WAVE-01 Rapid Revision — 5 Essential Bullets — Summary

SHM energy trap: KE = PE at x = A/√2 (≈0.707A), not A/2. Total energy E = ½mω^{2} $A^{2}$ is constant. Spring period T = 2π $\sqrt{m/k}$ is independent of amplitude and gravity.
Standing waves: Nodes at x = nλ/2; antinodes at x = (2n+1)λ/4. Consecutive nodes are λ/2 apart.
Pipe harmonics: Open pipe AND strings: all harmonics, $f_n$ = $\frac{nv}{2L}$ . Closed pipe: ODD harmonics only, $f_n$ = $\frac{nv}{4L}$ ; $f_{1}_{closed}$ = ½ $f_{1}_{open}$ (same L).
Doppler formula: f' = f $\frac{v ± v_O}{(v ∓ v_S)}$ . Toward = + numerator, − denominator → f' > f. Away → f' < f. $f_{beat}$ = | $f_{1}$ − $f_{2}$ |.
Wave speed order: $v_{solid}$ > $v_{liquid}$ > $v_{gas}$ . String: v = $\sqrt{T/μ}$ . Air: v ∝ √ $T_K$ . Pendulum free-fall: T = ∞ (no oscillation).