Waves

y(x,t) = A sin(kx − ωt) — particles oscillate ⟂.

Compressions / rarefactions along propagation.

Pulse moves at speed v across the medium.

v = fλ — see λ marker on the wave.

v = √(T/μ).

Constructive vs destructive at φ = 0° / 180°.

f_beat = |f₁ − f₂| — envelope visible.

λ_n = 2L/n — nodes & antinodes.

Highlighted N and A on a vibrating string.

Concentric wavefronts from a point source.

Air vs water vs steel vs diamond.

Pitch shifts as source approaches/recedes.

Observer's motion changes perceived frequency.

f' = f(v±v_o)/(v∓v_s).

f_n = nv/2L — first 6 harmonics.

Open: nv/2L. Closed: (2n−1)v/4L.

P = ½μvω²A² — power along a string.

φ = (2π/λ) Δx — visualize phase vs path.