What is the function of the body?

[JEStanek]

Bill, Dave has everyright to post. Further, I don't see that he's on a high horse, rather, I think he's frustrated that people aren't understanding the things he's said (repeatedly). Frankly, the physics is pretty advanced and doesn't lend itself to light conversation and there are people putting forth a lot of positions that aren't based on any data that has been previously presented.

I think someone could easily earn a doctoral dissertation out of a study on how the body functions in various stringed instrument systems. The method development alone to investigate them would likely earn a Masters or a PhD. I don't think you're going to find easy answers to the more specific questions of what is the body's function beyond providing a resonance chamber for stringed instruments without getting into a lot of physics involving theory and a lot of math. We're lucky to have Dave here to repeatedly talk about these complex aspects of dynamic systems.

Jamie

[Bertram Henze]

A string is mathematically 2-d - length.

Ah, but this is just a part of the eight dimensions * we live in:
- left
- right
- top
- bottom
- back
- front
- past
- future

OTOH, this is nothing compared to the ten dimensions of real string theory **

(*) That is apart from our own special eight dimensions:
- G
- G
- D
- D
- A
- A
- E
- E
(**) Caveat: common sense breach alert

[Tom Haywood]

I like the old ad showing Kim Breedlove drawing the shape of the body while looking at a nude female model. Maybe the function has to do as much with biology as physics.

[fatt-dad]

Between sblock and Bertram, I'm totally dismissed? O.K. I don't belong here, I guess. . . I do stand by my original comments, however and do not feel stupid for making them.

f-d

[billhay4]

I don't think sblock dismissed you f-d. He did address all three dimensions of a string and it vibration. But I understand your frustration.
Bill

[Mark Gunter]

But I understand your frustration.
Bill

And, ya gotta let go o' d' frustration. Anything you post here can be dismissed either philosophically, scientifically, semantically or otherwise - and will be. Best program is to just read the thread and try to learn something. There are axes to grind and egos to stroke, but plenty of interesting info along the way.

[fatt-dad]

Sorry, I got my feelings hurt. . .

My point being, the engineering behavior of of a string can be described by the line between the two points that form its ends along with the material properties described by density and moduli (elastic, shear and section). The dynamic response relates to the length. I'm a bit curious how to model the radial coordinates of motion (i.e., in cross-section). The longitudinal sine and cosign (amplitude and magnitude) make sense. I really don't know, I'm just reading through my filter as an engineer.

f-d

[Bertram Henze]

No offence meant, f-d.

A cylindrical object like a string is physically best described in polar coordinates length, radius and azimuth. Those sine waves it does are there, and more than one, the biggest base wavelength being twice the string length, then 1/2 of the base, 1/4 and so forth, making the overtones, their proportion depending on where the string is plucked.

P.S. thinking about this, I guess I get what f-d meant by 2 dimensions: the plane the string is supposed to oscillate in, is that it? Vibration in a plane is an assumption made for simplicity, and it works for understanding most of the process, but strike a string on your instrument and look at it from every angle: the string vibrates in all directions. With the pickstroke, we input not only momentum for side-to-side motion, but also angular momentum making the string rotate like a skipping-rope.

[fatt-dad]

. . . meaning a line is the intersection of two planes and that line is two dimensional. I'd think the mathematical description would be in Cartesian coordinates - at least that would be my inclination. . . The striking of the string will be in every plane that is consistent with the coordinates of the line. That motion seems to occur in radial coordinates.

With the description of section modulus we can remove the cylindrical properties of the actual string. Just like we'd do in beam analyses. The elastic modulus would inform lengthening as the string deforms and reforms from vibration. Not sure how shear modulus would play into this, but it could.

I'm just intellectualizing and thinking of how a FEM model may be developed. Simple modeling is often easier.

To all the builders out there, I know little about any of this and really should beg out. I'm just enjoying the thought process, science and engineering that comes to mind as I read this thread.

