Talk:String vibration

Bridge

Copied by me--Light current 22:48, 12 May 2006 (UTC)

This post of mine has been sitting on the Talk:Bridge (instrument) page for about 5 months with no replies. As we seem to have some very educated editors working on this page 8-), I thought some of you would like to comment.

Bearing in mind that there is a node on a vibrating string where it passes over the bridge, how exactly do vibrations get to the body. There arent any vibrations at the bridge cos its a node. Anyone know the answer to this paradox? 8-?--Light current 14:42, 12 November 2005 (UTC) The model of string vibration here violates Gauss's remarkable theorem because string action does not respect the measure of distance along the string. When a curve develops, the string bends but does not stretch. Then there is no x plus delta x. This is required because the inner product of kinetic and potential energy is a constant of motion. The result is an invariant sub manifold in which the image of the string is the same in every cross section of the manifold almost everywhere.

The tangent space of the inner product is the same at every point on the string. This makes the string time- and space averaged. Then the behavior of the string is dynamic converging on frequency as the Hamiltonian constant of motion. The Hamiltonian takes the string curve as the argument and the frequency as a result.

The differential equation for the string action needs to be re-written as an integral equation of the Liouville type. This is critical to pass from the string action to the coherent, laser-like frequency. Viewing the string action as a Hamiltonian flow formalizes the idea that unpredictable action on the string can settle into a predictable standing wave and it critical null points. The diagram assumes the string is shaped like a sine wave which is incorrect because the sine wave does not the least surface area manifold. The convex catenoid has that property.

This is critical to go from the string algebra to the algebra of many strings which form polyphonic union when they have concert frequency as a point in common. If we cannot understand string algebra then we cannot understand the algebra of tablature. That is why The Beatles Complete Scorestablature does not make sense. Terence B Allen (talk) 23:18, 27 August 2019 (UTC) --Light current 18:38, 1 May 2006 (UTC)

Resonance, I guess...--Army1987 13:09, 13 May 2006 (UTC)

Saying there is a node at each end of a string is a simplification. It would be true if the ends of the string were attached to something perfectly rigid, but they're not. The whole instrument is vibrating, including the bridge. Pfalstad 14:10, 13 May 2006 (UTC)

Yes I agree Paul. But can you propose a mechanism for transfer of the string vibrations to the body of the instrument that fits the obervations? 8-)--Light current 14:16, 13 May 2006 (UTC)

Do we agree that the end of the string is moving (not perfectly a node)? Well then, the string moves the bridge, and that moves the body... Pfalstad 14:25, 13 May 2006 (UTC)

Yes, but in which direction does it move the bridge-- up and down, back and forth, side to side, or some combination of all three?--Light current 14:29, 13 May 2006 (UTC)

[1] Pfalstad 14:39, 13 May 2006 (UTC)

Yes. I think I ve seen this one before.8-) This implies that its the up and down motion of the string that gets trasmitted to the belly. If this is true, it must mean that the 'side to side' vibrations of the string are somehow changed into 'up and down' vibrations. However, I have seen another expalnation that talks of the bridge being rocked back and forth (in a direction // to strings) thus tending to bend the belly slightly and transmit vibrations that way. BTW do you know the reasons for the peculiar shaped cutouts in the violin bridge?

Also, one thing I ve been wondering about is, when you have a vibrating string, set at a nominal tension T newtons say, then when the string is set into vibration, does the instantanoeous string tension change and could this be detected at the 'fixed' points (nut or bridge). The analysis on the page assumes that the tension is constant, but I bet it aint! 8-? In fact its obvious from the analysis that the string tension varies during the vibration cycle of the string. So, the answer to my question appears to be that these changes in tension should be able to be picked up by load cells measuning the tension. Now the more important question is: Are these tension changes the primary means of energy transmission to the bridge. If not, what other mechanisms are involved?--Light current 15:36, 13 May 2006 (UTC)

That's not the way I read it. It seems that both side-to-side and up-and-down motions will get transmitted. I have another reference which says that the bridge moves in many different ways. Pfalstad 19:18, 13 May 2006 (UTC)

