Physics behind the sound of guitars

[vcsharp2003]

Homework Statement

(a) How does plucking a taut string of a guitar produce sound?

(b) How can the same taut string of a guitar produce sounds of different frequencies?

(c) Does a taut string of a guitar always produce the fundamental mode i.e. the distance between where the finger presses the taut string ( one fixed end) and the other fixed end is always $\frac {\lambda} {2}$? I am wondering how the first overtone, second overtone etc. can be produced on a single taut string of the guitar, or maybe only the fundamental mode is possible.

Relevant Equations

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

(a) When a taut string is plucked with a finger then it starts vibrating with a transverse wave pattern in the string, which causes the air particles in the immediate vicinity of the vibrating string to oscillate. These oscillating air particles will result in a sound wave traveling in 3 dimensions that we hear as a guitar noise.

(b) I have little clue about how to answer this question. I am thinking that we can tighten a taut string with screw at the end of guitar. This increases tension $T$ in string, and since linear mass density $\mu$ is constant, so velocity increases as per the first equation. Higher $v$ means higher frequency if the wavelength is constant for a string as per the second equation ( not sure if wavelength is constant for a taut string).

(c) I am unable to answer this question.

 

[osilmag]

For b, when you press the fret board with your finger what changes?

A visual of the fundamental mode will help you answer c. Picture the string as the wave.

 

[vcsharp2003]

For b, when you press the fret board with your finger what changes?

The length $L$ mentioned in third equation i.e. the length of vibrating string changes ( decreased compared to string's orginal length).

A visual of the fundamental mode will help you answer c. Picture the string as the wave.

I can picture the fundamental mode for length $L$, but how can I explain how the first overtone, second overtone etc. are created provided they are possible?

Below is a rough sketch of the fundamental mode, if the taut wire is pressed at point F and then plucked between F and the fixed end on guitar's body.

 

[vcsharp2003]

For b, when you press the fret board with your finger what changes?

Or maybe the value of $n$ in third equation changes, whereas, $L$ remains constant which is the length of taut wire from one fixed end to the other.

 

[kuruman]

For part (c) you need to understand that when a string is plucked, it is deformed from straight and then released. Just before release, its shape is not half a sine as shown in post #3. More likely, it is some kind of triangular shape which can be reproduced mathematically as a superposition of sines. This is done by writing the initial shape of the waveform $y(x,0)$, as $$y(x,0)=\sum_{n=1}^{\infty}a_n\sin\left(\frac{n\pi x}{L}\right).$$Note that this expression is zero at the two ends of the string, $x=0$ and $x=L$ as it should be. The coefficients $a_n$ can be found if one knows the initial waveform but let's not worry about that now. All you need to know is that they get progressively smaller as $n$ gets larger.

Because this is a homework problem, I leave it up to you to explain why this means that you get overtones and not only the fundamental.

 

[vcsharp2003]

For part (c) you need to understand that when a string is plucked, it is deformed from straight and then released. Just before release, its shape is not half a sine as shown in post #3.

I get that. The plucking causes a sinusoidal wave of same frequency/wavelength/amplitude to travel in both directions from the point of plucking to the fixed ends These waves then interfere with their reflected waves from fixed ends of the string that will result in a stationary wave having the fundamental or first harmonic frequency. So, in my diagram for part c, I should have shown the half wave extending all the from one fixed end to the other. Is that correct?
A confusing part is if the guitar string is plucked without pressing any fret, then sinusoidal waves having a $\lambda = 2L$ travels in the string (this is very large wavelength and I am not sure about this), which on interference results in a fundamental harmonic.

 

[kuruman]

These waves then interfere with their reflected waves from fixed ends of the string that will result in a stationary wave having the fundamental or first harmonic frequency.

The picture is that at t = 0 (before plucking) you have set up the initial conditions for a superposition of standing waves which dissipate at their own pace after the string is released.

 

[vcsharp2003]

The picture is that at t = 0 (before plucking) you have set up the initial conditions for a superposition of standing waves which dissipate at their own pace after the string is released.

You mean the natural frequency of string and its tension/linear mass density are such that the wavelength of interfering sinusoidal waves is one-half the length of taut string.

 

[kuruman]

There is no interference, just many standing waves each doing its own thing with its own amplitude which could go on forever if there is no loss of energy. It's the other way around. You start with a superposition of wavelengths which set the frequencies that are multiples of the fundamental. For a fixed length $L$, you have multiples of half integer wavelengths. Thus
n = 1 gives $~1\times\frac{\lambda}{2}=L\implies \lambda_1 =\frac{2}{1}L$
n = 2 gives $~2\times\frac{\lambda}{2}=L\implies \lambda_2 =\frac{2}{2}L$
n = 3 gives $~3\times\frac{\lambda}{2}=L\implies \lambda_3 =\frac{2}{3}L$
n = k gives $~k\times\frac{\lambda}{2}=L\implies \lambda_k =\frac{2}{k}L$
For the same speed of propagation $v=\lambda~f$, different wavelengths correspond to different frequencies.

 

[Pyter]

(c) Does a taut string of a guitar always produce the fundamental mode i.e. the distance between where the finger presses the taut string ( one fixed end) and the other fixed end is always $\frac {\lambda} {2}$?

The question isn't very clear. Does it ask if it only produces the fundamental mode? Because the fundamental mode is always produced, together with the overtones.

Relevant Equations:: $v= \sqrt {\frac {T} {\mu} }$

I'm not sure about that one, shouldn't it be $\omega = \sqrt {\frac {T} {\mu} }$ for analogy with the harmonic oscillator?

