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

[FrankS]

No disrespect but a lot is mixed up here.

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

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.

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

Frank Sanns

Originally Posted by Alan Carruth

Having spend 'way too much time doing the experiments, I think I'm qualified to weigh in on this. So far I have not seen the right answer here (on further scrutiny, I see that Roger Knox did post the correct answer, and with less verbiage too; sorry Roger).

Think about what happens when you pluck a string: you push it down toward the top with the pick or your finger, and then let it go. Of course, you usually push it down at some angle to the left or right, but basically you displace the string at the plucking point, and it makes some angle with the saddle top as compared with un-displaced string. There will almost always be at least some downward component, so for the moment well concentrate on that.

When you push down on the string you're also pushing down on the saddle top. If you like you can measure this with a feeler gauge. You can also calculate the downward force. If you know the string tension and the downward angle you can find it on a trig table; it's the sine of the down angle times the tension, if I've got that right. Since the angle is usually less than about five degrees it will be small; something like 5% of the tension, more or less.

What happens when you let go of the string is interesting. The 'kink' that you made with the pick or your finger starts running out toward the ends of the string. It moves at the bending wave velocity of the string, which is set by the mass and tension. No information about a bend in the string can travel faster than this. What this means is that until the kink actually reaches the bridge, the bridge can't 'know' that you released the string. The string is still pushing the bridge into the top, just as it was before you released the string. It stays like that until the kink reaches the bridge, and then it flips upward, so that the string is pulling the top out.

When you let the string go the kink split to become two kinks, each one traveling toward opposite ends. The kink that went up toward the nut also reflects when it gets there, switching from 'down' to 'up', and starts traveling back down the string. If we're talking about the A string, tuned to 110 Hz, when 1/22oth of a second has gone by the two reflected kinks will meet, at a point as far from the nut as your original plucking point was from the bridge. At this instant the shape of the string is a reversed reflection of the original shape, and things just go forward in the same way, with all the forces reversed.

This, by the way, is called a 'time domain' view of the string. If you look at the up-and-down force on the saddle it's a square wave, with a duty cycle that is set by the plucking point. That is, if you plucked the string 1/5 of the way along it's length, then the string will push 'down' on the bridge 1/5 of the time, and pull it 'up' the other 4/5. A Fourier Transform of that wave form will show that it has all of the harmonics of the string in different amounts except for the 5th, 10th and so on. That's the usual 'frequency domain' way of looking at it. Both of these views give the same information. You can switch from one to the other depending on what you're interested in. The time domain is particularly nice for resolving the forces involved.

Of course, when you displace the string you stretch it, and the tension rises. This pulls the top of the bridge toward the nut. The actual rise in tension depends on the nature of the string and how far it's been displaced, but NOT on the initial tension. When you know some stuff about the string you can calculate this force as well. It varies, but averages about 1/7 of the up-and-down 'transverse' force that string displacement can put on the top of the saddle. The easier it is to 'bend' the string pitch, the larger the tension change is relative to the transverse force. Note that the tension rises twice for every full cycle of string vibration; once when the string is 'down', and once when it's 'up'.

Keep in mind that we build guitar tops to resist the deformation of bridge torque. Bridge torque also pulls the top up behind the bridge as it pushes it down in front, so any movement of the top is partly cancelled out as a sound producer. In the end bridge rocking does not produce much power, but can contribute to the timbre of the guitar. A taller saddle gives a bit more energy in the second harmonic due to bridge rocking, but, according to the measurements I've made, doesn't produce any more power.

If the string is moving sideways, parallel to the plane of the top, it doesn't push the top in and out, but the tension change is still there. To give another idea of the relative contributions of the two I've done a couple of different experiments where the string starts out vibrating either purely 'across' the top or purely 'perpendicular' to it. Perpendicular motion produces about 20dB more power: that's 100 times as much.

So, basically, forget bridge rocking. The actual sound of the guitar is produced by the fact that the string, in vibrating up and down, pulls the bridge up and down and makes the top move like a loudspeaker cone. Tension change is certainly there, but is not a major source of power. If it were, then flat top guitars, where the strings can torque the bridge, would sound an octave higher than archtops, where they can't, with the same strings at the same tension. Do they?

[Rodger Knox]

Originally Posted by FrankS

No disrespect but a lot is mixed up here.

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

That is correct, although I doubt it's anywhere near a sine wave. Ideal strings do not exist in the real world, which makes this more complicated that it should be. Plucking in the middle of the string results in the waves going in opposite directions reach the nut and saddle at the same time, and the reflected waves intersect each other at the middle of the string.

