the fundamental, but there will be some (not much) in the 3d and 5th partials as well, and maybe even some above that.

I'm not sure what you mean by a 'soft release'. If you're using a wide plectrum to spread the kink over a longer section of the string then you'll be suppressing the higher partials, of course.

"Plucking the string up or down does not have a 20db difference in volume. That is huge. The direction of the deflection of the string gives the same change in tension. These modes interchange even after the string is plucked. "

All I can tell you is what I saw (and yes, it's a huge difference). I ran the experiment in two ways. One used a 'wire break' pluck that can give very good control over the amplitude and location of the pluck as well as the initial direction of the string motion. In this case motion of the top does feed back into the string after a short time, but the initial level of the pluck was about 20 dB higher when the initial string direction was 'vertical' with respect to the plane of the top than it was when it was 'horizontal'.

The other used an electromagnetic drive: the string passes between the poles of a magnet, and an alternating current in the string causes it to move in one direction. This maintains it's polarization, but only contains the frequencies the driver puts in. Since I used a signal generator that produces sine waves I was seeing only one frequency. I don't have a unit that can produce square waves that is easily tunable it this time: heating of the string alters the pitch slightly, and with such a high-Q system you have to track it. This experiment, with the string being driven at it's fundamental frequency, is about as close as you'll get to your 'soft release in the middle'. It also produced about a 20 dB difference in output from the guitar, as measured with a dB meter.

"A flattop guitar does not have significant downforce on the bridge and that is in sharp contrast to the arch top which totally relies on it and BTW is much less loud."

Flat top guitars generally have a down force on the top of the saddle that keeps the string in contact. They also have an up force at the pins or tie block, depending on the bridge construction. You're correct in saying that flat top guitars don't need down force, but they do need some way of pinning the string at the top of the saddle so that the vertical transverse vibration of the string maintains contact.

In another post:
"An arch top guitar or a violin have quite significant downward forces on the bridges so their sound generating physics are quite different and they are not nearly as loud as a flat top guitar. "

First, let's not get mixed up between 'loudness' and 'power'. Basically, power is what you measure, with things like a dB meter, while loudness is subjective. It's very hard to devise a single method that relates loudness to power well in all cases.

Violins are certainly very different from guitars in many respects. I make both, and am pretty familiar with this. I don't know what to make of a statement that violins are 'not as loud' as flat top guitars: I've always found the opposite to be the case. Most guitar players are hard pressed to keep up.

Archtop guitars are not nearly as different from flat tops as people seem to think they are. Again, having made both, and studied them in some detail, I'm quite confident of this. The arching of the top adds a lot of stiffness, particularly at low frequencies, and reduces the output of the guitars, at least in principle. However, because of the added stiffness an arched top can be made significantly larger than a flat top without running into mechanical problems, and that enhances sound output. It's possible to make an arch top that has both plenty of power and is loud. The fact that so many don't and are not is not an indictment of the design.

Another post:

"The low E string should produce 82 Hz output of the guitar top but very few guitars do that."

That's because very few guitars have a strong resonance at that pitch. In most cases the lowest resonance of the guitar is the 'main air' mode, generally at around G~196. The low E is far enough down on the bandwidth curve that it doesn't get much help for the fundamental, so much of what you hear is the second partial.

I gave a talk about the string experiments I did some years ago at an ASIA Symposium, and I've posted it on my web site as a .pdf file entitled 'String Theory' (I couldn't resist). I've been re-doing some of those experiments recently to get some better data on some of the interesting quirks. I also did extensive experiments looking at the outcomes of changing the string height off the top and the break angle over the saddle, which basically only make sense if you accept the model I've been using, which is that set forth in Fltecher and Rossing (with some added extras). As I say, strings may be the simplest parts of the system, but they're not really simple. Didn't Einstein say something to the effect that everything should be made as simple as possible, but no simpler?

[Rodger Knox]

Originally Posted by jessupe

It should be noted that the movement shown is "side to side" which is what you would find in a bowed instrument. The guitar would demonstrate the same type of motion except it would be in an "up down" motion cycle

Not even close. That's a single wave initiated at a free end. The reflection is the same, but that's about it. A guitar string has two (2) waves starting at the pluck point and traveling in opposite directions.

[Alan Carruth]

On the violin the bow damps one of the kinks out and builds up the other. The driving force on the top of the violin bridge is thus (ideally) a saw tooth wave, with a duty cycle that depends on where the string is being bowed.

