OSCILLATIONS & WAVES

A circle’s shadow — the phasor shows why everything oscillates in cosines.

The period carries no m — mass cancels before the physics even starts.

Two travelers passing forever — their sum stands still.

Same speed, shorter spacing — pitch shift is geometry, not magic.

y₁+y₂ = 2A·cos(πΔf·t)·cos(2πf̄t) — the throb piano tuners count.

ma + cv + kx = 0 at every drawn frame — ring, settle, or crawl.

Huygens’ tautochrone: beads from any height reach the bottom in the same instant — exact SHM in arc length.

Push a damped spring: displacement peaks BELOW ω₀, velocity peaks AT ω₀ — the exact curve acres draws for RLC current.

Two carts, two normal modes — ω_s=√(k/m) in step, ω_a=√((k+2k_c)/m) opposed. Start one and their motion envelopes trade prominence and return.