Originally Posted by FrankS

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.

It's not the pluck that matters, it's the motion of the string after it is plucked. The vibration can be broken down into two components, perpendicular to the top and parallel to the top. Both of these components exist shortly after the pluck. They are not completely independent of the pluck, but any pluck will generate both components. The component perpendicular to the top produces more power.
I'm not sure what you mean by the modes interchange.

Originally Posted by FrankS

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

Frank Sanns

I'm not sure what the difference in static loading between a flattop and archtop makes, but it's the component of vibration perpendicular to the top that provides the lion's share of power to a flattop and all of the power to an archtop.

[AllThumbsBruce]

It is interesting that the idea that the sound is produced by changes in the string tension rocking the bridge and thus moving the top persists in the face of what seems like an obvious refutation by elegant experiments performed by Alan Carruth. Those that persist in this belief must a) not understand Alan's experiment, b) think he did it wrong, or c) think it does not lead to the conclusion that Alan has drawn.

In the simple terms, Alan excited string oscillations that were either parallel or perpendicular to the guitar top. (This does seem tricky, since, as Frank S points out, there is mixing of the polarizations and the pure polarization does usually not persist for long.) In any case, both would have the same change in tension and hence produce the same rocking of the bridge. However the string oscillations that are perpendicular to the top will produce an up and down force on the bridge, while the oscillations parallel to the top would not. Alan observed that the oscillations that were perpendicular to the top produced 100 times more sound, leading to the conclusion that it is not the rocking of the bridge that produces most of the sound, but the up and down force from the string moving the bridge up and down and hence moving the top and pumping air and making sound. In addition Alan points out that the rocking of the bridge occurs at twice the frequency of the string oscillation (see his post above), so the rocking of the bridge mechanism would produce a note at the first harmonic, not the fundamental.

(Alan please correct me if I got it wrong here).

In any case, I have read a lot of posts on both sides of this issue and Alan's experiments are the only actual evidence I have seen from either side.

[Cuki79]

Originally Posted by sirwhale

Thanks, I was speaking about this with my physics teacher colleague (I'm a biologist). He understood you answer, which he says supports what he was thinking, but, I'll need him to explain your answer a bit! haha

My colleague asks:

"Thanks for the reply! I am a teacher and I'm thinking of experiments research students could carry out in this area. I'd like to ask you something related.

From what I understand, some consequences would be:

- The longer the note, the closer to a "pure" note (in which the actual frequencies coincide with the frequencies of harmonics). This is because the frequencies further away from the harmonic dissipate energy to the bridge faster.

- With a frequency analyzer, we should see that in the first instant, there is a mess of frequencies and later the harmonic peaks appear.

The question is, am I right? And could these properties be found by using a frequency analyzer?

Also, if f is the frequency of the fundamental, you should get a peak intensity at f plus or minus some deltaf and a dip at the fundamental frequency f. Could that be detected?

Thanks again!"

I am not sure My answer led to that conclusion. If it was the case, the following video would end by a fundamental stationary mode and no left right propagation... It is not totally the case.



However those spectrograms from Yamaha go into your direction


On those the higher harmonics vanish faster than the fundamental ( I am not talking about the lowest frequency that involves the helmholz resonator due to the sound hole). Note that note only the strings harmonics but also the higher order modes from the tops play a role as previously described by Allan Carruth

I don t understand the last part of the post about delta f.

I tend to agree with your overall interpretation but it would mean that Taylor es2 miss the last part of sustain of every note... Did anyone observe that?

[Cuki79]

The fact that the bridge rocking is negligible does not totally contradict with how the es2 works. It only contradicts how the es2 is advertised and sold!

Probably the es2 benefits mostly from being not under too much tension. Piezo are usually nonlinear, and putting them under pressure under the saddle must not help.

I think the vertical vibration is coupled to the top and efficiently transfert to air. The horizontal vibration should be the least dissipated to improve sustain. It does not mean you can not take information from there. As far as the double frequency problem, when you watch the slow motion video, the left right movement is at the fundamental frequency so it is ok.

It just shows that es2 does not get any information from the top coupling to the air... But no other sensor does ( except may be the amulet)

Again: my 2 cents

[JeffreyAK]

Originally Posted by Alan Carruth

... I've done a couple of different experiments where the string starts out vibrating either purely 'across' the top or purely 'perpendicular' to it. Perpendicular motion produces about 20dB more power: that's 100 times as much.

