SHM & Wave Core Principles — Essential Key Points — Summary

SHM definition: Restoring force ∝ displacement: a = −ω^{2}x. Displacement: x = A sin(ωt + φ).
Velocity: v = ω√( $A^{2}$ − $x^{2}$ ); maximum Aω at mean position (x = 0); zero at extremes (x = ±A).
Acceleration: a = −ω^{2}x; zero at mean, maximum Aω^{2} at extremes. Velocity and acceleration are 90° out of phase.
Energy: E = ½mω^{2} $A^{2}$ = constant. KE and PE trade off; KE = PE at x = A/√2 (NOT x = A/2).
Spring-mass period: T = 2π√(m/k) — independent of amplitude and gravity; works in zero-g.
Series springs: k_eff = k_{1}k_{2}/(k_{1}+k_{2}) — softer, longer T. Parallel springs: k_eff = k_{1}+k_{2} — stiffer, shorter T.
Simple pendulum: T = 2π√(L/g) — independent of mass, depends on L and g only; valid for θ < 15°.
Pendulum in lift (up): T decreases (g_eff = g+a). Pendulum in lift (down): T increases (g_eff = g−a). Free fall: T = ∞.
Wave speed: v = fλ = ω/k. In string: v = √(T/μ). In air: v = √(γRT/M) ∝ √T_K.
Speed hierarchy: v_solid > v_liquid > v_gas (elastic modulus decreases across states).
Progressive wave equation: y = A sin(kx − ωt) for +x direction; k = 2π/λ is wave number.
Standing wave: y = 2A sin(kx) cos(ωt). Nodes at x = nλ/2; antinodes at x = (2n+1)λ/4.
Node spacing: Consecutive nodes are always λ/2 apart (not λ).
Open pipe/string: All harmonics f_n = nv/(2L), n = 1, 2, 3….
Closed pipe: Odd harmonics only f_n = nv/(4L), n = 1, 3, 5…. Fundamental = ½ that of open pipe.
Beat frequency: f_beat = |f_{1} − f_{2}|; one beat = one complete amplitude oscillation cycle.
Doppler formula: f' = f(v ± v_O)/(v ∓ v_S); toward = + numerator, − denominator; away = opposite.