Standing Waves
Two opposite waves make a pattern that oscillates in place.
Lesson at a glanceShow the key points
- A sin(ωt − kx) + A sin(ωt + kx) = 2A cos kx sin ωt (15.21): each particle oscillates in SHM with amplitude |2A cos kx|.
- Nodes (never moving) where cos kx = 0, i.e. x = (n + ½)λ/2; antinodes (largest amplitude) where |cos kx| = 1.
- Adjacent nodes, or adjacent antinodes, are λ/2 apart; no energy crosses a node.
- Unlike a travelling wave: the disturbance stays put, amplitudes differ from point to point, nodes are always at rest, all particles pass equilibrium together, particles between adjacent nodes move in phase, and energy is not transported.
Pluck a guitar string and it blurs into a shape that shimmers in place rather than running along. That stay-at-home pattern is two travelling waves added together.
Two waves in opposite directions
Take two sine waves of equal amplitude and frequency travelling in opposite directions on a long string (phase difference zero for now):
By superposition, and the identity sin a + sin b = 2 sin ½(a + b) cos ½(a − b):
Nodes and antinodes
The amplitude |2A cos kx| varies from point to point:
- Nodes: where cos kx = 0, i.e. , so . These points never move, though nothing holds them.
- Antinodes: where |cos kx| = 1; the amplitude is greatest, 2A.
Adjacent nodes (and adjacent antinodes) are λ/2 apart. When sin ωt = 1 every particle is at its maximum displacement (those with cos kx > 0 up, the others down); when sin ωt = 0 every particle is passing through its mean position. Because nodes never move, no energy crosses them.
A standing wave is y = (4.0 mm) cos[(π cm⁻¹)x] sin[(31.4 s⁻¹)t]. Find (a) the speed of the travelling waves that make it, (b) the first node and (c) the first antinode with x > 0, and (d) the amplitude at x = 1/3 cm.
Standing versus travelling waves
| Travelling wave | Standing wave | |
|---|---|---|
| Disturbance | moves at a definite speed | stays where it is produced |
| Amplitude | the same for all particles | different for different particles |
| Particles always at rest | none | the nodes |
| All particles at equilibrium together? | never | yes, twice per cycle |
| Phase | changes steadily from particle to particle | the same between adjacent nodes |
| Energy | carried along | stays in each region |
- 1Opposite equal waves: .
- 2Nodes (always still) and antinodes (amplitude 2A) alternate every λ/4; like neighbours are λ/2 apart.
- 3Standing waves don't transport energy.
Check yourself
Keep exploring
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