Superposition, reflection and standing waves

Principle of superposition: when two or more waves pass through the same place, each moves on as if the others were not there, and the displacement at any point is the sum of the separate displacements. If one wave lifts a particle 3 mm and another lifts it 2 mm, it rises 5 mm; if the second pushes it down 2 mm instead, it rises only 1 mm.

• Waves in phase reinforce (constructive interference); waves out of phase by π cancel (destructive).
• Two waves of the same frequency going the same way interfere; going opposite ways, they make a stationary wave; with slightly different frequencies, they make beats (next lesson).

When a wave meets a boundary it reflects:

• At a rigid end (a string tied to a wall), a crest comes back as a trough: a phase change of π.
• At a free end (a string ending in a ring that slides on a rod), a crest comes back as a crest: no phase change.
• Going from a light string into a heavy one is like a rigid end; going from a heavy into a light one is like a free end.

A wave and its own reflection travel in opposite directions and add to give a standing (stationary) wave, y = 2a sin kx cos ωt. Nothing moves along; the pattern stays in place:

• Nodes: points that never move, λ/2 apart. Antinodes: points of largest swing, 2a, halfway between the nodes. A node and the next antinode are λ/4 apart.
• All the particles in one loop (between two nodes) move in phase; the next loop moves the opposite way.
• No energy flows past a node, so the energy stays trapped in the loops.

A string fixed at both ends must have a node at each end, so a whole number of loops fits: L = nλ/2, giving ν_n = nv/2L with n = 1, 2, 3 ... The lowest, n = 1, is the fundamental (first harmonic); the rest are the 2nd, 3rd ... harmonics. The 0.65 m guitar string with v = 158 m/s has ν₁ = 158/1.3 ≈ 122 Hz, then 243 Hz, 365 Hz.

Air in a pipe works the same way, with an antinode at an open end and a node at a closed end:

Example, with v = 340 m/s and a 25 cm pipe: closed at one end, ν = 340/(4 × 0.25) = 340 Hz, then 1020 Hz and 1700 Hz; open at both ends, 680, 1360 and 2040 Hz. Note that where the air does not move (a displacement node), the pressure changes most (a pressure antinode).