Superposition and stationary waves
Cancelling sound with sound
- Noise-cancelling headphones play a wave that cancels the noise around you.
- Two waves can add up — or wipe each other out.
- This adding-up of waves is the idea behind this whole topic.
The principle of superposition
- Where waves overlap, the displacement is the sum of the separate displacements.
- Afterwards the waves carry on, unchanged.
When two waves overlap at a point, the resultant displacement is:
That is the principle of superposition — displacements add (as vectors) at each point.
Constructive and destructive
- In phase (crest meets crest) → amplitudes add → constructive.
- Out of phase (crest meets trough) → amplitudes cancel → destructive.
- Since $I \propto A^{2}$, two equal waves in phase give 4× the intensity of one.
Match what happens when the two waves meet.
In phase → a bigger wave; exactly out of phase → they cancel.
Two equal waves meeting in phase give four times the intensity of one wave alone.
The amplitude doubles to $2A$, and $I \propto A^{2}$, so $I \propto (2A)^{2} = 4A^{2}$.
Stationary waves
- Two identical waves travelling in opposite directions overlap to make a stationary wave.
- It happens when a wave reflects back on itself — on a string, or in an air column.
Nodes and antinodes
- A node never moves (the waves always cancel); an antinode has the biggest swing.
- Neighbouring nodes are $\dfrac{\lambda}{2}$ apart; a node and the next antinode are $\dfrac{\lambda}{4}$ apart.
A point on a stationary wave that is always at zero displacement is called a ____.
At a node the two waves always cancel. Halfway between nodes is an antinode (biggest swing).
Neighbouring nodes on a stationary wave are $0.30\ \text{m}$ apart. What is the wavelength?
Neighbouring nodes are $\dfrac{\lambda}{2}$ apart, so $\lambda = 2 \times 0.30 = 0.60\ \text{m}$.
Stationary vs progressive
- A stationary wave does not carry energy along, and its pattern stays put.
- A progressive wave moves along and carries energy with it.
A stationary wave transfers energy along its length.
No — the pattern stays put and no energy travels along it. A progressive wave is the one that carries energy.
Pipes and strings
- A closed pipe end is a node; an open end is an antinode.
- Closed-pipe fundamental: $L = \dfrac{\lambda}{4}$. Open both ends: $L = \dfrac{\lambda}{2}$. Measure node spacing → $\lambda$, then $v = f\lambda$.
In a pipe closed at one end, the fundamental fits a length of:
A node at the closed end and an antinode at the open end is a quarter of a wave, so $L = \dfrac{\lambda}{4}$.
You've got it
- superposition: overlapping displacements add (constructive) or cancel (destructive)
- a stationary wave has fixed nodes and antinodes, $\dfrac{\lambda}{2}$ apart
- it carries no energy along — unlike a progressive wave