beats

Tune two guitar strings to almost the same pitch, play them together, and you hear a single tone that throbs — swelling loud, then soft, then loud again, several times a second. That slow pulsing of loudness is the phenomenon of beats, and it is what forced vibration produces when the driving frequency is close to, but not equal to, the natural frequency.

Beats arise from adding two sinusoids of nearly equal frequency. The undamped forced oscillator driven near, but not at, resonance gives a solution that is a difference of cosines: cos(omega_1 t) - cos(omega_2 t) with omega_1 and omega_2 close. A trigonometric identity rewrites this as a product, 2 sin((omega_1 - omega_2)t/2) times sin((omega_1 + omega_2)t/2): a fast oscillation at the average frequency, multiplied by a slowly varying envelope at half the difference frequency. The slow envelope is the 'wah-wah' you hear; its rate is the difference of the two frequencies.

Beats are the bounded cousin of resonance — what you get just off resonance rather than exactly on it. As the driving frequency is tuned closer to the natural frequency, the beat envelope stretches longer and longer, and in the exact limit the envelope period becomes infinite and the oscillation grows without bound: beats become resonance. Musicians use beats to tune by ear (adjust until the throbbing stops); the same mathematics explains amplitude modulation in radio and the slow pulsing in coupled pendulums.