Stationary Waves, Nodes and Antinodes: MDCAT Physics notes
Stationary Waves, Nodes and Antinodes notes for MDCAT: how standing waves form, node–antinode spacing, energy and contrast with progressive waves.
Formation of stationary waves
A stationary (standing) wave is formed by the superposition (interference) of two progressive waves of the same frequency, same amplitude and same speed travelling in opposite directions along the same line. Usually one is the incident wave and the other its reflection from a boundary, so reflection is part of the set-up, but the pattern itself is produced by interference.
Waves of unequal amplitude do not give a true stationary wave: complete cancellation (true nodes) is impossible.
Stationary waves can be set up in strings and ropes, in air columns (sound in pipes), on water surfaces and with electromagnetic waves. There is no class of these waves in which they cannot be produced.
Nodes and antinodes
- Node (N): point of zero amplitude; always at rest, so its displacement, velocity and acceleration are all zero. Nodes do not move along the string.
- Antinode (A): point of maximum amplitude (twice the amplitude of each component wave).
- At a fixed end of a string there is always a node; at an open end of a pipe there is an antinode.
| Separation | Distance |
|---|---|
| Node to next node | $\lambda/2$ |
| Antinode to next antinode | $\lambda/2$ |
| Node to next antinode | $\lambda/4$ |
One "loop" (segment between two nodes) has length $\lambda/2$.
Example: if adjacent nodes are 30 cm apart, then $\lambda = 60\ \text{cm}$ and each antinode lies 15 cm from the nearest node. If the wave speed is $120\ \text{m s}^{-1}$, the frequency is $120/0.6 = 200\ \text{Hz}$.
Properties
- The waveform does not travel: there is no net transfer of energy along the wave. Energy is stored, shuttling between kinetic and potential forms within each loop.
- All particles between two adjacent nodes vibrate in phase; particles in neighbouring loops vibrate in antiphase.
- Amplitude varies from zero (node) to maximum (antinode) along the medium.
Progressive vs stationary
| Feature | Progressive | Stationary |
|---|---|---|
| Energy | Transferred | Not transferred |
| Amplitude | Same for all particles | Zero at nodes, maximum at antinodes |
| Phase | Changes continuously with distance | Same within a loop |
| Nodes | None | Present |
An ordinary progressive sound wave travelling through open air has no nodes; nodes appear only when sound forms a stationary wave, as in organ pipes.
Key formulas
- N–N or A–A: $\lambda/2$
- N–A: $\lambda/4$
- Antinode amplitude $= 2a$
Common MDCAT traps
- Consecutive nodes are $\lambda/2$ apart, not $\lambda$.
- Node to adjacent antinode is $\lambda/4$.
- Unequal amplitudes do not produce stationary waves.
- Zero acceleration is at nodes, not antinodes.
- Stationary waves arise by interference of oppositely travelling waves.
Quick revision
- Fixed end of a string: node.
- No energy flows along a stationary wave.
- Particles in one loop vibrate in phase.
- Progressive waves have no nodes.