Again, I do apologize for getting unnecessarily provoked. Deep down, I know we are all friends and brothers/sisters in music and for that I'm thankful!

f-d

[Dave Cohen]

Carl, Bertram was referring to what is sometimes called "cylindrical polar coordinates", a 2D polar plane described by a radius vector and a polar angle, plus a Z-axis, as in cartesian coordinates. The Z-axis is what he called an azimuth. That is the most friendly coordinate system for a string, which is essentially a very long, thin cylinder. In the extreme of a string with no volume,i.e., a line, the motion is still described in the polar plane. You might have been thinking of spherical polar coordinates, which would indeed be clumsier for describing string motion.

[sblock]

. . . meaning a line is the intersection of two planes and that line is two dimensional. ...

Again, I do apologize for getting unnecessarily provoked. Deep down, I know we are all friends and brothers/sisters in music and for that I'm thankful!f-d

Yes, cylindrical polar coordinates (theta, phi, z) are the most convenient for developing analytical expressions of 3D cylinders, if you want to treat the string in all three dimensions as a long, thin cylinder. Most (but not all) computer models of fairly complex shapes, such as FEM models, are numerical and don't need to worry about analytical simplicity for the math, and they tend to use Cartesian coordinates (x, y, z), where all three coordinates are equivalent. But the coordinate system is a trivial matter, and one of convenience: simple formulas can be used to convert from one system to another.

That said, the intersection of any two planes forms a LINE, and such a line is a one-dimensional object, not a two-dimensional one. This goes back to Euclid. An abstract line has length, but no breadth.

[Bertram Henze]

a 2D polar plane described by a radius vector and a polar angle, plus a Z-axis, as in cartesian coordinates. The Z-axis is what he called an azimuth.

Actually, what I called azimuth is the polar angle. The Z-axis is what I called length (for the lack of a better word, but Z-axis is of course correct).
This leads me back to my diploma thesis when I spent time with vortices (think: tornadoes), which are the home of cylindrical coordinates.

A line with no thickness is 1-dimensional, because you need only one coordinate number to describe a position on it (positive or negative, counting from some point of origin). Defining a line as the intersection of two planes is interesting, because intersection always takes away one dimension: intersecting two 3-dim spaces makes a 2-dim plane, intersecting two 2-dim planes makes a 1-dim line, intersecting two 1-dim lines makes a 0-dim point.

[Wes Brandt]

Just to get with the spirit of the conversation, isn't a 1 dimensional line only possible in theory because if you represent it with something, that something has to have thickness and width, no matter how minimal?

[billhay4]

Well, I'll observe that your getting your feelings hurt led to a fascinating discussion (albeit over my head) of the math and modelling involved here. This thread has had over 8700 views and 200 posts partially because some people have gotten their feelings hurt and I find it a very valuable thread.
Bill

[Bertram Henze]

Just to get with the spirit of the conversation, isn't a 1 dimensional line only possible in theory because if you represent it with something, that something has to have thickness and width, no matter how minimal?

in reality yes, but physics is all about models, i.e. simplified abstraction of reality. The art lies in knowing what to leave out, but if your assumptions were too simple nature is setting you right during your next experiment...

[CarlM]

Dave
That wasn't me. I understand cylindrical vs spherical. I've been staying out of this food fight

[billhay4]

physics is all about models, i.e. simplified abstraction of reality

Bill

[fatt-dad]

CarlM. Dave is using my given name, which we share.

f-d

[CarlM]

Welcome to the conspiracy

[Dave Cohen]

in reality yes, but physics is all about models, i.e. simplified abstraction of reality. The art lies in knowing what to leave out, but if your assumptions were too simple nature is setting you right during your next experiment...

Physics isn't all about models. Physics, like all branches of science, has experiment at the top of the hierarchy. We build models to help interpret the experimental results. The simplified models relate the more complex experimental systems to simpler systems for which we can more easily visualize the behaviors. For example, we model a string instrument body as a system of coupled masses on springs (the two-mass and the three-mass models). It is possible to build that model because guitar plates are elastic, i.e., springy, and they have mass. Same deal for the masses of air in the soundhole(s). They are essentially attached to a "spring" of air inside the instrument body cavity. Nor are those simple models necessarily complete abstractions. You can actually make physical two-mass and three-mass -and even multi-mass- models for demonstration purposes and directly observe the resonance behavior. Iirc, Tom Rossing did just that at an ASA meeting many years ago.

