WAVE KINEMATICS

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.

Snell twice per color — and at minimum deviation, the prism measures its own glass.

Darkness made of light — d·sinθ = mλ, and only ⌊d/λ⌋ fringes exist.

One open slit still forbids directions — a·sinθ = mλ dark, the central hub twice a side band, every d/a-th double-slit order missing.

Soap-film colors: 2nt = (m−½)λ — and a hidden half-flip makes a vanishing film go black, not bright.

At 0.6c the light ahead exactly doubles — and abeam, time dilation shows its bare face.

d·sinθ = mλ run backwards — measure an angle, recover the light, residual zero.

The bow is a caustic — dD/dθ = 0 on a sphere: red 42.3° outside violet 40.6°, and a dark band above.