The analysis on the page is very weird. I've seen other analyses that make more sense, and they say that they are making many assumptions, including assuming the tension is constant, but I don't think it is. My guess is that this is the primary means of energy transmission. I'll check my reference to see if it says more. (It's a very technical reference so I had trouble finding good quotes in it, but I'l check again.) Pfalstad 19:18, 13 May 2006 (UTC)

THanks. I look forward to seeing what your book says.8-)--Light current 19:21, 13 May 2006 (UTC)

This is how you settle a debate, with verifiable observations. My take on the current physics is that it explains pitch and greater surface area on a string to produce lower frequencies. Anything else is a matter of speculation because there are no observations to go any further. When you get a verifiable observation, only then can you expalin it. If that doesn't work, you need to come up with a new theory that does explain it.

1. If you place a vibrating tuning fork against any part of the neck or body, you get a response.

2. If you place a rod to reach both the nut and saddle, and place a vibrating tuning fork against it, you get a response.

3. If you place a lump of soft clay at the end of a violin neck, the tone changes - a tv show from the 90's filmed this.

4. Placing a transducer against any part of a guitar neck or body will pick up vibrations. This implies that the neck is the transmitter, making the body only a box. The only reason for a greater response at the bridge is due to the flexing of the body.

5. My latest observation is that if you create harmonics all along the string's length, you get 3ds,5ths,7ths, octaves, and even d on the e string. If you try to get harmonics off a dead instrument, you hardly get a response. This implies that tone is dependent on overtones that help to sustain the base frequency. My personal test for a fretted instrument is to create harmonics at the 2d and 3d fret, if I don't get the harmonics, I know by experience that the tone is inferior. This is also the reason why audiofiles prefer vinyl, the overtones are not picked up in current digital recording.

6. Placing transducers against different parts of any instrument will show the source note's origin and supporting tonal surfaces.Radical man 7 17:18, 9 January 2007 (UTC)

7. This is a repeatable observation, tapping harmonics will get you the following: E-last string- base frequecy. all octaves; 12th fret-E 11th fret-F 10th fret-Gb 9th fret-Ab 8th fret-E 7th fret-B 6th fret-D 5th fret-E 4th fret-B# 3d fret-B 2d fret-F# 1st fret-F Not counting all the frequencies between frets.

Here is another way of settling this matter, as E is the base frequency, take into account the placing of the neck pickup. The octaves/frequencies/overtones will be those of B and E, hence the fat tone. Now, the bridge pickup will get E, B. and a lot more octaves/frequencies/overtones resulting in a sharper tone. Another way of looking at this would be that B represents the E string's vibration divided into 3 parts(7th & 19th fret harmonics), the vibration being readily picked up by the Neck pickup. Taking another look at the bridge, tapping the 12th fret will be picked up by both pickups. accessing higher harmonics at the bridge is really a question of the string's vibration being dividing into so many parts that are not easily picked by a neck pickup, for example, the 1st octave-2 parts, 2d octave-4 parts, 3d octave-8 parts, etc,...

If you are still not convinced, reproduce the same situation with a synthesizer, start with a base note and start adding frequencies, specifically the higher octaves-by the way, the presence control on many guitar amplifiers are set at 10khz. The other way is to start in the opposite direction, the second octave of E(4th string, 2d fret) and start adding lower octaves and associated frequencies at lower pitch this will guarantee a fat tone. Not only do you have a repeatable observation, you have a mechanical working model, everything else is speculation. Feel free to put this to the test, tell me what you think...Radical man 7 06:39, 4 May 2007 (UTC)

The physics of wind instruments?

