Short on time? This page is the recap. For Figures 1 and 2 and the worked examples, use the full study guide.
Recap
- Interference is what happens when two or more pulses or waves overlap. They travel through each other; they do not bounce off.
- Superposition: while they overlap, add the displacements, with signs.
- Constructive: displacements in the same direction. Destructive: opposite directions.
- Beats: two waves of slightly different frequency drift in and out of phase, so the loudness rises and falls periodically.
- Standing wave: two waves travelling in opposite directions in a confined region. Nodes never move; antinodes have the largest amplitude. Neighbouring nodes are λ/2 apart.
- Boundary conditions: fixed string end or closed pipe end → node; loose string end or open pipe end → antinode.
- Fundamental (first harmonic): the longest possible wavelength. Harmonic number n = f/f₁.
Key relationships
| Relationship | When to use it |
|---|---|
| f_beat = |f₁ − f₂| | two nearby frequencies sounding together |
| v = fλ | every standing-wave calculation |
| v = √(F_T/μ) | wave speed on a string |
| λₙ = 2L/n, fₙ = n·v/(2L), n = 1, 2, 3 … | fixed–fixed string, open–open pipe, loose–loose string |
| λₙ = 4L/n, fₙ = n·v/(4L), n = 1, 3, 5 … | pipe closed at one end, string fixed at one end and loose at the other |
| node-to-node = λ/2; node-to-antinode = λ/4 | reading lengths from a diagram |
Assumptions behind the numbers
- Waves in a given medium all travel at the same speed, whatever their frequency.
- Pipes are treated as ideal: the antinode sits exactly at the open end.
- The speed of sound is 340 m/s unless the question gives another value (it rises with air temperature).
Mistakes to avoid
- Adding sizes, not signed displacements. An upward 3 cm pulse and a downward 1 cm pulse give +2 cm, not 4 cm.
- Beat frequency as a sum or an average. It is the difference.
- Closed end as an antinode. It is a node.
- Even harmonics in a closed pipe. Only odd ones exist.
- Nodes a whole λ apart. They are λ/2 apart.
- Forgetting v. Frequency depends on v as well as on L.
Quick self-check
- Two notes of 440 Hz and 437 Hz sound together. What is the beat frequency? (3 Hz)
- A pipe closed at one end is 0.17 m long and v = 340 m/s. Find its two lowest resonant frequencies. (500 Hz and 1500 Hz)
- A string fixed at both ends is 0.80 m long. Give the wavelengths of its first two harmonics. (1.6 m and 0.80 m)
- Two pulses pass through each other. What are their shapes afterwards? (Exactly as before.)
Next: practice questions.