Units 6-7 cover oscillations (simple harmonic motion) and waves — how energy propagates through media without net transfer of matter.
SHM: restoring force proportional to displacement (F = -kx). Examples: mass-spring (T = 2π√(m/k)) and simple pendulum (T = 2π√(L/g)). Energy: oscillates between KE (max at equilibrium) and PE (max at extremes). Total mechanical energy constant (no damping). Position: x = A cos(ωt), velocity: v = -Aω sin(ωt). Maximum speed at equilibrium; zero speed at amplitude.
Mechanical waves need a medium. v = fλ. Transverse: displacement ⊥ propagation. Longitudinal: displacement ∥ propagation (compressions and rarefactions). Superposition: waves add algebraically. Constructive (in phase) → larger amplitude. Destructive (out of phase) → cancellation. Reflection: fixed end → inverted; free end → upright.
Standing waves: superposition of two waves travelling in opposite directions. Nodes (zero displacement) and antinodes (maximum). String fixed at both ends: λₙ = 2L/n, fₙ = nv/2L. Open pipe: fₙ = nv/2L (all harmonics). Closed pipe: fₙ = nv/4L (odd harmonics only). Sound: longitudinal wave. Speed depends on medium. Beats: two close frequencies → |f₁-f₂| = beat frequency.
A closed pipe has a node at the closed end (air cannot move) and an antinode at the open end (maximum displacement). The fundamental fits one quarter of a wavelength in the pipe (L = λ/4). The next possible standing wave has three quarters (L = 3λ/4, which is the 3rd harmonic); then five quarters (5th harmonic), etc. Only odd multiples of the fundamental frequency work because even harmonics would require a node at the open end — which contradicts the boundary condition there. An open pipe has antinodes at both ends, so all harmonics are possible.
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