# Fundamental note vs fundamental frequency of string

#### somecelxis

1. Homework Statement
A guitr player changes the frequncy of the note produced by a guitar string by pressing his fingers along the string. The fundamental frequency of the string is 264hz. What are the frquncies of the fundamental note if the player plucked the string at 1/4 of the way from one end?

2. Homework Equations

3. The Attempt at a Solution
my ans is fundamental freuqncy of note = fundamental frequncy of string which is 264hz. but the ans is 352hz. why is it so?
what's the diffrence between fundamental freuqncy of note and fundamental frequncy of string actually?

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#### Simon Bridge

Homework Helper
I think the problem has a typo - they do not mean "plucked".
If the string is pressed 1/4 of the way from the end - what frequency does it play?

#### rude man

Homework Helper
Gold Member
Two answers are required. One is the answer you were given. The other is > 352 Hz.

Assume no change in string tension when the string is pressed or plucked.

#### somecelxis

what's the diffrence between fundamental freuqncy of note and fundamental frequncy of string actually?

#### somecelxis

I think the problem has a typo - they do not mean "plucked".
If the string is pressed 1/4 of the way from the end - what frequency does it play?
If the string is pressed 1/4 of the way from the end , the frequency that's being played is 264 x 2 =528hz (first overtone occur)

#### NascentOxygen

Mentor
If the string is pressed 1/4 of the way from the end , the frequency that's being played is 264 x 2 =528hz (first overtone occur)

#### somecelxis

when the string is plucked at 1/4 of its end , first overtone( 2nd resonsnace ) occur. when the string is plucked at 1/2 of its end , fundamnetal node produced. (1st resonance)

#### rude man

Homework Helper
Gold Member
If the string is pressed 1/4 of the way from the end , the frequency that's being played is 264 x 2 =528hz (first overtone occur)
You now have two separate strings, effectively.

Unfortunately, neither one gives 528 Hz.

What formula relates frequency to length, tension and linear mass density?
Then, realize that when you push on a string the only thing that changes is length. And as I said there are now effectively two strings so possibly two separate frequencies.

#### AlephZero

Homework Helper
when the string is plucked at 1/4 of its end , first overtone( 2nd resonsnace ) occur.
If you pluck at the 1/4 point, you excite both the first and second resonances (and also higher ones).

If you pluck at the 1/4 and also touch the string at the mid point, you prevent the vibration of the fundamental frequency and the sound is mainly the 2nd resonance.

But that has nothing to do with the OP's question.

#### Simon Bridge

Homework Helper
If the string is pressed 1/4 of the way from the end , the frequency that's being played is 264 x 2 =528hz (first overtone occur)
No - I mean: what is the physical effect on the string?

A standing wave happens when waves reflect off the ends of the string, come back, and interfere with each other. You should have this in yur notes.

When you press part of the string to a surface, your finger prevents the vibration reaching some of the string. The waves reflect off your finger instead of reflecting off the far end of the string.

The effect is to have made the string shorter.

The question is making a distinction between the fundamental for the whole string and the fundamental for the note that is being played by pressing on the string while plucking.

#### olivermsun

Two answers are required. One is the answer you were given. The other is > 352 Hz.

Assume no change in string tension when the string is pressed or plucked.
It's a guitar. When you "stop" the string 1/4 from the end, you don't pluck it behind the (finger) stop.

Edit: Although I suppose you could be playing a note very high on the fingerboard, in which case a physicist might interpret 1/4 from the end as being 1/4 from the bridge instead of the nut. I don't think a guitar player would ever interpret it that way, though.

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#### olivermsun

If you pluck at the 1/4 point, you excite both the first and second resonances (and also higher ones).
The "plucking" question is more interesting than the "stopping" question… but then the original question ought to ask what are the fundamental frequencies produced.

#### Simon Bridge

Homework Helper
If you pluck at the 1/4 point, you excite both the first and second resonances (and also higher ones).

If you pluck at the 1/4 and also touch the string at the mid point, you prevent the vibration of the fundamental frequency and the sound is mainly the 2nd resonance.

But that has nothing to do with the OP's question.
It's a different way of describing it - i.e. in terms of the modes of vibration of the overall string.
It is important to distinguish between the effect of plucking and pressing+plucking.

I want to emphasize some of your points that may get lost on a quick read through:
- In a guitar string, there is significant enough damping for overtones to quickly vanish ... so you have to press then pluck.
- Press first and it is possible to excite tones which are not modes of the whole string - though it is possible to describe them as a sum of those modes.
- any wave on the string can be represented as a sum of modes. When described like this, pressing on the fret-board is, mathematically, selecting available modes.

However:
Could it be more direct to consider that pressing on the fret-board changes the effective length of the string? Then you can treat the string as playing the fundamental for the effective length?

#### AlephZero

Homework Helper
- In a guitar string, there is significant enough damping for overtones to quickly vanish
That's not true. The way you excite individual harmonics is by touching the string at a nodal point (1/2, 1/3, 1/4, etc, along the length) when you pluck, not by pressing the string onto the fingerboard. You can then release the "touching" finger and the harmonic will continue almost as long as a normal note.

Here's a demo. (Skip to about 1:30 if you want to cut to the chase). I think somecelxis was getting confused about exciting harmonics in this way.

any wave on the string can be represented as a sum of modes. When described like this, pressing on the fret-board is, mathematically, selecting available modes.
You can to represent the shape of a string that is partly vibrating and partlly static as a Fourier series, but the you can't represent the vibration at a different frequency in that way. (Well, not without getting way ahead of the level of the OP's question).

