Stationary Waves

CIE A-Level Physics· Unit 8: Superposition, Topic 2· 15 min read

1. Formation and Key Features★★☆☆☆⏱ 4 min

📘 Definition

Stationary (Standing) Wave

A wave that forms when two identical progressive waves of the same frequency and amplitude travel in opposite directions along the same medium, superpose, and produce no net propagation of energy

Example:

Vibrations on a guitar string held fixed at both ends

When the two waves superpose, points of permanent destructive interference are called nodes (zero amplitude), and points of permanent constructive interference are called antinodes (maximum amplitude). All points between two adjacent nodes vibrate in phase with each other.

📐 Worked Example

A stationary wave forms on a string of length 1.2 m fixed at both ends, with 3 antinodes between the fixed ends. How many nodes are present in total on the string?

1
Fixed ends of a string are always nodes, so we start with 2 nodes at each end.
2
Between 3 antinodes, there are 2 additional internal nodes, between each pair of adjacent antinodes.
3
Total nodes = 2 (ends) + 2 (internal) = 4 nodes total

Exam tip:

Always remember: fixed ends of strings and closed ends of air tubes are always nodes; open ends of air tubes are always antinodes.

2. Wavelength and Harmonic Calculations★★★☆☆⏱ 5 min

A core relationship for all stationary waves is that the distance between two consecutive nodes (or two consecutive antinodes) is always $\frac{λ}{2}$ . The distance between a node and its adjacent antinode is $\frac{λ}{4}$ . For a string fixed at both ends, the length of the string $L$ relates to wavelength by:

L = \frac{n\lambda}{2}, \quad n = 1, 2, 3,...

$n$ is the harmonic number, where $n = 1$ is the first (fundamental) harmonic.

📐 Worked Example

A string of length 0.8 m fixed at both ends vibrates in its 3rd harmonic. Calculate the wavelength of the stationary wave.

1
For a string fixed at both ends, use the relation: $L = \frac{nλ}{2}$
2
Substitute $L = 0.8$ m and $n = 3$ :
3
$\lambda = \frac{2L}{n} = \frac{2 \times 0.8}{3} = \frac{1.6}{3} \approx 0.53 \, \text{m}$

✓ Quick check

Test your understanding:

What is the distance between a node and the next adjacent antinode?
- $λ /4$
- $λ /2$
- $λ$
- $3 λ /4$

3. Stationary Waves in Air Columns★★★☆☆⏱ 5 min

✓ Calculator OK

Sound waves reflect off the ends of air tubes to form stationary waves. There are two common cases: open-ended tubes (open at both ends) and closed-ended tubes (closed at one end, open at the other). The end conditions and wavelength relations are different for each case.

Open at both ends: Antinodes at both ends, $L = \frac{nλ}{2}$ , $n = 1, 2, 3...$ (all integers)
Closed at one end: Node at closed end, antinode at open end, $L = \frac{nλ}{4}$ , $n = 1, 3, 5...$ (only odd harmonics)

📐 Worked Example

A 25 cm long closed-end air tube produces its 1st harmonic. The speed of sound is 340 m/s. Calculate the frequency of the harmonic.

1
Convert length to meters: $25 cm = 0.25 m$
2
1st harmonic for closed-end tube: $n = 1$ , so $L = \frac{λ}{4}$
3
Rearrange to get $λ = 4 L = 4 \times 0.25 = 1 m$
4
Use wave equation $v = f λ$ , rearrange for $f$ :
5
$f = \frac{v}{\lambda} = \frac{340}{1} = 340 \, \text{Hz}$

Exam tip:

Always confirm if the air column is closed at one end or open at both before starting calculations, this is a common source of lost marks.

4. Stationary vs Progressive Waves★★☆☆☆⏱ 3 min

Property	Stationary Wave	Progressive Wave
Net energy transfer	No net transfer	Transfers energy
Amplitude	Varies from 0 at nodes to max at antinodes	Same for all points
Phase	All points between 2 nodes are in phase	All points have different phase
Wavelength relation	$λ = 2 \times$ inter-nodal distance	$λ =$ distance between consecutive in-phase points

📐 Worked Example

State two differences between a stationary wave and a progressive wave of the same frequency, and state the net speed of energy transfer for a stationary wave.

1
Two valid differences are: (1) Stationary waves have no net energy transfer, while progressive waves transfer energy along the medium. (2) Amplitude varies along a stationary wave, but is constant for a progressive wave.
2
The net speed of energy transfer for a stationary wave is 0 m/s, since energy is trapped in the oscillations of each segment.

5. Common Pitfalls

Wrong move:

Assuming open ends of air columns are nodes

Why:

Confusion with fixed ends of strings, which are always nodes

Correct move:

Open ends of air columns allow maximum air movement, so they are always antinodes

Wrong move:

Using $L = nλ /2$ for the fundamental harmonic of a closed-end air column

Why:

Ignoring the different boundary condition for a closed end

Correct move:

Use $L = nλ /4$ for one-closed-end air columns, with only odd values of n

Wrong move:

Taking inter-nodal distance equal to a full wavelength $λ$

Why:

Mixing up wavelength definitions for progressive and stationary waves

Correct move:

Distance between two consecutive nodes is $λ /2$ , so $λ = 2 \times$ inter-nodal distance

Wrong move:

Claiming stationary waves have no energy at all

Why:

Misinterpretation of the 'no net transfer' rule

Correct move:

Energy is stored in the oscillations of the wave; there is just no net propagation of energy along the medium

Wrong move:

Stating all points on a stationary wave have the same amplitude

Why:

Confusing the property of stationary waves with progressive waves

Correct move:

Amplitude ranges from zero at nodes to maximum at antinodes along a stationary wave

6. Quick Reference Cheatsheet

Medium Type	End Conditions	Wavelength Relation	Allowed Harmonics
String fixed both ends	Nodes at both ends	$L = \frac{nλ}{2}$	$n = 1, 2, 3...$
Air column open both ends	Antinodes at both ends	$L = \frac{nλ}{2}$	$n = 1, 2, 3...$
Air column one closed end	Node at closed, antinode at open	$L = \frac{nλ}{4}$	$n = 1, 3, 5...$ (only odd)
Consecutive nodes/antinodes		distance = $\frac{λ}{2}$
Node to adjacent antinode		distance = $\frac{λ}{4}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 12
Count nodes on a stationary wave
2023 · 22
Calculate frequency for closed air column
2021 · 11
Compare stationary and progressive waves

Going deeper

syllabusCIE 9702 A-Level Physics Syllabus

What's Next

Stationary waves are a core application of the superposition principle, and underpin the behavior of all musical instruments, resonance, and acoustic systems. They are one of the most commonly tested topics in CIE A-Level Physics, appearing in both multiple choice and structured questions, so mastering boundary conditions and harmonic relations is critical for full marks. Next, we will extend the superposition principle to two dimensions to cover interference and diffraction, which build directly on the wave superposition concepts you learned here.