 

[kuruman]

I'm not sure about that one, shouldn't it be $\omega = \sqrt {\frac {T} {\mu} }$ for analogy with the harmonic oscillator?

No. Angular frequency is different from linear velocity and have different dimensions. Besides, in the harmonic oscillator (spring-mass system) the angular frequency is $\omega=\sqrt{\frac{k}{m}}$ where
$k$ is the spring constant, as in $F=-kx$, not to be confused with the wave number number $k$, as in $y(x,t)=A\sin(k~x-\omega~t).$ These $k$'s can be confusing at times and need to be taken in context, especially when they appear in front of $x$.

 

[hutchphd]

Because the fundamental mode is always produced, together with the overtones.

Unless one damps the fundamental (with a soft finger at the string midpoint) to mute the fundamental and encourage only the harmonics.

 

[Pyter]

Unless one damps the fundamental (with a soft finger at the string midpoint) to mute the fundamental and encourage only the harmonics.

I also thought about that, but actually when you do it, for instance at the 12th fret, you play the same note as the fully pressed string, only with a different timbre (overtones).

 

[hutchphd]

But you don't get the fundamental at all. And if you do this on the 7th fret the harmonic will be an octave above the note generated by simply depessing the string on that fret. Details.

 

[Pyter]

@hutchphd you're right. So I guess this answers question c).

 

[Pyter]

One thing I've always wanted to discuss about: why do you have to adjust the scale of the single string individually?

I mean if you tune all the free strings to exact pitch and adjust the saddles at the bridge all flush, then you play them at the 12th fret, some sound flat and some sharp with respect to the note one octave above the free string pitch.

I would expect that each one of them played exactly double the frequency of the free string, since they were all pressed at mid length.

 

[hutchphd]

I think mainly because the slight displacement of the string, when pressed to the fret, changes the tension slightly. Different diameter strings (and maybe the windings) are affected differently.

 

[Pyter]

It could be so if the amount of saddle adjustment were proportional to the string's diameter, but this isn't always the case.

 

[Jodo]

Hutchphd is describing intonation. Rumour has it Pythagoras calculated this for musicians/scholars way back when mankind adopted 12 tone equal temperament.
I don't know the math but intonation is more pronounced because of 2 factors, string diameter and length.
Intonation becomes more of a factor on short/small diameter strings
On a piano intonation has minimal effect on the lower bass strings but as you cross the octaves it gets progressively worse.
As a guitarist I notice the effects a lot on my cheaper guitars. My Martin D28 must have been set up by a genius..it is damn near perfect across the whole fretboard.
I am going to make a bold statement and say that perfect intonation is impossible on stringed instruments.
Would love to see one of the mentors write an insight about intonation.

 

[Pyter]

The exact name for this phenomenon is Inharmonicity. But the Wikipedia article only explains it qualitatively, not with a rigorous mathematical treatment.

This is also the reason why the pianos are tuned with stretched octaves.

 

[Steve4Physics]

As a guitarist I notice the effects a lot on my cheaper guitars. My Martin D28 must have been set up by a genius..it is damn near perfect across the whole fretboard.
I am going to make a bold statement and say that perfect intonation is impossible on stringed instruments.

Unlike many electrics, acoustic guitars rarely have bridges which allow intonation adjustment. However such bridges do exist, e.g. https://i9.photobucket.com/albums/a53/Slsaville/Adam Merry/Patent_zpsaabaedd2.jpg
(image from this forum: https://www.acousticguitarforum.com/forums/showthread.php?t=282762 ).
These should allow excellent intonation to be achieved.

Presumably sufficiently skilled players of unfretted stringed instruments (e.g. violinists) can achieve effectively perfect intonation.

 

[Jodo]

Unlike many electrics, acoustic guitars rarely have bridges which allow intonation adjustment. However such bridges do exist, e.g. https://i9.photobucket.com/albums/a53/Slsaville/Adam Merry/Patent_zpsaabaedd2.jpg
(image from this forum: https://www.acousticguitarforum.com/forums/showthread.php?t=282762 ).
These should allow excellent intonation to be achieved.

Presumably sufficiently skilled players of unfretted stringed instruments (e.g. violinists) can achieve effectively perfect intonation.

I couldn't imagine altering my Martin. But truth be told, I did mount a Floyd Rose to a cheap Yamaha acoustic. Tone was horrible but I could play Eruption properly on it

 

[pbuk]

Unlike many electrics, acoustic guitars rarely have bridges which allow intonation adjustment.

Not true - you just need a file and a skilled luthier instead of a screwdriver

Presumably sufficiently skilled players of unfretted stringed instruments (e.g. violinists) can achieve effectively perfect intonation.

It's a lot more complicated than that - there is the rest of the orchestra to think about, as well as the instrument's other strings; in general, intonation is always a compromise.

 

[Steve4Physics]

Not true - you just need a file and a skilled luthier instead of a screwdriver

I was referring to the fact that adjustment of intonation on an acoustic guitar can't usually be done in the same way as on an electric guitar. Not that it can't be done at all.

Having to use a luthier (e.g. after changing strings for different gauge ones) may not be a practical option.

It's a lot more complicated than that - there is the rest of the orchestra to think about, as well as the instrument's other strings; in general, intonation is always a compromise.

Agreed in general. But for a specified tuning system (e.g. ‘equal temperament’ or ‘just intonation’) a string quartet, for example, by using only stopped strings, could play every note in a piece at the precise frequency specified by the tuning system.

I guess that would be an example of (single and multiple) instruments playing with perfect intonation.