When you press the string against the fingerboard, you change the boundary conditions of the vibration problem, and it is much easier to
consider that pressing on the fret-board changes the effective length of the string? Then you can treat the string as playing the fundamental for the effective length

#### somecelxis

No - I mean: what is the physical effect on the string?

A standing wave happens when waves reflect off the ends of the string, come back, and interfere with each other. You should have this in yur notes.

When you press part of the string to a surface, your finger prevents the vibration reaching some of the string. The waves reflect off your finger instead of reflecting off the far end of the string.

The effect is to have made the string shorter.

The question is making a distinction between the fundamental for the whole string and the fundamental for the note that is being played by pressing on the string while plucking.
thanks for telling me that When you press part of the string to a surface, your finger prevents the vibration reaching some of the string. The waves reflect off your finger instead of reflecting off the far end of the string.
if the question ask for 2nd resonance of string (first overtone) . isn't that euqla to 528hz?

#### somecelxis

No - I mean: what is the physical effect on the string?

A standing wave happens when waves reflect off the ends of the string, come back, and interfere with each other. You should have this in yur notes.

When you press part of the string to a surface, your finger prevents the vibration reaching some of the string. The waves reflect off your finger instead of reflecting off the far end of the string.

The effect is to have made the string shorter.

The question is making a distinction between the fundamental for the whole string and the fundamental for the note that is being played by pressing on the string while plucking.
by saying that The waves reflect off your finger instead of reflecting off the far end of the string.

do you mean the frequncy of sound played is only affected by the length of string from the finger plucked to the right node? this is not affected by the length of left node to the hand plucked?

#### NascentOxygen

Mentor
thanks for telling me that When you press part of the string to a surface, your finger prevents the vibration reaching some of the string. The waves reflect off your finger instead of reflecting off the far end of the string.
if the question ask for 2nd resonance of string (first overtone) . isn't that euqla to 528hz?
No. No. No. The string has now changed. By holding it down firmly it at some point along its length you have created a new string. This new string is shorter than the old string, so it has a new fundamental frequency of vibration (and, in turn, it follows that it has also a new overtone).

#### olivermsun

The fundamental frequency corresponds to the mode with only 2 nodes — one at each end of the vibrating string. Thus the fundamental frequency is only affected by the distance between the nodes, which you have altered by pushing the string down behind a fret with your left hand. Now you have a shorter string which has its left end at the fret.

Overtones correspond to modes with extra nodes in the middle—they will have frequencies which are multiples of the (new) fundamental. How the string was plucked tends to change which overtones get excited, but it doesn't change the list of overtone frequencies.

#### somecelxis

my book give wehn the string is plucked at l / (2n) , n$f_0$ produced , which n$f_0$ = first ,second overtone and etc , n= 2,3,4.....
this is making me confused. what's the diffrence between the note in photo and what you've said earlier?

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#### olivermsun

All the overtones produced are multiples of $f_0$, the fundamental frequency. At the moment of release (after plucking), the string contains a mix of all the overtones (in varying amounts).

#### somecelxis

If you pluck at the 1/4 point, you excite both the first and second resonances (and also higher ones).

If you pluck at the 1/4 and also touch the string at the mid point, you prevent the vibration of the fundamental frequency and the sound is mainly the 2nd resonance.

But that has nothing to do with the OP's question.
what do you mean by pluck the string only , plucked and touch on string ?

#### rude man

Homework Helper
Gold Member
"Touch" is not a good word. "Press" is better. You need to effectively stop the vibrations at the point of touch.

If you press at the 1/4 point you now have 2 strings instead of one. For each segment the basic formula for f applies. If L was the original length you now have a string of length L/4 and a second string of length 3L/4. So you get two frequencies, one for each segment. Usually the longer segment is the only one plucked (I think).

It does not matter where along a given segment the string is plucked. The frequency is the same.

#### olivermsun

"Touch" is not a good word. "Press" is better. You need to effectively stop the vibrations at the point of touch.
I believe what AlephZero was talking about is playing a "harmonic" note, in which case the object is to induce a node at a certain point on the string but not kill the vibrations behind the finger. If so, it is definitely "touch" and not "press."

#### rude man

Homework Helper
Gold Member
I believe what AlephZero was talking about is playing a "harmonic" note, in which case the object is to induce a node at a certain point on the string but not kill the vibrations behind the finger. If so, it is definitely "touch" and not "press."
Not being a guitar or similar player, I'm not aware of this subtlety. I would think we are dealing with the physics of the sonometer, which deals with strings characterized solely by tension, mass per unit length, and length. See attached. 'Touching' a string does not seem to fit in with this theory, and would not seem to be the subject of an introductory physics course in any case.

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#### Simon Bridge

Homework Helper
if the question ask for 2nd resonance of string (first overtone) . isn't that euqla to 528hz?
... depends: 1st overtone of what?

by saying that The waves reflect off your finger instead of reflecting off the far end of the string.

... do you mean the frequency of sound played is only affected by the length of string from the finger plucked to the right node? this is not affected by the length of left node to the hand plucked?
That's correct - and the key to doing this problem.

There are fancy fingering things you can do, but they won't form part of your course.

"Fundamental note vs fundamental frequency of string"

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