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Confused about second overtone/third harmonic; finding the frequency of 2nd overtone

by teggenspiller
Tags: frequency, harmonics, overtone, string, vibration
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teggenspiller
#1
May21-11, 05:23 PM
P: 94
1. The problem statement, all variables and given/known data
If a guitar string has a fundamental frequency of 500 Hz, what is the frequency of its second overtone?

A. 250 Hz
B. 750 Hz
C. 1 000 Hz
D. 1 500 Hz


2. Relevant equations
fundamental frequency is 2times wavelenth, lambda

and the second overtone=the third harmonic.

3. The attempt at a solution


well, 3*fundamental frequency would make it 1500. its wrong.
then i was thinking the freqency of fundamental is 2*length of string, so i did 500Hz/2 = wavelength of string and multiplied by 3.

ahh. Futhermore, but foremost, i need to understand what in the heck an overtone is. and why there are 1st, second, 3rd. if i have a string and im on its first harmonic, how do i get to it's second? by plucking it again? or with time it increases? i dont understand that conceptual part of harmonics and so its preventing me from applying the concepts. but i also need help with that question. thanks!
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Fewmet
#2
May21-11, 08:04 PM
P: 329
Quote Quote by teggenspiller View Post
1. The problem statement, all variables and given/known data
If a guitar string has a fundamental frequency of 500 Hz, what is the frequency of its second overtone?

A. 250 Hz
B. 750 Hz
C. 1 000 Hz
D. 1 500 Hz


2. Relevant equations
fundamental frequency is 2times wavelenth, lambda

and the second overtone=the third harmonic.

3. The attempt at a solution


well, 3*fundamental frequency would make it 1500. its wrong.
then i was thinking the freqency of fundamental is 2*length of string, so i did 500Hz/2 = wavelength of string and multiplied by 3.

ahh. Futhermore, but foremost, i need to understand what in the heck an overtone is. and why there are 1st, second, 3rd. if i have a string and im on its first harmonic, how do i get to it's second? by plucking it again? or with time it increases? i dont understand that conceptual part of harmonics and so its preventing me from applying the concepts. but i also need help with that question. thanks!
I think the answer is 1500 Hz. The only think I can think of is that there is a variation in terminology and the test maker is counting differently.

About overtones: when you pluck a string, you get a standing wave making the string vibrate with a note at the bridge and at the nut (i.e., the two ends of the string). You also get the string vibrating with a nodes at the bridge, nut and center (over a guitar's 12th fret). That is the first overtone (a.k.a. the second harmonic). Having half the wavelength of the the fundamental, it has twice the frequency (i.e., the octave above the fundamental). You can easily demonstrate the existence of the first overtone by plucking the string and then lightly touching it at the 12th fret. That prevent there being an antinode there, preventing the fundamental from sounding. When you do this, it allows the first overtone to keeps sounding (along with other odd-numbered overtones), so you hear the sound leap up an octave.

Plucking the strong also give the other overtones as well. Here is an excellent animation of multiple standing waves on a violin string. It lets you look at individual harmonics or the sum effect.
teggenspiller
#3
Jun6-11, 06:54 PM
P: 94
thank you so much. i do not completely understand harmonics still, but you provided an EXCELLENT explanation. Thanks!

Fewmet
#4
Jun6-11, 08:20 PM
P: 329
Confused about second overtone/third harmonic; finding the frequency of 2nd overtone

Quote Quote by teggenspiller View Post
thank you so much. i do not completely understand harmonics still, but you provided an EXCELLENT explanation. Thanks!
Let me try again, because it is not that difficult an idea. Have a look at this diagram.

When a string vibrates like it is shown in the top of the image, we say it is one half-wavelength. Let's say the string length is L. The wavelength is then 2L. Now suppose the wave moves along the string at speed V. That means the frequency f is given by
f=V/(2L)
That is the fundamental (or the first harmonic).

While the string vibrates in one half-wavelength, it can also simultaneously vibrate in two half-wavelengths. (The second line of the diagram shows the two half-wavelengths, but not superimposed on the one half-wavelength.) That vibration of the string has a wavelength L and a frequency
f'=V/L = 2f
and it is call the first overtone (or second harmonic).

Carrying on to the next step, we get
f''=V/(2L/3)= 1.5V/L=3f

Next next comes f'''=4f, f''''=5f, etc.

All those sounds blends together with various volumes to make the string's sound. When two strings sound different (like a guitar string and a banjo string) it is because different overtones are accentuated or supressed.


It that any better?


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