Oscillations and Waves
Oscillations and Waves covers simple harmonic motion, energy in SHM, pendulums and springs, then superposition, standing waves in strings and pipes, beats and the Doppler effect.
Core rules
- SHM: x = A sin(ωt + φ), v = ω√(A² − x²), a = −ω²x, and ω = 2π/T.
- Energy: K = ½mω²(A² − x²), U = ½mω²x², total ½mω²A², constant in time.
- Pendulum T = 2π√(l/g); spring–mass T = 2π√(m/k).
- Wave speed v = νλ; on a stretched string v = √(T/μ).
- Standing waves: string fixed at both ends and open pipe give ν = nv/2L; a closed pipe gives ν = (2n−1)v/4L, odd harmonics only.
- Beats: ν(beat) = |ν₁ − ν₂|.
- Doppler: ν′ = ν(v ± v(o))/(v ∓ v(s)), signs chosen so that approach raises the observed frequency.
Worked example
A simple pendulum has l = 1.0 m. Then T = 2π√(1.0/9.8) = 2π × 0.319 ≈ 2.0 s. On the Moon, where g is about one-sixth of Earth's, the period grows by √6 ≈ 2.45 times, to roughly 4.9 s, because T depends on √(1/g).
Traps
Using degrees in ω = 2π/T; expecting even harmonics from a closed pipe; choosing the wrong Doppler sign when source and observer approach; believing the total energy in SHM changes with displacement.