On a plucked string the kink runs out in both directions from the plucking point to reflect off the nut and saddle. This gives the square wave driving force at the top of the saddle of the guitar that I talked about. In effect this is the sum of two saw tooth waves like what you see on a violin.

The violin bow only allows the string to move 'across' the top, more or less. Driving the top that way doesn't produce much sound. The sound post, however, imposes a (nearly) stationary point on the top at the treble foot of the bridge. This converts the tall bridge into a bell crank, pivoting on the top of the post, and pushing downward on the bass bar, which then helps spread the vibration to the rest of the top. This is a much different system than the guitar, and makes drawing parallels hazardous.

[JeffreyAK]

Originally Posted by Rodger Knox

A guitar string has two (2) waves starting at the pluck point and traveling in opposite directions.

Initially, yes, but very quickly you get a mess of wave motions superimposed, including most prominently the fundamental mode that (ideally) has nodes at the saddle and nut and maximum amplitude in the middle. Some of this motion is side-to-side, some is up-down, and it all changes over time as the higher frequency motion damps out first.

Originally Posted by Alan Carruth

...motion of the top does feed back into the string after a short time, but the initial level of the pluck was about 20 dB higher when the initial string direction was 'vertical' with respect to the plane of the top than it was when it was 'horizontal'.

Interesting, how short is a short time, in your experiment? For sure after a good fraction of a second, my ears don't hear any significant difference, but it's possible there's a short transient that I'm not noticing. Sounds like there must be, if that's what you are measuring with instruments.

[Rodger Knox]

Originally Posted by JeffreyAK

Initially, yes, but very quickly you get a mess of wave motions superimposed, including most prominently the fundamental mode that (ideally) has nodes at the saddle and nut and maximum amplitude in the middle.

There's actually two points of maximum amplitude, one at the pluck point and another the same distance from the nut as the pluck point is from the saddle. That's because there's two waves going in opposite directions. There are the fundamental and all the harmonics superimposed on one another, with both up/down and side/side motion.

I'd also like to know how the 20dB was measured...

[LSemmens]

Thank you all for your contributions to this discussion, The Canons and physics of stretched strings have fired my imagination since I was in High School.

[Alan Carruth]

I wrote:
"In most cases the lowest resonance of the guitar is the 'main air' mode, generally at around G~196. "

That's what happens when you get in a hurry. I looked at my pitch-frequency chart and G~196 popped out. The main air mode on most guitars is actually closer to G~98 Hz. That's still far enough above the low E that the ~82 Hz pitch is well out of the bandwidth. Thanks to Howard Klepper for catching that.

Roger Knox asked:
"I'd also like to know how the 20dB was measured... "

With a dB meter (my trusty old RadShack unit) set on the 'C' (flat) weighting scale and 'fast' response. The meter was about 3 feet away from the top, iirc. The needle jumps right up and drops down, so that was the max reading. I had to use a different scale with far higher gain to get the crosswise pluck. This was a long time ago, and sort of a toss-off experiment, so I'm not as clear on the details as I'd like to be.

I described the magnetic driver. The wire pluck consists of using a length of fine magnet wire (~#44) to pull the string aside. You loop the wire under the string and pull until it breaks. This stuff is so consistent that it breaks within about 2% of the same force every time. A researcher who works for one of the string manufacturers says it's the most consistent mechanical pluck he's been able to find. You do have to be careful to check that the wire broke behind the string, and not someplace else. With care you can get virtually the same plucking force at the same location and initial string direction every time.

It is a large difference, but makes some sense when you keep in mind that:
1) the 'transverse' force averages 7 times the amplitude of the tension change signal,
2) the bridge moves far more in the 'vertical' direction that the transverse force is driving than in the 'longitudinal' direction of the tension change, and,
3) vertical bridge motion is far more effective at producing sound than bridge rocking for a given amount of motion.

To get back to the original question, the answer is that the bridge is almost a stationary node. That, in fact, is it's 'job'; to tell the string how long it is so that it 'knows' what pitch to make. Again, if it really were a node you would not hear anything from the guitar, but the closer it can come to being one the better the string is likely to sound. So, as so often happens on the guitar, we're trying to find a balance between a string sound with lots of harmonic partials, and a top and bridge that allow for lots of sound. It's a tricky balancing act, but fortunately the traditional designs tend to establish decent limits for everything.

