When Frequencies Clash
If you add two waves with slightly different frequencies, you get Beats. The volume of the sound will slowly grow and shrink.
\[f_{\text{beat}} = |f_1 - f_2|\]
Phase Velocity vs Group Velocity
- Phase Velocity: How fast the individual "ripples" of the wave are moving.
- Group Velocity: How fast the entire "packet" of waves is moving.
Worked Examples
Example 1: Tuning a Guitar
If you play a string and a tuning fork (440 Hz) and you hear 3 "beats" every second, your string is either at 437 Hz or 443 Hz. As you tighten the string and the beat frequency slows down to zero, you know you are perfectly in tune.
The Bridge to Quantum Mechanics
In Quantum Mechanics, a "Particle" is actually a Wave Packet—a collection of waves of different frequencies added together. The Group Velocity of this packet is exactly the velocity of the particle as seen by a human. The Phase Velocity, however, is often faster than the speed of light! This doesn't break physics, because the individual ripples don't carry information—only the group (the particle) does. This distinction is how we resolve the paradoxes of wave-particle duality.