I have a question about 'beats' between two sound waves of slightly different frequencies. Basically, I think I understand everything on this page (and have read a few textbooks on this): http://www.animations.physics.unsw.edu.au/jw/beats.htm . And in practice, I work with this concept all the time. However, I still do not entirely understand why the beat frequency is the absolute value of the difference between the two frequencies. I understand the mathematical explanation on the page that I linked. However, I still do not grasp, physically, why the peaks or troughs of two waves moving at different frequencies would coincide that many times per second. Let me elaborate. Hartmann writes in Principles of Musical Acoustics: " Suppose that we measure the two waves from the two tuning forks (at 500 Hz and 501 Hz) when each is at a positive maximum. They are “in phase.” Then the sum will be as loud as it can be. As time progresses the two waves will become out of phase because the forks have different frequencies. After half a second the 500-Hz wave will have executed 250 complete cycles and it will be at a positive maximum again. However, the 251-Hz wave will have executed 250.5 cycles, and the extra half a cycle will mean that this wave is at a negative maximum. The positive maximum and negative maximum will cancel and the sum will be quiet. After a full second, the 500-Hz wave will have gone through 500 cycles and be at a positive peak; the 501-Hz wave will have gone through 501 complete cycles and also be at a positive peak. As a result, the waves will once again be perfectly in phase and there will be a maximum again. This example shows that there is one maximum and one minimum per second when the frequency difference is 1 Hz." So far so good. Then he gives the example of waves moving at 500 Hz and 510 Hz: Now there are ten beats per second. Ten beats per second is rather fast but one can still hear them as a rapidly varying loudness. Again, we choose to start the clock when both waves are together. After 50 ms (0.05 s) the 500-Hz wave will have gone through 25 cycles but the 510-Hz wave will have gone through 25.5 cycles. The waves will be out of phase and will cancel. After 100 ms the two waves will have gone through 50 and 51 complete cycles and will be in phase again. This makes sense to me. However, I think that this only works because 10 is the greatest common factor of the two numbers he has chosen. If we combine sound waves moving at 179 Hz and 189 Hz, the beat frequency would still be 10 Hz but the physical explanation would be much less clear. After 100 ms, the waves would have gone through 17.9 cycles and 18.9 cycles, respectively, meaning that neither one would be at its peak. So why would there be a 'maximum' here? In fact, I don't think the two waves' peaks would coincide more than once per second. So why do we get 10 beats/s in this case (other than because trigonometry says so)?