[jaybones]

Originally Posted by 3piece

You ever plunge a toilet? Same thing...

I didn't know you'd heard me play!

[sirwhale]

Originally Posted by Alan Carruth

I wrote:
"In most cases the lowest resonance of the guitar is the 'main air' mode, generally at around G~196. "

That's what happens when you get in a hurry. I looked at my pitch-frequency chart and G~196 popped out. The main air mode on most guitars is actually closer to G~98 Hz. That's still far enough above the low E that the ~82 Hz pitch is well out of the bandwidth. Thanks to Howard Klepper for catching that.

My Lucky 13 has it's lowest resonance around F sharp, and I have the characteristic thuddy note there. It has great bass for its size and compared to much bigger guitars.

But my question is why more guitars are not built to have the resonance around D below the low E.

My guitar has a thin carbon fibre top and no bridge plate, which must help as it only has a 14inch lower bout.

Maybe with more composite guitars being made, this can be achieved (if it is a structural problem).

[printer2]

Originally Posted by Alan Carruth

... With a dB meter (my trusty old RadShack unit) set on the 'C' (flat) weighting scale and 'fast' response. The meter was about 3 feet away from the top, iirc. The needle jumps right up and drops down, so that was the max reading. I had to use a different scale with far higher gain to get the crosswise pluck. This was a long time ago, and sort of a toss-off experiment, so I'm not as clear on the details as I'd like to be. ....

The good old RS meter surprised me. There was some disagreement in whether using two speakers on a guitar amp results in the same SPL as using one speaker. I set up the test using a RS meter and a professional Yamaha amp that has switched level controls in 2dB increments along with some nice large dB meters. The RS response matched what the amp said it put out. Measured the amp output with volt meters to make sure the Yamaha was putting out what it said it was.

[Alan Carruth]

sirwhale asked:
"But my question is why more guitars are not built to have the resonance around D below the low E. "

The short answer is that most people probably refer the sound of the usual setup. As to why..?

One guess has to do with the way the guitar works in the low range. Any box with a hole in it will have a 'Helmholtz' type of resonance; the sort of thing you hear when you blow across the top of the wine bottle. On a guitar shaped box with rigid walls this would be at around 125 Hz; say about B on the A string.

Of course, the walls on a guitar are not particularly 'rigid', and the top, in particular, flexes a fair amount. If you look at a top on a 'rigid' rim, but with no back, it will tend to move like a loudspeaker with a resonance at around 180 Hz; about F or F# on the D string.

On the assembled guitar the air moving in and out of the hole changes the pressure and pushes on the top. At the same time the top moving in and out changes the pressure too, and pumps air in and out of the hole. The two resonances are 'coupled', so that in the low range the guitar is a 'bass reflex enclosure'. The coupling shifts the pitches of the two resonances, often to about G on the low E string (the 'main air resonance), and the open G string pitch (the 'main top resonance').

These are the lowest two resonances of the guitar that can produce sound effectively, and there's usually nothing in between. Notes near a resonance are generally enhanced in power, and ones that are further from the peak can be quite a bit weaker.

As Frank pointed out, many guitars don't actually produce much power in the fundamental of their low E note. That's not because the string isn't pushing the top at that frequency, but rather it's due to the low efficiency of the box at making sound at that low a pitch off resonance. It would help that note a lot if the 'main air' resonance was at that pitch. The problem with that is that once you start getting off that pitch the output drops off fast. In loudspeakers they get around that by using a lot of 'damping' to make the resonant peak broader, but that's inefficient. They get away with it by dumping in lots of power, but we're working with a driver that's limited to a few millewatts, so we can't afford that. Since there's not another resonance until you get to the 'main top' mode that would leave a big hole in the output.

Putting the 'air' resonance as low as E also is a waste of power. The resonance has a 'band width' on either side of the peak, so if you put the peak at E then the bandwidth below that is not doing you any good. The better option is to move it up to some pitch just above the lowest note on the instrument. Since you don't play that note all the time you don't need it to be as strong, and putting the resonance higher in pitch gets the most use out of it, and leaves the smallest 'hole' between it and the next one up. Putting the 'air' resonance down at D would more or less waste it all. At that point why bother to put in a hole?

