Key Idea: When two identical waves travel in opposite directions they superpose into a standing wave — a fixed pattern that does not move along and carries no net energy. It has nodes (points that never move) and antinodes (points that swing the most), and neighbouring nodes sit exactly half a wavelength apart.
Form it
- Two identical waves, opposite directions
- Nodes fixed, antinodes swing most
- Same loop in phase; across a node antiphase
Fit it
- Fixed or closed end: node
- Free or open end: antinode
- String, open pipe 2L/n; closed pipe 4L/n, odd n
Drive it
- Resonance: driving f = natural f₀
- Damping: lower, broader peak
- Critical: back fastest, no swing
A string or air column only resonates at certain frequencies, its harmonics, where a standing wave fits the length. This is examined on both papers. Paper 1A is usually quick: spot how a standing wave is made, compare phase on a standing vs travelling wave, or read the wavelength condition for a string or pipe. Paper 2 is longer structured work: determine a wavelength from a measured node/antinode spacing and then a frequency from the given v = fλ, or calculate the harmonic frequencies of a string or pipe.
Key formulas
Only the wave equation carries the data-booklet badge (look for it). The harmonic wavelength conditions and the node-spacing rule are not printed — they come from how the standing wave fits between the ends, so you memorise them.
- wave speed — how fast the wave travels (m s⁻¹)
- frequency — waves per second (Hz)
- wavelength — length of one full wave (m)
Not in the booklet: they come from fitting the wave between the ends.
- wavelength of the standing wave (m)
- distance between two neighbouring nodes (or two neighbouring antinodes) (m)
- length of the string or pipe (m)
- harmonic number (1, 2, 3 …; for a closed pipe only odd: 1, 3, 5 …)
Standing wave vs travelling wave
| Feature | Travelling wave | Standing wave |
|---|---|---|
| The pattern | Moves along, carrying the shape with it | Stays put — does not move along |
| Energy | Transfers energy from place to place | No net energy transfer along it; energy stays stored in place |
| Amplitude | Every point has the same amplitude | Varies: zero at the nodes, maximum at the antinodes |
| Phase | Shifts smoothly from point to point | Points are ONLY ever in phase (same loop) or antiphase (across a node) |
The boundary types — which condition to use
| Boundary | Ends are… | Wavelength condition | Which harmonics |
|---|---|---|---|
| String, both ends fixed | node — node | λ = 2L ÷ n | all: n = 1, 2, 3, … |
| Pipe open at both ends | antinode — antinode | λ = 2L ÷ n | all: n = 1, 2, 3, … |
| Pipe closed at one end | node — antinode | λ = 4L ÷ n | odd only: n = 1, 3, 5, … |
| String, one end fixed, one end free | node — antinode | λ = 4L ÷ n | odd only: n = 1, 3, 5, … |
| String, both ends free | antinode — antinode | λ = 2L ÷ n | all: n = 1, 2, 3, … |
| Pipe closed at both ends | node — node | λ = 2L ÷ n | all: n = 1, 2, 3, … |
A string or open pipe matches at its ends (both the same type), so a half-wave fits → 2L/n, all harmonics. A closed pipe is lopsided (node one end, antinode the other), so a quarter-wave fits → 4L/n, odd harmonics only — its 'second harmonic' is really n = 3, and the frequencies run 1 : 3 : 5, never 1 : 2 : 3.
Resonance and damping
Driven at its natural frequency f₀, a system resonates: its amplitude is a maximum. More damping gives a lower, broader peak at a slightly lower frequency. Resonance is useful in instruments and radio tuning, and destructive in bridges and buildings.
| Damping | After release | Example |
|---|---|---|
| Light | many oscillations, amplitude dies slowly, period almost unchanged | pendulum in air |
| Critical | back to equilibrium in the shortest time, no oscillation | car suspension |
| Heavy | no oscillation, slow return to equilibrium | pendulum in thick oil |
Worked exam-style questions
A standing wave is set up on a stretched wire vibrating at 50 Hz. The distance from one node to the next node is measured as 0.18 m. (a) Find the wavelength of the wave. (b) Find the speed of the wave on the wire.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A guitar string of length 0.60 m is fixed at both ends. A wave travels along it at 300 m s⁻¹. (a) Find the wavelength of its first harmonic. (b) Find the first-harmonic frequency.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A pipe of length 0.25 m is closed at one end and open at the other. The speed of sound in the air inside is 340 m s⁻¹. Find the frequencies of its first two harmonics.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A bar of chocolate is heated in a microwave oven with the turntable removed. An evenly spaced row of melted spots appears, with neighbouring spots 6.1 cm apart. Microwaves travel at c = 3.0 × 10⁸ m s⁻¹. (a) Find the wavelength of the microwaves. (b) Hence find their frequency.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Important: A pipe closed at one end has only odd harmonics. After the first harmonic the next resonance is the third, at 3f₁.
Quick self-check
Tap each card to reveal the answer.
How is a standing wave made? By two identical waves travelling in opposite directions superposing — often a wave and its reflection off a fixed end. The result is a fixed pattern that does not travel along.
What is a node? An antinode? How far apart are neighbouring nodes? A node never moves (zero displacement); an antinode swings the most. Neighbouring nodes (and neighbouring antinodes) are half a wavelength apart, so λ = 2 × the spacing.
Does a standing wave carry energy along it? How do its points move? No net energy is transferred along it — the energy stays stored in place. Points in the same loop move in phase; points across a node move in antiphase. Never anything in between.
Wavelength condition for a string fixed at both ends? λ = 2L/n for n = 1, 2, 3, … — n half-wavelengths fit into the length L. A pipe open at both ends uses the same condition.
Why does a pipe closed at one end have only odd harmonics? Its ends differ — a node at the closed end and an antinode at the open end — so only odd quarter-wave patterns fit: λ = 4L/n with n = 1, 3, 5, … The next resonance after the first harmonic is n = 3.
How do you turn a wavelength into a frequency? Use the given wave equation v = fλ, rearranged to f = v ÷ λ (v is the speed of sound for a pipe, or the speed of light for microwaves).
What does damping do to a resonance peak? More damping: the peak is lower and broader, and sits at a slightly lower frequency than f₀.
Exam tips
Exam tips
- A standing wave needs two identical waves going opposite directions; nodes never move, antinodes swing the most, and the pattern carries no net energy along it.
- Neighbouring nodes (or antinodes) are HALF a wavelength apart, so λ = 2 × the spacing. This is not in the data booklet — double the measured spacing, never report it as the wavelength itself.
- String / open pipe: λ = 2L/n (all n). Closed pipe: λ = 4L/n (odd n only) — its second resonance is the THIRD harmonic, and the frequencies run 1 : 3 : 5.
- Always finish by turning the wavelength into a frequency with the given v = fλ, rearranged f = v/λ (use the speed of sound for a pipe, the speed of light for microwaves).
- On a standing wave points are only ever in phase or antiphase, unlike a travelling wave whose phase shifts smoothly — a classic Paper 1A comparison.
- Convert every length to metres first (cm = 10⁻² m), and remember a string fixed at both ends in its nth harmonic has n loops and (n + 1) nodes, counting the two end nodes.
- Resonance: driving frequency = natural frequency, amplitude a maximum. Heavier damping lowers and broadens the peak and shifts it slightly below f₀; critical damping returns a system to rest fastest without oscillating.