OCR A Level Physics A: Stationary waves, harmonics and resonance tubes (H556) – Revision Notes

These revision notes condense section 4.4.4 (Stationary waves) of the OCR Advanced GCE in Physics A specification (H556), Version 3.0 (March 2026), for first assessment in 2017. Read the study guide for full explanations, then try the practice questions.

Course pages: OCR A Level Physics hub | checklist | free diagnostics.

Key terms

Term	Meaning
Stationary (standing) wave (4.4.4(a))	Pattern formed when two progressive waves of the same frequency travelling in opposite directions superpose; it does not move along and transfers no energy along its length
Node (4.4.4(d))	Point of zero amplitude; the two waves are always in antiphase there
Antinode (4.4.4(d))	Point of maximum amplitude; the two waves are always in phase there
Fundamental mode (4.4.4(g))	Lowest-frequency stationary wave for the system, also called the 1st harmonic
Harmonics (4.4.4(g))	The fundamental frequency and the higher frequencies at which the system also gives stationary waves
Resonance	Large-amplitude oscillation when the driving frequency matches a frequency at which a stationary wave fits the system

Key relationships

Idea	Relationship
Wave speed	v = f lambda
Adjacent nodes (or adjacent antinodes), 4.4.4(f)	separation = lambda/2
Node to next antinode	lambda/4
Fixed string or tube open at both ends	L = n lambda/2, f = n v / 2L, every harmonic present
Tube closed at one end	L = (2n - 1) lambda/4, f = (2n - 1) v / 4L, odd harmonics only
Resonance tube	lambda = 2 (L2 - L1), v = f lambda

Stationary or progressive: quick comparison (4.4.4(c))

	Progressive	Stationary
Energy	Transferred along the wave	Not transferred along the wave
Amplitude	Same everywhere	Zero at nodes, maximum at antinodes
Phase	Changes along the wave	Same inside a loop, antiphase across a node
Profile	Moves	Fixed in place

Demonstrations (4.4.4(a))

• Microwaves: transmitter facing a metal plate; probe receiver moved between them. Maxima are antinodes and minima are nodes.
• String: vibration generator at one end, mass over a pulley at the other. Adjust the signal-generator frequency until large loops appear.
• Air column: loudspeaker or tuning fork above a tube. Loud sound means resonance.

Reading a graph of a stationary wave (4.4.4(b))

• Displacement against position at one instant is a sine curve. A quarter of a period later it is flat.
• The envelope shows the largest displacement at each position.
• Antinode amplitude = sum of the two progressive amplitudes.
• Same loop: in phase. Neighbouring loops: antiphase.
• Displacement against time at one position: same period everywhere, different amplitude.

Patterns (4.4.4(e)(i))

Letters: N = node, A = antinode.

System	Fundamental	Next harmonics
String fixed at both ends	N A N, L = lambda/2	2nd: N A N A N; 3rd: three loops
Tube open at both ends	A N A, L = lambda/2	2nd: A N A N A; same frequency series as string
Tube closed at one end	N A (closed end node, open end antinode), L = lambda/4	3rd: N A N A, L = 3 lambda/4; 5th: L = 5 lambda/4

Method in steps: resonance tube (4.4.4(e)(ii), PAG5)

1. Set up a transparent tube in a tall cylinder of water with a tuning fork of known frequency above it.
2. Lower the tube while the fork sounds. Record the length L1 of the first loud sound.
3. Continue to the second loud sound at L2.
4. Calculate lambda = 2 (L2 - L1).
5. Find v = f lambda. Repeat with other forks and average.

Why use the difference: the antinode is slightly above the tube mouth, and that offset appears in both readings, so it cancels.

Worked reminders

A. A tube open at both ends is 0.80 m long (speed of sound 340 m/s). f1 = v / 2L = 340 / 1.60 = 213 Hz (212.5). Closed at one end: f1 = 340 / 3.2 = 106 Hz, then 319 Hz for the 3rd harmonic.

B. A string 1.20 m long vibrates in its 4th harmonic at 300 Hz. lambda = 2L / 4 = 0.600 m. v = 300 x 0.600 = 180 m/s. The fundamental is 300 / 4 = 75 Hz.

