Time period of Beat (acoustics)

In summary, the equation $$y=2 A \cos 2 \pi\left(\frac{\nu_{1}-\nu_{2}}{2}\right) t \sin 2 \pi\left(\frac{\nu_{1}+\nu_{2}}{2}\right) t$$ represents the combination of two sinusoids at different frequencies, which is known as beats. It can be derived from the sum of two sine waves at different frequencies. The time period for the formation of beats is determined by the difference frequency of the two sine waves, which is given by $$T=\dfrac{1}{\nu_{1}+\nu_{2}}$$. The beat frequency is given by $$n=\nu
  • #1
Huzaifa
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$$y=2 A \cos 2 \pi\left(\frac{\nu_{1}-\nu_{2}}{2}\right) t \sin 2 \pi\left(\frac{\nu_{1}+\nu_{2}}{2}\right) t$$

Can you explain me the significance of the above equation in the context of waves and oscillations? It's something to do with 'beats,'.
 
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  • #2
Where did you get it from? They probably derive it from the combination of the two sinusoids at different frequencies, no?
 
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  • #3
https://en.wikipedia.org/wiki/Beat_(acoustics)

1641320559632.png
 
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  • #4
berkeman said:
Where did you get it from? They probably derive it from the combination of the two sinusoids at different frequencies, no?
Yes sir, they derived it from $$y_{1}=A \sin \omega_{1} t ; y_{2}=A \sin \omega_{2} t$$
 
  • #5
$$y=y_{1}+y_{2}=A\left[\sin \omega_{1} t+\sin \omega_{2} t\right]=A\left[2 \sin \left(\frac{\omega_{1}+\omega_{2}}{2}\right) t \cos \left(\frac{\omega_{1}-\omega_{2}}{2}\right) t\right]$$

How to proceed ahead from here?
 
  • #6
Huzaifa said:
$$y=y_{1}+y_{2}=A\left[\sin \omega_{1} t+\sin \omega_{2} t\right]=A\left[2 \sin \left(\frac{\omega_{1}+\omega_{2}}{2}\right) t \cos \left(\frac{\omega_{1}-\omega_{2}}{2}\right) t\right]$$

How to proceed ahead from here?
Sorry, what do you mean? Are you familiar with common trig identities like the sum of two sine waves?

https://www.purplemath.com/modules/idents.htm

1641321981618.png
 
  • #7
Duplicate thread starts merged into one thread
Consider two two sinusoids at different frequencies $$y_{1}=A \sin \omega_{1} t ; y_{2}=A \sin \omega_{2} t$$
$$y=y_{1}+y_{2}=A\left[\sin \omega_{1} t+\sin \omega_{2} t\right]=A\left[2 \sin \left(\frac{\omega_{1}+\omega_{2}}{2}\right) t \cos \left(\frac{\omega_{1}-\omega_{2}}{2}\right) t\right]$$

For the formation of waxing, A is maximum, what will be the time period? And for the formation of wanning, A is minimum, what will be the time period?
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This is from my class notes. Unfortunately, I missed my class and I am not able to understand now. So I have turned to Physics Forums for help.

Please explain me how the time period for the formation of beat T, $$T=\dfrac{1}{\nu_{1}+\nu_{2}}$$
and the beat frequency $$n=\nu_{1} \sim \nu_{2}$$
 
  • #9
Huzaifa said:
How to solve for time period now?
Which time period? The sound you hear when listening to audio beats is the sum frequency, modulated at the difference frequency. So the period of the beats (the louder-softer characteristic) is at the difference frequency of the two sine waves.
 
  • #10
berkeman said:
Which time period? The sound you hear when listening to audio beats is the sum frequency, modulated at the difference frequency. So the period of the beats (the louder-softer characteristic) is at the difference frequency of the two sine waves.
Yes, I think the time period of the beats. Does the time period depend on A? When the A is maximum, $$t=\dfrac{n}{\nu_1-\nu_2}$$, When A is minimum $$t=\dfrac{2n+1}{2 \left( \nu_1-\nu_2 \right)}$$.
 

Related to Time period of Beat (acoustics)

What is the time period of beat in acoustics?

The time period of beat in acoustics refers to the time it takes for two sound waves with different frequencies to interfere with each other and create a perceivable variation in loudness. It is usually measured in seconds.

How is the time period of beat calculated?

The time period of beat can be calculated by taking the inverse of the difference between the frequencies of the two sound waves. For example, if one sound wave has a frequency of 100 Hz and the other has a frequency of 105 Hz, the time period of beat would be 1/5 or 0.2 seconds.

What factors can affect the time period of beat?

The time period of beat can be affected by the difference in frequencies, the amplitude of the sound waves, and the distance between the sound sources. It can also be influenced by the direction and speed of the sound waves.

How is the time period of beat used in music?

In music, the time period of beat is used to create a rhythmic effect by deliberately introducing two sound waves with slightly different frequencies. This creates a pulsating or "beating" sound that can add depth and texture to a musical piece.

Can the time period of beat be perceived by the human ear?

Yes, the time period of beat can be perceived by the human ear as a change in loudness or pulsating sound. Our brains are able to detect and interpret these variations in sound waves, making the time period of beat audible to us.

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