Superposition of Waves
This chapter covers the superposition principle, stationary waves in strings and pipes, sonometer and resonance-tube experiments, beats, and the Doppler effect. Expect questions on harmonics, end correction, beat frequency and apparent frequency.
Core rules
- Superposition: resultant displacement is the vector sum of the individual displacements; the waves must be coherent for a stable pattern.
- Stationary waves: two identical waves travelling in opposite directions give nodes (zero amplitude) and antinodes (maximum amplitude) spaced λ/2 apart.
- String fixed at both ends: f(n) = nv/2L, n = 1, 2, 3...; all harmonics present. v = √(T/m), where m is mass per unit length.
- Pipe open at both ends: all harmonics, f(n) = nv/2L. Pipe closed at one end: only odd harmonics, f(n) = (2n−1)v/4L.
- End correction e = 0.6r for an open end; the effective length becomes L + e.
- Resonance tube: v = 2f(L₂ − L₁) using the first and second resonance lengths.
- Beats: frequency = |f₁ − f₂|; loudness varies at the beat frequency.
- Doppler: f' = f(v ± v(o))/(v ∓ v(s)), signs chosen so approaching raises the pitch.
Worked example
A string of length 1.0 m and linear density 0.01 kg/m is under tension 100 N. Then v = √(100/0.01) = 100 m/s, and the fundamental is f₁ = v/2L = 50 Hz. The third harmonic is 150 Hz. Two tuning forks of 256 Hz and 260 Hz give 4 beats per second.
Traps
- Giving even harmonics for a closed pipe - only odd ones exist.
- Forgetting the end correction when the tube radius is not negligible.
- Adding frequencies for beats instead of taking the difference.