Stationary waves on strings

Two waves, one string

Send a single pulse along a rope tied to a wall and it reflects and comes back. Drive the rope continuously instead, and the reflected wave has to travel back through the incoming one. The rope now carries two waves at the same time. They have the same frequency and the same speed, their amplitudes are similar, and they move in opposite directions.

Wherever two waves overlap, the displacement of the rope is the sum of the displacements each wave would produce on its own. That sentence is the principle of superposition. Two waves of the same frequency and similar amplitude travelling opposite ways sum to a pattern that stays where it is, a stationary wave, also called a standing wave.

On a driven string a second idea joins the first. Only at the frequencies whose pattern fits the fixed ends do the repeated reflections reinforce cycle after cycle, building the pattern to a size worth seeing. That build-up is resonance, the driven string answering at one of its natural frequencies, and it is a separate idea from the superposition that shapes the pattern.

Describe the drawing

Two identical waves, one travelling each way (faint), and their point-by-point sum (cyan). The sum never travels: it pumps in place, pinned at the amber nodes, which never move; between them the string pumps within the envelope. Every three periods the travelling pair fades and the dashed envelope appears, the second figure this animation replaces.

Be careful with the name. The rope itself is still moving, in places more violently than either travelling wave would move it alone. What has stopped is the pattern. The crests no longer travel along the rope. Instead, each point of the rope oscillates about its rest position with a fixed amplitude of its own.

Nodes and antinodes

At certain points the two waves always arrive in antiphase, so they cancel and the rope there never moves at all. These points are called nodes. Midway between them the waves always arrive in phase and reinforce, so the rope swings with the largest amplitude found anywhere on the pattern. These points are antinodes.

That cancellation is total only when the two waves carry equal amplitudes, which is the condition for a pure stationary wave and the case every exam question means. A real support absorbs a little, so the returning wave comes back slightly weaker and the pattern is a partial one, its minima shallow rather than dead still. The spacing is unaffected: nodes and antinodes sit where they always sat, and only the depth of the minima changes.

Two distances are worth learning. Adjacent nodes are λ/2$\lambda/2$λ/2 apart, because one complete wavelength contains two loops of the pattern, not one. The distance from a node to the nearest antinode is λ/4$\lambda/4$λ/4.

A stationary wave also transfers no net energy along the rope. Each section keeps its own energy, which passes back and forth between kinetic and potential forms as the rope oscillates. Contrast this with a progressive wave, which carries energy from one place to another.

Harmonics on a stretched string

A string fixed at both ends must have a node at each end, because a fixed point cannot move. That single condition restricts the wavelengths the string can support: a pattern only fits if it places a whole number of loops between the two ends.

Each loop is half a wavelength, so a pattern of n loops on a string of length L has

The simplest pattern, n = 1, is the first harmonic. Many books call it the fundamental, and the two names mean the same pattern. It has the longest wavelength the string allows, 2L, and therefore the lowest frequency. The string fixes the wave speed, so v = fλ$v = f\lambda$v = fλ forces the harmonic frequencies into whole-number multiples of the first, f_n = nf₁$f_n = nf_1$f_n = nf_1.

Two things fix that speed, and neither of them is the oscillator. A transverse wave travels along a stretched string at

where T is the tension in newtons and μ$\mu$μ is the mass per unit length in kg m⁻¹. Tighten the string and the wave runs faster; hang the same tension on a heavier string and it runs slower. Be clear about which speed this is. It is the speed of the travelling waves the pattern is built from, moving along the string, and not the speed of any point of the string, which is bobbing across it and passes through zero at every node. Edexcel prints this form and sets questions straight on it; the other boards print only the frequency it leads to.

That frequency is one substitution away. The first harmonic has λ₁ = 2L$\lambda_1 = 2L$λ_1 = 2L, so f₁ = v/2L$f_1 = v/2L$f_1 = v/2L, and putting the speed in gives

Raising the tension or using a lighter string raises the frequency. A guitarist works both handles, tightening the tuning pegs to change T and reaching for the thicker, heavier strings to play low notes.

Phase along a stationary wave

On a progressive wave, every point oscillates with the same amplitude, and there is a steady phase difference between one point and the next. A stationary wave differs on both counts. The amplitude depends on position, running from zero at a node to a maximum at an antinode, and only two phase relationships exist anywhere on the pattern.

All the points between one pair of adjacent nodes move in phase. They reach their maximum displacements at the same instant, though each has its own amplitude. Points on opposite sides of a node move in antiphase, a phase difference of π$\pi$π rad or 180°. Cross two nodes and the motion is back in phase.