C. Microwave minima are 1.6 cm apart. lambda = 3.2 cm = 0.032 m. f = 3.00 x 10^8 / 0.032 = 9.4 x 10^9 Hz (9.375 x 10^9).

D. A fixed string of length L: harmonics and their features.

Harmonic	Loops	Nodes (with ends)	Wavelength	Frequency
1st	1	2	2L	f1
2nd	2	3	L	2 f1
3rd	3	4	2L/3	3 f1
4th	4	5	L/2	4 f1

E. Graph of f against 1/L for a fixed string: the line passes through the origin and its gradient is v / 2. If the gradient is 36 m/s, then v = 72 m/s.

Model answer: how a stationary wave forms

Use this skeleton for any “explain” question on a string or tube.

1. A progressive wave travels to a boundary and reflects.
2. The reflected wave has the same frequency and speed but travels in the opposite direction.
3. The incident and reflected waves superpose.
4. At nodes they are always in antiphase, so the resultant displacement is always zero.
5. At antinodes they are always in phase, so the amplitude is a maximum.
6. Only frequencies for which the pattern fits the end conditions give a steady pattern, and these are the harmonics.

F. Closed tube, harmonics up to the 5th: frequencies f1, 3 f1 and 5 f1. For f1 = 150 Hz these are 150 Hz, 450 Hz and 750 Hz. The gap between consecutive resonances is always 2 f1 = 300 Hz.

Must-know distinctions

• Wavelength versus node spacing: nodes are lambda/2 apart, never lambda.
• Closed versus open tube: a closed tube needs a node at the closed end and has only odd harmonics. Opening the other end doubles the fundamental frequency for the same length.
• Harmonic number versus overtone count: the 3rd harmonic of a closed tube is the second resonance. Use the harmonic number.
• Node versus rest: particles at nodes do not move; particles elsewhere do, all at the same frequency.
• Number of loops: on a string the nth harmonic has n loops, n + 1 nodes (including the ends).

Quick self-test

1. State the condition for a stationary wave to form on a string.
2. What is the amplitude at a node?
3. Two adjacent nodes in a microwave experiment are 1.4 cm apart. What is lambda?
4. A string of length 0.50 m is fixed at both ends. What is the wavelength of its fundamental mode?
5. How many nodes (including the ends) does the 4th harmonic have on a fixed string?
6. A closed tube has a fundamental of 150 Hz. State the next two frequencies at which it resonates.
7. Write down one difference between the way energy moves in progressive and stationary waves.
8. A stationary wave has an antinode amplitude of 8.0 mm. What is the amplitude of each progressive wave?
9. In a resonance tube experiment with a 600 Hz fork, resonances occur at 13.6 cm and 41.9 cm. Find v.
10. Why does the experiment use L2 - L1 and not L1 alone?

Answers

1. Superposition of two progressive waves that share a frequency but move in opposite directions, usually an incident wave and its reflection at a fixed end.
2. Zero.
3. lambda = 2 x 1.4 = 2.8 cm.
4. lambda = 2L = 1.0 m.
5. n + 1 = 5.
6. 3 x 150 = 450 Hz and 5 x 150 = 750 Hz.
7. Energy is transferred along a progressive wave but not along a stationary wave.
8. 4.0 mm.
9. L2 - L1 = 0.283 m = lambda/2, so lambda = 0.566 m and v = 600 x 0.566 = 340 m/s (339.6).
10. The offset between the antinode and the tube mouth is the same in both readings, so it cancels in the difference.

Where marks are usually lost

• Calling the node spacing lambda rather than lambda/2.
• Counting minima rather than gaps between them.
• Writing “the wave stops” or “no energy” without explaining the cancellation in terms of antiphase superposition.
• Describing the pattern without saying where the nodes and antinodes are, especially the end conditions of a tube.
• Including even harmonics for a closed tube.
• Forgetting that the second resonance in a resonance tube is the 3rd harmonic.
• Quoting a final answer without a unit, or with too many significant figures for the data.
• Describing a stationary wave as having a single phase throughout, instead of noting the change of 180 degrees across a node.
• Giving a vague practical improvement such as “be more careful” rather than repeat readings, several forks and a mean.

Try the practice questions next.

Official syllabus

OCR Advanced GCE in Physics A (H556), Version 3.0 (March 2026), for first assessment in 2017, published by OCR. Section 4.4.4, Stationary waves.