Oscillations and Waves, Acoustics
Core concepts
Simple harmonic motion satisfies a = −ω²x with x(t) = A cos(ωt + φ). Know the spring–mass (ω = √(k/m)) and simple pendulum (ω = √(g/L)) results, and the energy exchange between kinetic and potential. Damped and forced oscillations: resonance at driving frequency near natural frequency. For waves: v = fλ, superposition, standing waves on strings (both ends fixed: L = nλ/2) and pipes (open–open vs closed–open). Sound intensity level β = 10 log(I/I₀) with I₀ = 10⁻¹² W/m². Doppler effect: f' = f(v ± v_o)/(v ∓ v_s).
Traps
- Using degrees where radians are required in ωt.
- Forgetting that a closed pipe supports only odd harmonics.
- Sign errors in the Doppler formula - draw the geometry.
- Confusing amplitude with displacement in energy calculations.
- Assuming small-angle pendulum works for large amplitudes.
Worked example
A string of length 1.0 m fixed at both ends vibrates in its fundamental. λ = 2L = 2.0 m. If the wave speed is 200 m/s, f = v/λ = 100 Hz. Rule of thumb: always identify the boundary conditions first - they determine which wavelengths fit.