Are there any articles on how wind instruments, or their reeds, work? Or does someone know about books on the matter (I have learned physics so it is OK for the books to be academical) so I can maybe write one? --Satúrnus 20:26, 11 December 2006 (UTC)

String velocity

If possible, can anybody who has the knowledge to do so, please clarify the string velocity section as it is difficult to follow. The f=feta x t part is especially puzzling. thanks. —Preceding unsigned comment added by Owen Graham (talk • contribs) 03:43, 22 February 2008 (UTC)

Derivation

The force balance is incorrectly stated in this discussion and the linearization is glossed over. If applying Newton's 2nd law to the free body shown, the sum of the forces is in fact 2*T*sin(theta) which after linearization leads to 2*T*theta. Furthermore, the "F-vector" in the diagram is confusing as it is not a body force. Explanation of the kinematics is also rough. Suggest using a derivation similar to: http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/waveq.html#c2 -bingjeff-71.184.194.138 (talk) 00:28, 25 June 2008 (UTC)

I agree that the derivation was incorrect, its use of centrifugal force assumes the wave is an object, or that points on the string have a substantial longitudinal movement. I've changed it to be like the one you've suggested, and expect that the diagram will need to be replaced. Jonathan48 (talk) 08:41, 18 September 2008 (UTC)

I've recovered the old derivation based on the balance of centrifugal force and elastic force directed towards the center of curvatrue of the string due to the tension of the string. Of course the old derivation is correct. You just have to reason like if the string were made of many dots, each with mass dl * mu. In my opinion the old derivation is much more easy to understand and more convincent so I advise not to remove it again. Instead I'd suggest to keep both derivations, even the one based on the differential equation.Alberto Orlandini (talk) 09:31, 1 October 2008 (UTC)

No, the old derivation is not correct. In the linear approximation, each piece of string only moves up and down, not left and right, so its acceleration is not centripetal. You can assume that the force still equals v²/R on dimensional analysis grounds, but that would not be a rigorous argument. -- Army1987 (t — c) 12:53, 19 October 2008 (UTC)

Hi I'm going to edit this derivation for clarity. It seemed bogus to me for ~15 minutes but now that I get it I'm going to make some minor changes. Specifically, I'm going to highlight the assumptions being made. —Preceding unsigned comment added by 130.113.32.61 (talk) 20:48, 9 January 2010 (UTC)

Visible

I don't know how, but I can sometimes see string vibrations during daylight with my peripheral vision. In plain sunlight.... Mike, EU — Preceding unsigned comment added by 91.148.80.161 (talk) 03:23, 21 October 2011 (UTC)

Amplitude

Are there equations relating the amplitude to linear mass density at constant pitch? Thanks.CountMacula (talk) 02:10, 19 June 2014 (UTC)

Velocity derivation

If someone has understood this completely, can you edit it and write what all symbols mean? Amey.bhavsar (talk) 16:54, 5 July 2018 (UTC)

Explanation of inharmonic overtones?

At https://music.stackexchange.com/questions/112007/what-is-the-explanation-for-higher-harmonics-not-being-integral-multiples-of-the there is a discussion of why higher piano string overtones are apparently not multiples of the fundamental. The accepted answer refers to this article to justify its claim that tension and density cause this deviation, but if I understand the formula $f_{n}={\frac {n}{2L}}{\sqrt {\frac {T}{\mu }}}$ right, $T$ and $\mu$ do not depend on $n$, so that this formula predicts multiples of the fundamental. It would be well worth having an explanation of the phenomenon in this article (or linked by it – I have not checked other articles so far); perhaps the approximations mentioned in the derivation are responsible? Since some approximations break down at higher amplitudes, perhaps it has to do with the attack (music) of the piano note, which is infamously complicated. PJTraill (talk) 21:27, 25 March 2021 (UTC)

In addition to the tension on the string there is a resistance to bending due to the stiffness of the material which is neglected in the derivation of the wave equation. Higher frequency harmonics bend the string more sharply so the effect is greater for them.