Interesting. If I pluck a string sideways vs. up/down, with a small approximately constant amplitude somewhere near the middle (to keep things simpler, this produces fewer harmonics), I don't hear any significant difference in volume for about the same initial amplitude of motion. This is true on guitar and also mandolin, an arch-top with a tailpiece. How do you measure 100x more power perpendicular vs. parallel to the top?

[FrankS]

The downward force on the bridge is not very high and in fact can be zero.

This is often confused with the break angle that keeps the strings in close contact with the saddle. There is no net force there as it is just like trying to lift yourself up while seated in a chair. There is an equal and opposite force between the ball ends of the strings pulling upward with the same force as the strings pushing down on the saddle. No net up or down on the top of the guitar.

The force upward or downward on the guitar top are only due to neck angle and the hight of the nut to the saddle. This is typically a very small number like 0.25 inches out of 25 inches. Plugging in the trig function gives on the order of 1 pound of downward force for an average dred. The neck angle can actually be dropped down so there is no angle and the guitar will still play quite loud at a zero degree angle. While this is not ideal for playing, it demonstrates that it is not the up and down force on the top from the horizontal or vertical plucking of the strings that causes the majority of the volume from the top.

Next is polarization issues and the reported differences in amplitude or harmonic content of a string gently plucked at the 12th fret. Yes, I know strings are not exactly idea but let's not get caught up in the minutia while there are still large factors to consider. Polarization has a great deal to do with how the string passes over the saddle. It is not a symmetric system as the strings are in a groove cut by the strings. This is a different geometry than the path that the strings come over the saddle. All saddles have a different radius for the strings to cascade over. Just like in school, if you have horizontal blinds, a string will oscillate well in the horizontal direction but not the vertical. Go with some vertical freedom and the string will prefer to go vertically. This is complicated in the guitar because the higher harmonics are either allowed to be there or filtered out by the non symmetry of the anchoring of the ends of the strings.

The saddle is really a direction changer. It would do the same thing as a pulley for the string tension if it were anchored to another structural member above the guitar top. The is not the case so the saddle/bridge arrangement has a net torque on it. Changing string tension when plucking a note will change both the torque on the top of the guitar and downward force. Since a zero neck angle will give no downward force no matter what the string tension, the sound has to be generated by some other mechanism, namely changing torque with changing string tension from the pluck.

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.

Frank Sanns

[Rjlipton]

As a physicist I am always interested in these discussions. I, of course, went to the physics papers archive (arXiv) to look for recent papers on the subject. It turns out that there are several papers by 2004 Nobel Prize winner David Politzer on the acoustics of bridges. But they are all about BANJOS!

That just got me depressed. Anyway the bridge must move to transfer energy to the soundboard. It is only approximately a fixed node. In addition to the vertical motion there is also be some torque that is much more signifcant in a banjo than a guitar due to the different shape of the bridges and compliance of the soundboard vs banjo head.

[Cuki79]

Originally Posted by FrankS

The downward force on the bridge is not very high and in fact can be zero.

This is often confused with the break angle that keeps the strings in close contact with the saddle. There is no net force there as it is just like trying to lift yourself up while seated in a chair. There is an equal and opposite force between the ball ends of the strings pulling upward with the same force as the strings pushing down on the saddle. No net up or down on the top of the guitar.

The force upward or downward on the guitar top are only due to neck angle and the hight of the nut to the saddle. This is typically a very small number like 0.25 inches out of 25 inches. Plugging in the trig function gives on the order of 1 pound of upward force for an average dred. The neck angle can actually be dropped down so there is no angle and the guitar will still play quite loud at a zero degree angle. While this is not ideal for playing, it demonstrates that it is not the up and down force on the top from the horizontal or vertical plucking of the strings that causes the majority of the volume from the top.

Next is polarization issues and the reported differences in amplitude or harmonic content of a string gently plucked at the 12th fret. Yes, I know strings are not exactly idea but let's not get caught up in the minutia while there are still large factors to consider. Polarization has a great deal to do with how the string passes over the saddle. It is not a symmetric system as the strings are in a groove cut by the strings. This is a different geometry than the path that the strings come over the saddle. All saddles have a different radius for the strings to cascade over. Just like in school, if you have horizontal blinds, a string will oscillate well in the horizontal direction but not the vertical. Go with some vertical freedom and the string will prefer to go vertically. This is complicated in the guitar because the higher harmonics are either allowed to be there or filtered out by the non symmetry of the anchoring of the ends of the strings.

