Reflection of waves, standing waves and organ pipes

Reflection and standing waves

Send a pulse along a slinky tied to a wall. At the fixed end, the support pushes back, and the crest returns as a trough: a phase change of π. At a free end, a crest returns as a crest: no phase change. Sound waves behave in a similar way. At a closed end (a denser boundary), a compression returns as a compression. At an open end (a rarer boundary), a compression returns as a rarefaction. Where two media meet, part of a wave is reflected and the rest is transmitted into the second medium, with no change of phase. When a wave meets its own reflection, two equal waves travel in opposite directions along the same line. Their superposition is a standing (stationary) wave. Its shape does not move forward; the loops only swell and shrink. Points that never move are nodes (N). Points that vibrate the most are antinodes (A). Unlike a travelling wave, the amplitude changes from point to point, and no energy flows past any point on average.

Derivation: equation of a standing wave

1. 1
   Incident wave along +x: y₁ = a sin(ωt − kx).
2. 2
   Reflected wave along −x from a fixed (denser) end, with a phase change of π: y₂ = −a sin(ωt + kx).
3. 3
   By superposition: y = y₁ + y₂ = a[sin(ωt − kx) − sin(ωt + kx)].
4. 4★
   Using sin C − sin D = 2 cos[(C + D)/2] sin[(C − D)/2], this becomes y = −2a sin kx cos ωt.
5. 5
   Every point vibrates with angular frequency ω, but its amplitude 2a sin kx depends on its position x. The shape does not travel.
6. 6
   Nodes: sin kx = 0, so kx = 0, π, 2π ..., giving x = 0, λ/2, λ ...
7. 7
   Antinodes: sin kx = ±1, so x = λ/4, 3λ/4, 5λ/4 ..., where the amplitude is 2a.
8. 8
   So, nodes are λ/2 apart, and a node and the next antinode are λ/4 apart.

★ key step

Harmonics of organ pipes (v = speed of sound, l = length)

How to draw the modes of open and closed pipes

• Draw tall narrow rectangles for the pipes. Close the bottom of a closed pipe with a thick line; leave both ends of an open pipe open.
• Closed pipe, fundamental: write N at the closed end and A at the open end. Draw two curves widening from N to A. Write l = λ/4 and ν = v/4l.
• Closed pipe, first overtone: from the closed end mark N, A, N, A. Write l = 3λ/4 and ν = 3v/4l.
• Open pipe, fundamental: mark A at both ends and N in the middle. Write l = λ/2 and ν = v/2l.
• Open pipe, second harmonic: mark A, N, A, N, A. Write l = λ and ν = v/l.
• Keep the curves narrow at nodes and widest at antinodes.

Worked example: standing waves and pipes

Activity: resonance column for the speed of sound, and comparing two tuning forks

1. 1
   Set the resonance tube upright. Fill the reservoir so that the water stands near the top of the tube.
2. 2
   Strike a tuning fork of known frequency ν on a rubber pad and hold it just above the mouth of the tube.
3. 3
   Lower the water slowly. At a length L₁ the sound is loudest: the first resonance. The air column is closed by the water and open at the top, so L₁ ≈ λ/4.
4. 4
   Lower the water to about 3L₁ to find the second resonance, L₂ ≈ 3λ/4. Take each reading three times and find the average.
5. 5
   L₂ − L₁ = λ/2, so λ = 2(L₂ − L₁) and v = 2ν(L₂ − L₁). Using the difference cancels the small error from the antinode forming just above the open end (the end correction).
6. 6
   To compare two forks, find ΔL = L₂ − L₁ for each. The speed of sound is the same for both, so ν₁/ν₂ = ΔL₂/ΔL₁.
7. 7
   Sample result: a 512 Hz fork gives L₁ = 16.0 cm and L₂ = 49.5 cm, so v = 2 × 512 × 0.335 m ≈ 343 m s⁻¹. A second fork gives ΔL = 44.7 cm against 33.5 cm for the first, so ν₁/ν₂ ≈ 1.33 and ν₂ ≈ 384 Hz.