Standing Waves

Standing Waves

nodes and antinodes

Two waves travelling opposite ways add to a pattern that stays in place, with motionless nodes and big-swinging antinodes.

Example

Two waves travelling opposite ways add to a pattern that stays in place, with motionless nodes and big-swinging antinodes. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Two waves going opposite ways

Send identical waves along a string in opposite directions and they overlap everywhere. At some points the two always cancel; at others they always add. The combined pattern no longer travels — it just swings up and down in place. That is a standing wave.

opposite travelling waves  →  a pattern that stands\text{opposite travelling waves} \;\to\; \text{a pattern that stands}
A standing-wave patternA standing wave on a string with fixed points (open rings) that never move and bulges between them.

Nodes stay still; antinodes swing most

The points that never move are nodes; they sit half a wavelength apart. Halfway between each pair, the antinodes swing with the biggest amplitude. Because identical opposite waves cancel exactly at the nodes, the nodes are pinned in place.

node spacing=λ2\text{node spacing} = \tfrac{\lambda}{2}
Nodes (open rings) and antinodes (bulges)The standing wave with its motionless nodes marked as open rings and the swinging antinodes bulging between them.

Higher harmonics put nodes closer together

Each harmonic still has nodes half a wavelength apart. Higher harmonics have shorter wavelengths, so their nodes are packed closer together along the same string.

nλnnode spacing112 m6 m26 m3 m34 m2 m\begin{array}{c|c|c}n & \lambda_n & \text{node spacing} \\ \hline 1 & 12\ \text{m} & 6\ \text{m} \\ 2 & 6\ \text{m} & 3\ \text{m} \\ 3 & 4\ \text{m} & 2\ \text{m}\end{array}
waves Nodes sit exactly a half wavelength apart, marked as open rings where the waves always cancel.