The saddle is really a direction changer. It would do the same thing as a pulley for the string tension if it were anchored to another structural member above the guitar top. The is not the case so the saddle/bridge arrangement has a net torque on it. Changing string tension when plucking a note will change both the torque on the top of the guitar and upward force. Since a zero neck angle will give no downward force no matter what the string tension, the sound has to be generated by some other mechanism, namely changing torque with changing string tension from the pluck.

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.

Frank Sanns

Thanks Frank for this nice answer

Cuki

[Cuki79]

Here are reported 3-axis measurement just behind the bridge using some mems.

http://www.analog.com/en/analog-dial...c-pickups.html

[FrankS]

Sorry but I had two typos in my post. After starting out calculating the DOWNWARD force on a normal guitar top, I switch to saying upward. Oh the hazards of quickly written late night posts. Cuki would you mind editing your repost of mine to correct those two so nobody gets confused. That is of course assuming anybody will read this thread this far. lol The first statement is correct but then I inadvertently switched nomenclature on the next two occurrences.

Thanks!

Frank Sanns

[John Arnold]

I am with Al on this analysis. If tension change (rocking bridge) were a significant component of sound generation, then the flat top guitar would sound an octave higher, since the tension change is twice the fundamental frequency of the string. I first discovered this fact over 40 years ago. IMHO, the Kasha designs that were based on the rocking bridge principle were a long way from a rousing success.

Rather than saying that flat top guitars are louder than archtops, I suggest that the main difference is the sound spectrum. Flat tops generate more sound pressure in the lower frequencies. That is because archtops are inherently stiffer.

String grooves in an ebony bridge may be cut by the strings in extreme cases, but for the most part, the grooves on a Martin guitar were intentionally cut to clear the bridge pins. That contiinued even after slotted pins were introduced, because the slots were not deep enough to accomodate the larger strings. Deep groove bridge pins were not introduced on Martin guitars until the late-1980's. At that point, Martin stopped slotting bridges, and those guitars with ebony bridges rarely experience any more than the slightest dimple on the edge of the hole.

[FrankS]

Originally Posted by John Arnold

I am with Al on this analysis. If tension change (rocking bridge) were a significant component of sound generation, then the flat top guitar would sound an octave higher, since the tension change is twice the fundamental frequency of the string. I first discovered this fact over 40 years ago. IMHO, the Kasha designs that were based on the rocking bridge principle were a long way from a rousing success.

Rather than saying that flat top guitars are louder than archtops, I suggest that the main difference is the sound spectrum. Flat tops generate more sound pressure in the lower frequencies. That is because archtops are inherently stiffer.

String grooves in an ebony bridge may be cut by the strings in extreme cases, but for the most part, the grooves on a Martin guitar were intentionally cut to clear the bridge pins. That contiinued even after slotted pins were introduced, because the slots were not deep enough to accomodate the larger strings. Deep groove bridge pins were not introduced on Martin guitars until the late-1980's. At that point, Martin stopped slotting bridges, and those guitars with ebony bridges rarely experience any more than the slightest dimple on the edge of the hole.

But the output frequency IS double the string frequency at the bottom end. The low E string should produce 82 Hz output of the guitar top but very few guitars do that. The vast majority of guitar low E is actually 164 Hz. It is evident in any transducer or microphone measurement. In fact, the manufacturer of the guitar can be identified by the ratio of the fundamental and the next three harmonics.

While it is true that some guitars rumble with fundamental frequency bass, they are not the ones people perceive as the ones that sound bassy. The ear does not really hear 82 Hz very well at all so that energy is pretty much wasted. The ear does hear 164 Hz many times better. Fortunately, that is where most guitars are producing most of their energy.

Things do change on the way up in frequency but at the bottom, that answer is mostly double for most guitars.

Frank Sanns

[Rjlipton]

I found a really nice article on the physics of strings, bridges, etc. (with minimal math) by David Politzer (2004 Nobel prize for quantum chromodynamics):
http://www.its.caltech.edu/~politzer/zany.pdf

It is focussed on the banjo, but much of it applies equally well to the guitar. One important point is "The key idea is that, whenever there are two things that can oscillate at the same frequency, any small interaction between them, no matter how small, can eventually have a huge effect"

So a bridge, even though an approximate nodal point, still can transmit significant energy.

Ron

[jessupe]

Originally Posted by Cuki79

On that video around 1:00 they show the vibration transfert between two springs with different rigidity. You have clearly reflexion at the impedance mismatch. It is very similar to the case In which the end was fixed (earlier in the video). The point where the two different springs are attached does not vibrate much.

Cuki

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