Waves
Core concepts
Wave questions require the wave equation v = fλ, superposition, standing waves, resonance in strings and pipes, beats, and the Doppler effect. You should understand transverse versus longitudinal waves, phase and path difference, and conditions for constructive and destructive interference.
Traps
- Confusing wave speed with particle speed; particles oscillate, the wave propagates.
- Forgetting that a wave's speed is set by the medium, not the frequency.
- Mixing boundary conditions: a string fixed at both ends has nodes at both ends; a pipe open at both ends has antinodes at both ends.
- Using the wrong harmonic formula for closed pipes (only odd harmonics).
- Applying the Doppler formula with the wrong sign convention for source versus observer motion.
Rule of thumb
For a string of length L fixed at both ends, the nth harmonic is f_n = nv/(2L). For a pipe closed at one end, only odd harmonics occur: f_n = nv/(4L), n odd.
Worked example
A string of length 0.8 m and mass 4 g under 100 N tension. μ = 0.004/0.8 = 0.005 kg/m. Speed v = √(T/μ) = √(100/0.005) = √20000 ≈ 141 m/s. Fundamental frequency f₁ = v/(2L) = 141/1.6 ≈ 88 Hz.