Question #8c553

1 Answer
Oct 26, 2016



We have 2 sound waves of same amplitude #A#, with frequencies #f_1 approx f_2#, so

  • #y_1 = A sin 2 pi f_1 t#

  • #y_2 = A sin 2 pi f_2 t#

The resultant wave is

#y = y_1 + y_2 = A sin 2 pi f_1 t+ A sin 2 pi f_2 t#

And by the sum to product identity:

#y = 2 A * color(gren)(sin pi ( f_1 + f_2) t) * color(red)( cos pi (f_1 - f_2) t )#

The red cosine term is the beat frequency we can perceive, ie #f_b = abs (f_1 - f_2)#. It acts as an amplitude envelope for the green sine term.

The amplitude of the resultant beat can therefore be seen to be

#A(t) = 2 A * cos pi (f_1 - f_2) t #

And that is what is perceived when, say, 2 musical notes that are very close in pitch are plucked on a guitar: in the model proposed above, a sinusoid pattern to its amplitude.

Because Intensity #I propto A^2#, we can see that at max amplitude, ie when #cos pi (f_1 - f_2) t = pm1#, intensity is quadrupled as amplitude is doubled from A to 2A.