See: https://csclub.uwaterloo.ca/~pbarfuss/The_Physics_of_Musical_Instruments.pdf 130.44.188.49 (talk) 07:44, 29 August 2024 (UTC)

Real-world example

Reason for removal

I removed this section, which contained original research. However, I felt it was important to archive it here for further reference. SpiralSource (talk) 16:55, 3 September 2024 (UTC)

Archive

See also: Guitar tunings, 12 equal temperament, and A440 (pitch standard)

A Wikipedia user's Jackson Professional Soloist XL electric guitar has a nut-to-bridge distance (corresponding to $L$ above) of 255⁄8 in. and D'Addario XL Nickel-wound Super-light-gauge EXL-120 electric guitar strings with the following manufacturer specs:

D'Addario EXL-120 manufacturer specs

String no.	Thickness [in.] ($d$)	Recommended tension [lbs.] ($T$)	$\rho$ [g/cm3]
1	0.00899	13.1	7.726 (steel alloy)
2	0.0110	11.0	"
3	0.0160	14.7	"
4	0.0241	15.8	6.533 (nickel-wound steel alloy)
5	0.0322	15.8	"
6	0.0416	14.8	"

Given the above specs, what would the computed vibrational frequencies ($f$) of the above strings' fundamental harmonics be if the strings were strung at the tensions recommended by the manufacturer?

To answer this, we can start with the formula in the preceding section, with $n=1$:

$f={\frac {1}{2L}}{\sqrt {\frac {T}{\mu }}}$

The linear density $\mu$ can be expressed in terms of the spatial (mass/volume) density $\rho$ via the relation $\mu =\pi r^{2}\rho =\pi d^{2}\rho /4$, where $r$ is the radius of the string and $d$ is the diameter (aka thickness) in the table above:

$f={\frac {1}{2L}}{\sqrt {\frac {T}{\pi d^{2}\rho /4}}}={\frac {1}{2Ld}}{\sqrt {\frac {4T}{\pi \rho }}}={\frac {1}{Ld}}{\sqrt {\frac {T}{\pi \rho }}}$

For purposes of computation, we can substitute for the tension $T$ above, via Newton's second law (Force = mass × acceleration), the expression $T=ma$, where $m$ is the mass that, at the Earth's surface, would have the equivalent weight corresponding to the tension values $T$ in the table above, as related through the standard acceleration due to gravity at the Earth's surface, $g_{0}=980.665$ cm/s². (This substitution is convenient here since the string tensions provided by the manufacturer above are in pounds of force, which can be most conveniently converted to equivalent masses in kilograms via the familiar conversion factor 1 lb. = 453.59237 g.) The above formula then explicitly becomes:

$f_{\mathrm {Hz} }={\frac {1}{L_{\mathrm {in} }\times 2.54\ \mathrm {cm/in} \times d_{\mathrm {in} }\times 2.54\ \mathrm {cm/in} }}{\sqrt {\frac {T_{\mathrm {lb} }\times 453.59237\ \mathrm {g/lb} \times 980.665\ \mathrm {cm/s^{2}} }{\pi \times \rho _{\mathrm {g/cm^{3}} }}}}$

Using this formula to compute $f$ for string no. 1 above yields:

$f_{1}={\frac {1}{25.625\ \mathrm {in} \times 2.54\ \mathrm {cm/in} \times 0.00899\ \mathrm {in} \times 2.54\ \mathrm {cm/in} }}{\sqrt {\frac {13.1\ \mathrm {lb} \times 453.59237\ \mathrm {g/lb} \times 980.665\ \mathrm {cm/s^{2}} }{\pi \times 7.726\ \mathrm {g/cm^{3}} }}}\approx 330\ \mathrm {Hz}$

Repeating this computation for all six strings results in the following frequencies. Shown next to each frequency is the musical note (in scientific pitch notation) in standard guitar tuning whose frequency is closest, confirming that stringing the above strings at the manufacturer-recommended tensions does indeed result in the standard pitches of a guitar:

Fundamental harmonics as computed by above string vibration formulas

String no.	Computed frequency [Hz]	Closest note in A440 12-TET tuning
1	330	E4 (= 440 ÷ 25/12 ≈ 329.628 Hz)
2	247	B3 (= 440 ÷ 210/12 ≈ 246.942 Hz)
3	196	G3 (= 440 ÷ 214/12 ≈ 195.998 Hz)
4	147	D3 (= 440 ÷ 219/12 ≈ 146.832 Hz)
5	110	A2 (= 440 ÷ 224/12 = 110 Hz)
6	82.4	E2 (= 440 ÷ 229/12 ≈ 82.407 Hz)