# Guitar Strings & Tuning Forks: Investigating Beats

• Granger
In summary: Therefore, the time between consecutive maxima is half the period of the amplitude cycle, which is 1/|f1 - f2|. In summary, the problem involves a guitar string vibrating at frequencies that are multiples of 109 Hz and a tuning fork of 440 Hz. The beats between these two signals can be calculated by finding the difference in frequencies, and the time interval between consecutive maximum sound intensity can be found by taking the inverse of the beat frequency. The 4th harmonic is responsible for these beats, and the time between consecutive maxima is half the period of the amplitude cycle, which is 1/|f1 - f2|.
Granger

## Homework Statement

A guitar string with 0.60 m and 0.012 of mass vibrates with frequencies that are multiples of 109 Hz. Approaching to the string a tuning fork of 440 Hz we verify beats between the sound signals of the string and the tuning fork. Calculate the time interval between consecutive maximum of sound intensity. Identify the harmonic responsible for these beats.

## Homework Equations

$$f=\abs{f_2-f_1}$$

## The Attempt at a Solution

I don't have any idea on how to determine the harmonic responsible for these beats. But without that there is no way I can determine the frequency and therefore I can not determine the period (time interval between to consecutive maximum).
There's something here that is escaping me, can someone clarify me please...

HEllo Granger,

Beats have to do with small differences in frequencies that cause a low-frequency amplitude envelope. So in your case you would expect something to happen with the third harmonic (436 Hz) of the string and the ground tone of the fork. See if you can write the sum of these two tones as a product as in the second link. And think carefully what the beat frequency value means for the time between consecutive maxima

Granger
BvU said:
HEllo Granger,

Beats have to do with small differences in frequencies that cause a low-frequency amplitude envelope. So in your case you would expect something to happen with the third harmonic (436 Hz) of the string and the ground tone of the fork. See if you can write the sum of these two tones as a product as in the second link. And think carefully what the beat frequency value means for the time between consecutive maxima

Thanks! Oh ok I didn't understand the concept. It makes sense it's small frequencies. After calculation I concluded it's the 4th harmonic (436 Hz).
And I think I understand now. Because we have a product of cosines (so (-1)(-1)=1) we will have 2 maxima in a period (inverse of the beat frequency). Therefor the time between consecutive maxima is half the inverse of the beat frequency, right?

Last edited:
Granger said:
4th harmonic
sorry, my mistake (counted: ground tone - 1st - 2nd - 3rd instead of ground - 2nd - 3rd - 4th)

Nevertheless, I think you understand it quite well !

Granger said:
Thanks! Oh ok I didn't understand the concept. It makes sense it's small frequencies. After calculation I concluded it's the 4th harmonic (436 Hz).
And I think I understand now. Because we have a product of cosines (so (-1)(-1)=1) we will have 2 maxima in a period (inverse of the beat frequency). Therefor the time between consecutive maxima is half the inverse of the beat frequency, right?
Actually, the time between consecutive maxima is 1/|f1 - f2|. This is because a maximum occurs twice in each amlitude cycle which has a frequency of |f1 - f2|/2.

## 1. What is a beat in relation to guitar strings and tuning forks?

A beat occurs when two sound waves with slightly different frequencies interfere with each other. In the case of guitar strings and tuning forks, the strings and forks produce sound waves that interact and create a pulsing or beating sensation.

## 2. How do you investigate beats with guitar strings and tuning forks?

To investigate beats, you will need a guitar, a tuning fork, and a tuner. Start by plucking a string on the guitar and holding the tuning fork near the guitar's sound hole. Adjust the tuning fork until the beats stop, indicating that the tuning fork is producing the same frequency as the guitar string.

## 3. What factors affect the number of beats produced between guitar strings and tuning forks?

The number of beats produced is affected by the difference in frequency between the two sound sources. The larger the frequency difference, the more beats will occur. Additionally, the distance between the guitar string and the tuning fork can also impact the number of beats.

## 4. How does changing the tension of a guitar string affect the production of beats with a tuning fork?

Changing the tension of a guitar string can alter its frequency, which in turn can affect the production of beats with a tuning fork. A tighter string will produce a higher frequency, resulting in fewer beats, while a looser string will produce a lower frequency, resulting in more beats.

## 5. What are some real-world applications of studying beats with guitar strings and tuning forks?

The study of beats with guitar strings and tuning forks can be applied to various fields, such as music and acoustics. It can help musicians tune their instruments more accurately and understand the principles of harmony and resonance. It can also be used in engineering to design and optimize sound systems and in medical fields to diagnose and treat hearing impairments.

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