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