This is one of those things where we can use physics to explain what's happening, but not to decide on the best thing to do. 'Best' is an artistic judgement, and in something like the design of the guitar it's arrived at by concensus. The 'air' resonance is not at D below the low E because most folks prefer it to be higher. You can find that peak over a range of pitches, but I've seldom seen 'normal' guitars where it's much lower than F# or much higher than A. That's part of what makes a flat top guitar sound like a flat top guitar, and not an arch top, for example.

[JohnW63]

In a nutshell, from reading this, and trying to recall my physics classes, a guitar makes sound by vibrations transferred to the saddle by the strings motion. What I am not quite sure of is if there is agreement on HOW. Is it the changing in string tension on the vertical plane, as I think Alan has stated, or is it vibrations induced because the saddle is not at a perfect node point, or because of torques on the bridge/saddle that make the top move sort of up and down.

I can see how some can say the NET forces on the bridge could be zero ( The same amount downward as upward ) , but there is NOT a zero force of the string downward on the saddle. In fact, if the bridge and saddle had a zero sum , then flat top guitars would not need to be glued down, once the strings were up to full tension. Therefore, I think the zero figure is not in all directions. Perhaps just in rotational around an axis. There is certainly a force toward the nut. So, as the string vibrates or changes it's tension the force toward the nut changes and THOSE changes pull on the top. Because the strings start moving upward from the bridge plate and THEN move in a straight line toward the nut, the amount of force is not just a function of the angle of the strings at the bridge. It is the sum of the force applied by the string angle and the forces upward until it passes out of the bridge holes.

I have no problem standing corrected, but I might need some cool diagrams to pick a winner among the real experts.

Since it's not arm wrestling, and no one has raised their big meaty paw in celebration, I'm not sure who's winning.

[JeffreyAK]

We should be careful, when nothing is moving then there are no net forces anywhere. Everything in all directions sums to zero, as it must because if it did not then something would be accelerating. What there can be are static tensions and compressions, in strings, wood, truss rods, etc. We can add energy to the system by deflecting a string under tension, and then in the absence of dissipation things will move and eventually reach a stable state of harmonic motion determined by the string and guitar parameters and their resonant frequencies. But we've added energy by applying an external force to a string under tension.

[rick-slo]

Originally Posted by JohnW63

In a nutshell, from reading this, and trying to recall my physics classes, a guitar makes sound by vibrations transferred to the saddle by the strings motion. What I am not quite sure of is if there is agreement on HOW. Is it the changing in string tension on the vertical plane, as I think Alan has stated, or is it vibrations induced because the saddle is not at a perfect node point, or because of torques on the bridge/saddle that make the top move sort of up and down.

I can see how some can say the NET forces on the bridge could be zero ( The same amount downward as upward ) , but there is NOT a zero force of the string downward on the saddle. In fact, if the bridge and saddle had a zero sum , then flat top guitars would not need to be glued down, once the strings were up to full tension. Therefore, I think the zero figure is not in all directions. Perhaps just in rotational around an axis. There is certainly a force toward the nut. So, as the string vibrates or changes it's tension the force toward the nut changes and THOSE changes pull on the top. Because the strings start moving upward from the bridge plate and THEN move in a straight line toward the nut, the amount of force is not just a function of the angle of the strings at the bridge. It is the sum of the force applied by the string angle and the forces upward until it passes out of the bridge holes.

I have no problem standing corrected, but I might need some cool diagrams to pick a winner among the real experts.

Since it's not arm wrestling, and no one has raised their big meaty paw in celebration, I'm not sure who's winning.

Not changes in string tension. It's changes in the vibrating string's angular momentum (at the saddle and nut) that provides the energy for sound production. For example, if you hold one end of a thick rope and someone at the other end of the rope waves it up and down you will feel the rope shake your hand (no changes in overall rope tension required).

[Bruce Sexauer]

I only read the first page, and then I saw Alan had posted on this page, so it probably been revealed already, but I think there is a general misunderstanding about what is driving the top of the guitar. It may be partly the string swinging back a forth, but what that really represents is the string gaining and losing tension. It is less tension when straight, and more tension when longer, which it is when at either end of its swing. This is why it rocks the bridge fore and aft rather than side to side.

Short and simple . . . Then there's the little matter of tone, but that's another story.

Edited because I see that Rick-slo, who has the post before this one, and I disagree. Just goes to show how poorly the guitar is understood after all these years of scrutiny. Remarkable!

I think the rope is quite a different kettle of fish as it is NOT a resonant vibrating structure, though if pulled hard enough it could be. But don't try to hold your end of it!