Stationary Waves

Stationary wave — formed when two progressive waves of the same frequency and similar amplitude travel in opposite directions and superpose. In practice this is usually a wave and its own reflection.

No net energy transfer — energy is stored in the oscillation rather than carried along, which is the fundamental difference from a progressive wave.

Nodes — points of permanently zero displacement, where the two waves always arrive in antiphase and cancel.

Antinodes — points of maximum displacement, where the waves always arrive in phase.

Spacing — adjacent nodes are half a wavelength apart, and so are adjacent antinodes. A node and its neighbouring antinode are a quarter of a wavelength apart. Forgetting the factor of a half here is one of the most common errors in this topic.

Comparison with a progressive wave

Phase behaviour — every point between two adjacent nodes moves in phase, reaching its maximum displacement at the same instant. Points on opposite sides of a node are in antiphase. This is quite unlike a progressive wave, where phase changes steadily with distance.

Harmonics on a string fixed at both ends — the ends must be nodes, which allows only certain wavelengths.

First harmonic — one antinode, and the string holds half a wavelength, so λ = 2L and

f₁ = v / 2L

Second harmonic — two antinodes, λ = L, and the frequency is 2f₁. In general the nth harmonic has n antinodes and frequency nf₁.

Example: a string of length 0.80 m carrying waves at 200 m s⁻¹ has a first harmonic of 200 ÷ (2 × 0.80) = 125 Hz. Its next harmonics are 250 Hz, 375 Hz and so on.

Air columns — a pipe closed at one end has a node at the closed end and an antinode at the open end, so it holds a quarter of a wavelength in its first harmonic. An open pipe has antinodes at both ends and holds half a wavelength, like a string.