beat frequency

A phenomenon that can be heard with two constant tones which are close to, but don't exactly match each other. The individual tones can be heard, and a third frequency can be heard between the two. It's easiest to hear when the "real" frequencies are off by one or two Hz; in this case, the third frequency (the beat frequency) can be heard as a slow wavering in the intensity of the waves. If you get two very different tones loud enough, the "wavering" creates a third, audible note.

Bear with me, this is a lot easier to explain with a picture...The two frequencies start out in phase, reinforcing each other. As one frequency moves out of phase with the other, they start to cancel each other out until they are completely out of phase, causing the total amplitude to be zero. Your ear perceives this as a combination of the tones which goes from loud to soft to loud at the frequency of the difference of the "real" frequencies. It can be expressed as:

f_beat = f_a - f_b

A way to hear this clearly for yourself is to get two tone generators, set one to 250 Hz and the other to 252 Hz. You will hear two tones going from loud to soft twice every second. Turn the second generator down to 251 Hz, and the beat frequency will go to once a second. Yay.

How does the beat frequency (f_a-f_b)/2 come into being? That most hated part of the high school Mathematics curriculum, trigonometry, holds the keys!

Suppose we have frequencies f+w and f-w, and we sound them together. If there's no phase difference between them (and if there is any, the same thing holds and we still get the same result), then we have the waveforms sin(2π(f+w)t) and sin(2π(f-w)t) sounding together. Retrieving some trigonometric identities from distant memory, we have that

sin(a+b)+sin(a-b) = sin(a)cos(b)+cos(a)sin(b) + sin(a)cos(b)-cos(a)sin(b) =
= 2sin(a)cos(b)

(that wasn't so bad, was it?).

So sin(2π(f+w)t)+sin(2π(f-w)t) = 2sin(2πft)cos(2πwt). In flamingweasel's example, f=251Hz and w=1Hz. We're going to be hearing a pure sine wave at frequency 251Hz (that's 2sin(2πft)), only it will be modulated (in exactly the same way that an AM signal is modulated) by a frequency of 1Hz. In other words, over the first half-second the 251Hz wave will have its amplitude squished down to 0, over the next half-second it will go back to full strength, and it will continue to cycle like that.