Models are good to the extent that they work, i.e., describe real behavior. So do the two-mass and three-mass models work? In fact, they work quite well for the lowest frequency body resonances and for the Helmholtz air resonance - well enough that for another 30 yrs or so, few if any payed much attention to deriving a more complex model.

[sblock]

Just to get with the spirit of the conversation, isn't a 1 dimensional line only possible in theory because if you represent it with something, that something has to have thickness and width, no matter how minimal?

Well, yes and no. As I'm sure you realize, ALL physical objects in our real world are always three-dimensional (ignoring time, and the possibility of higher numbers of compactified dimensions that string theorists sometimes contemplate). No more and no less. Actually, there are no such things as one- or two-dimensional objects -- not physical ones, anyway. Such things are purely theoretical constructs, but they are incredibly useful, and they go back to the time of the ancient Greeks (like Euclid) and even before that. Points (0D), lines (1D) and planes (2D) do not actually exist. All matter takes up space (it's all 3D). But we still study points, lines, and planes in elementary and junior high school. These are not very hard concepts to grasp, in fact, and children usually get it. It's much harder to try to "understand" (i.e., develop a physical intuition for) 4 spatial dimensions -- like the tesseract (4D analog of the cube).

In physics, it is incredibly useful, and mathematically much easier, to develop certain theories in less than 3 dimensions. The results from these can apply, and sometimes in excellent approximation, to real-world objects. 1D waves in lines can model pretty accurately (but not perfectly) the vibrations of strings, which are hundreds-to- thousands of times longer than they are wide. 2D waves on surfaces can model pretty accurately (but not perfectly) things like ripples on a pond. Or the vibrations of the top plate of an instrument, which is tens-to-hundreds of times thinner than it is across.

But the answer to your real question is definitely "NO." If, in a mathematical theory, you choose to represent a string as being 1D, or a surface as being 2D, then NO, you are not required to "represent it with something that has to have thickness and width." The width or thickness become precisely zero in these mathematical approximations. Despite that, these approximations can be quite accurate. Reduced dimensionality plays an important and time-honored role in physics. Physicists understand esoteric things like superconductivity, phase transitions, and relativity from models with reduced dimensionality, as it happens. Einstein's famous theory of special relativity was first worked out in a single dimension, as a matter of fact.

[David Lewis]

CarlM. Dave is using my given name, which we share. f-d

Did someone call me?

[fscotte]

This is all good and stuff, but a 1923 Lloar still sounds as good as anything. What else is there after that?

[plinkey]

In order to know the eagle, one shall become an eagle himself.
Bill, I am certainly not suggesting that you transform yourself into a mandolin.
However, one obvious way to get a deeper understanding of the mandolin is to keep building them and/or play music on them.
That is clearly a difficult and lengthy process, and many spend their lifetime on it, and there is still not a definitive answer given at the end of it all. Yet, most often, it should be a very gratifying process too.

[Larry Simonson]

This is a follow-up of my posting back in February (#107) where I wrote:

" Now I have a drummer friend who tells me about the "sweet spots" on his drums so I was wondering if such spots may exist on mandolin bellies. Maybe if the bridge position was moved to such a sweet spot by strategically locating the fingerboard there might be some improvement in the efficiency of transmitting plucked string energy to sound. I have no idea how this will turn out but I find it interesting and am in the position to give it a try, so that is what I am planning."

I have glued the top of my current build onto the rim and last night my drummer friend came by and thunked the top along center line from one end to the other and declared that the best sound came from a point at the intersection of a line between the inner notches of the f-holes and the centerline, exactly where the bridge is normally located. So maybe Loar had this figured out long ago.