Musical instruments - beats phenomenon

[CAF123]

Can anyone give me a description of the beats phenomenon associated with musical instruments (or in general, any waves with which beats are associated).
I have looked at numerous textbooks however I feel I don't completely understand the concept still.
Thanks

 

[torquil]

Suppose you are hearing two sinusoid signals of slightly different frequencies f and f'. Say the phases are such that you are hearing the time-dependent waveform f(t) = sin(f*t)+sin(f'*t).

Mathematically, this is the same as

f(t) = 2 * sin(0.5*(f+f')*t) * cos(0.5*(f-f')*t)

Since f-f' is small, the cos() will be a slowly varying function of time, while the sin() will vary quickly in time. The cos() will be heard as beats, i.e. a slowly varying amplitude of the quickly oscillating signal with frequency 0.5*(f+f') which is the average of f and f'.

It will be very similar for real signals cantaining harmonics.

 

[K^2]

Just a small correction, if frequency is f, the waveform is sin(2π f t).

 

[The_Duck]

To put a picture to what torquil said: http://www.wolframalpha.com/input/?i=plot+sin(x)+++sin(1.1x),+x=0..200

 

[PJC01]

The beats phenomenon is a perceptual effect that occurs when two sound waves of slightly different frequencies are played simultaneously. The result is a periodic variation in the overall loudness of the sound, known as beats. This can be observed with musical instruments such as guitars, where two strings of slightly different frequencies are played together. The beats are heard as a pulsating sound, with the loudness increasing and decreasing at regular intervals.

The reason for this phenomenon lies in the way our ears and brain process sound waves. When two sound waves of different frequencies reach our ears, they are converted into electrical signals and sent to the brain. The brain then interprets these signals and perceives the difference in frequencies as beats.

The frequency of the beats is equal to the difference between the two original frequencies. For example, if one string of a guitar has a frequency of 440 Hz and another has a frequency of 442 Hz, the resulting beats will have a frequency of 2 Hz. As the difference between the frequencies decreases, the beats become slower and eventually disappear when the frequencies are the same.

The beats phenomenon can also be observed with other types of waves, such as light and water waves. In general, any two waves with slightly different frequencies can produce beats when combined.

I hope this explanation helps you better understand the concept of beats phenomenon. It is a fascinating phenomenon that has been studied and utilized by musicians and scientists alike. If you have any further questions, please do not hesitate to ask.