Wave velocity ≠ particle velocity
● VERBATIM — examined word-for-wordWave velocity: constant, = fλ, direction of propagation. Particle velocity: SHM variable — v = y₀ω cos(ωt), equal to the gradient of the y–t graph, zero at crest/trough, max (= ωy₀) at equilibrium. Raising intensity raises PARTICLE max speed, never wave speed — wave speed is a property of the medium.
What this actually means
Two different speeds live on the same picture. The profile slides along at a constant v = fλ. Each bead does SHM: momentarily at rest at the crest, fastest through equilibrium at ωy₀.
Particle velocity is the gradient of the y–t graph, so it changes at every instant. Wave velocity never changes while the medium is fixed.
Turning up the volume raises the amplitude, so particles hit higher top speeds, but the sound still crosses the room at the same speed. Wave speed belongs to the medium, not to the source.
Using v = fλ for a particle, or claiming the crest particle is fastest — it is momentarily at rest.
Prove it — watch it be true
- Turn on the velocity arrows: longest at the equilibrium crossings, zero at the crests
- Watch the ghost profile: it never speeds up or slows down
- Raise the amplitude: particle arrows grow, profile speed unchanged