Pushing more often than once each period disrups the motion. On the other hand, if you pushed the child every other swing, your pushes would match the child's motion and amplitude would increase. So submultiple pushes will not disrupt motion. Similarly with sound. Waves of the same frequency can interfere destructively or constructively, depending on their relative phase, but to alternate between constructive and destructive interference, two waves have to have different frequencies.
Beats arise from such alternation between constructive and destructive interference.
The piano tuner should loosen the piano string. When 3 beats per second is first heard, the tuner knows he was 3 hertz off the correct frequency. But this could be either 3 hertz above or 3 hertz below. When he tightened the string and increased its frequency, a lower beat frequency would have told him he was on the right track. But the greater beat frequency told him he should have been loosening the string.