Explain how standing waves are formed. Derive the positions of nodes and antinodes, and use them to find the frequencies of a pipe closed at one end.

1. 1 · Define. A standing wave forms when two equal waves travel in opposite directions along the same line and superpose.
2. 2 · Write the two waves. Incident: y₁ = a sin(ωt − kx). Reflected at a fixed end: y₂ = −a sin(ωt + kx).
3. 3 · Add them. y = −2a sin kx cos ωt: the amplitude 2a sin kx depends on x, and the shape does not move.
4. 4 · Find the nodes and antinodes.
   • Nodes: sin kx = 0, so x = 0, λ/2, λ ...
   • Antinodes: sin kx = ±1, so x = λ/4, 3λ/4 ...
   • Node to node = λ/2; node to antinode = λ/4
5. 5 · Draw the closed pipe modes. [Closed pipe with N at the closed end and A at the open end, for l = λ/4, 3λ/4 and 5λ/4]
6. 6 · Write the frequencies. ν = v/4l, 3v/4l, 5v/4l ...: only odd harmonics.
7. 7 · Real life. Blowing across a bottle gives a closed-pipe note; adding water raises the pitch.
8. 8 · Close in one line. So, a closed pipe gives a fundamental of v/4l and only odd harmonics.

Scoring tip: Mark N and A on every pipe diagram, and write l in terms of λ beside each sketch.