Oscillations and Waves
What this domain tests
Oscillations and Waves tests simple harmonic motion, pendulum, wave equation, superposition, beats, standing waves and Doppler effect. Questions often ask period, frequency, phase or wave speed from an equation.
Core rules
- SHM: acceleration a = -ω²x, directed opposite displacement. Displacement x = A sin(ωt + φ); velocity leads displacement by π/2; acceleration leads velocity by π/2.
- Time period: T = 2π/ω. For a spring, T = 2π√(m/k); for a simple pendulum, T = 2π√(L/g).
- Wave relation: v = fλ = ω/k. For y = A sin(ωt - kx), ω = 2πf and k = 2π/λ.
- Superposition gives interference; beats occur when frequencies differ slightly, beat frequency = |f1 - f2|.
- In a closed pipe, fundamental frequency f = v/4L; in an open pipe, f = v/2L.
Worked example
Find the period of a simple pendulum of length 1 m. Take g = π² m/s². T = 2π√(L/g) = 2π√(1/π²) = 2π(1/π) = 2 s. The reasoning converts g to π² so the square root is exact; the result is independent of bob mass for small oscillations.
Traps
The pendulum formula is valid only for small angular amplitudes. Do not confuse angular frequency ω with frequency f; ω = 2πf. In a wave equation, the sign before kx gives direction of propagation. Beat frequency is a difference, never a sum. Resonance occurs at a natural frequency, not at any driving frequency.