How a stationary wave forms

Send a pulse along a rope and let it reflect from the far end. Now the rope carries two waves: the one you sent, and its reflection, travelling in opposite directions.

The two waves have the same frequency and speed, and roughly the same amplitude. Where they overlap, they superpose.

The result looks nothing like a wave sliding along the rope. Instead the pattern stays in one place and oscillates up and down. We call it a stationary wave (or standing wave).

Anything that reflects a wave can make one: a string, the air in a pipe, microwaves bouncing off a metal plate.

A stationary wave is just superposition, done at every point at once. Here is the recipe:

1. Draw wave 1 at one instant.
2. Draw wave 2 at the same instant, travelling the other way.
3. Add their displacements point by point — that is the resultant.
4. Repeat a quarter of a period later, then a half period later.

You find two special kinds of point:

• Points where the waves are always in antiphase: displacements always cancel, so the point never moves. These are nodes.
• Points where the waves are always in phase: displacements always reinforce, so the point oscillates with maximum amplitude. These are antinodes.

The diagram shows the result. At $t = 0$ the pattern has its full height. At $t = \dfrac{T}{4}$ the two waves are in antiphase everywhere and the string is flat. At $t = \dfrac{T}{2}$ the pattern has flipped over.

A node is a point of zero displacement at all times. A antinode is a point of maximum amplitude.

The spacing follows from the pattern:

• Neighbouring nodes are $\dfrac{\lambda}{2}$ apart.
• Neighbouring antinodes are also $\dfrac{\lambda}{2}$ apart.
• A node and the nearest antinode are $\dfrac{\lambda}{4}$ apart.

Between a node and an antinode, the medium oscillates with an amplitude somewhere between zero and the maximum — but every point oscillates with the same frequency.

Unlike a progressive wave, a stationary wave transfers no net energy along the medium. Energy sloshes back and forth between kinetic energy of the moving string and elastic potential energy of the stretched string.