Oscillations and waves, including sound and superposition
Oscillations and waves
This domain tests simple harmonic motion, pendulums and springs, wave speed, superposition, standing waves, sound, beats, resonance and the Doppler effect. Questions often require you to relate a displacement graph to phase, or to find a harmonic frequency on a string or in a pipe.
Core rules
- SHM: a=−ω²x and x=A cos(ωt+φ); ω=2π/T=2πf. For a mass–spring system T=2π√(m/k); for a small-angle pendulum T=2π√(L/g). Period is independent of amplitude for small oscillations.
- Energy in SHM: total energy is ½kA²; KE and PE interchange, with maximum speed v_max=ωA at equilibrium.
- Wave relation: v=fλ. On a string v=√(F/μ), where μ is linear mass density. Sound speed in air is about 343 m/s at 20 °C and depends on temperature, not frequency.
- Superposition: displacements add. Constructive interference occurs when path difference is mλ; destructive when it is (m+½)λ.
- Standing waves: a string fixed at both ends has nodes at both ends, with λ_n=2L/n. Beats occur at f_beat=|f1−f2|.
- Doppler: observed frequency rises when source and observer approach, and falls when they separate.
Worked example
A string of length 0.80 m has tension 80 N and linear density 0.0050 kg/m. Find the fundamental frequency. The wave speed is v=√(80/0.0050)=√16000=126 m/s. For a string fixed at both ends, the fundamental has λ=2L=1.60 m. Then f=v/λ=126/1.60=79 Hz. The next harmonic is 158 Hz. Doubling the tension multiplies the frequency by √2, not by 2.
Traps
Using degrees in ωt; thinking period depends on amplitude for a pendulum at large angles, where the small-angle approximation fails; placing an antinode at a fixed end; confusing beat frequency with the average frequency; using 3×10⁸ m/s for sound; forgetting that the Doppler shift depends on relative motion.