AQA A-Level Physics: Waves (7408) – Revision Notes

These notes condense section 3.3 Waves of the AQA AS and A-level Physics specification (7407/7408, version 1.4, AS and A-level exams June 2016 onwards): 3.3.1 Progressive and stationary waves and 3.3.2 Refraction, diffraction and interference. It is AS and A-level content, assessed at A-level in Paper 1 and assumed in Paper 2; it is not one of the Paper 3 Section B options (sections 3.9 to 3.13). For full explanations and worked examples, use the Waves study guide.

Test yourself afterwards with the Waves practice questions. The course hub and printable checklist list every topic.

Equations for this topic

Equation	Symbols	Use
c = fλ	c speed, f frequency, λ wavelength	Any progressive wave
f = 1/T	T period	Period from frequency
phase difference = 2πx/λ rad	x separation along the wave	Two points on one wave
f = (1/2l) √(T/μ)	l length, T tension, μ mass per unit length	First harmonic on a string
w = λD/s	w fringe spacing, D slit–screen distance, s slit separation	Young’s double slits
d sin θ = nλ	d slit spacing, n order	Grating at normal incidence
n = c/c_s	c_s speed in the substance	Refractive index
n₁ sin θ₁ = n₂ sin θ₂	angles from the normal	Snell’s law
sin θc = n₂/n₁	n₁ > n₂	Critical angle

Recall: n for air ≈ 1. All EM waves travel at the same speed in a vacuum, 3.00 × 10⁸ m s⁻¹.

3.3.1.1 Progressive waves – definitions

• Amplitude: maximum displacement from equilibrium.
• Wavelength: distance between adjacent points in phase.
• Frequency: oscillations per second. Period: time for one oscillation.
• Phase difference: how far one point is ahead of another in its cycle. Give it as a fraction of a cycle, in degrees, or in radians: ¼ cycle = 90° = π/2 rad; ½ cycle = 180° = π rad.

The particles oscillate about fixed positions; energy, not matter, moves along.

3.3.1.2 Longitudinal vs transverse

	Transverse	Longitudinal
Oscillation vs energy direction	Perpendicular	Parallel
Examples	Waves on a string, all EM waves	Sound
Can be polarised?	Yes	No

Polarisation confines oscillations to one plane. It is evidence that a wave is transverse.

• Polaroid filters: crossed at 90° they block light; sunglasses reduce glare from reflected light.
• Aerials: a receiving aerial must line up with the plane of polarisation of the transmitted signal.
• Malus’s law is not required.

3.3.1.3 Stationary waves

Formation (method in steps)

1. Two progressive waves of the same frequency travel in opposite directions (often a wave and its reflection).
2. They superpose: resultant displacement = sum of the two displacements.
3. Where they are always in antiphase: node, zero amplitude.
4. Where they are always in phase: antinode, maximum amplitude.
5. Draw the two waves a quarter period apart and add them to show this graphically.

Must-know distinctions: progressive vs stationary

Feature	Progressive	Stationary
Energy	Transferred	Stored, not transferred
Amplitude	Same everywhere (no losses)	Zero at nodes, maximum at antinodes
Phase	Changes continuously along the wave	Same within a loop; antiphase in adjacent loops
Spacing	Crest to crest = λ	Node to node = λ/2

Harmonics on a string fixed at both ends

Harmonic	Loops	Wavelength	Frequency
1st	1	2l	f₁
2nd	2	l	2f₁
nth	n	2l/n	nf₁

AQA uses “first harmonic”, not “fundamental”, and does not use “overtone”.

Proportionalities from f = (1/2l) √(T/μ): f ∝ 1/l, f ∝ √T, f ∝ 1/√μ. Quadruple T and f doubles; quadruple μ and f halves.

Microwaves and sound: a reflector (metal plate, hard wall) sets up a stationary wave; a probe or microphone finds nodes λ/2 apart.

Worked reminder: a probe moves 6.0 cm from the 1st node to the 4th node. That is 3 spacings, so 3 × λ/2 = 6.0 cm and λ = 4.0 cm. Then f = c/λ = 3.00 × 10⁸ / 0.040 = 7.5 × 10⁹ Hz. Count the gaps, not the nodes.

Required practical 1: frequency of stationary waves on a string versus length, tension (hanging masses, T = mg) and mass per unit length. Change one variable at a time; plot f against 1/l or √T.

3.3.2.1 Interference – method in steps

1. Check the sources are coherent: same frequency, constant phase difference.
2. Find the path difference to the point.
3. Path difference = nλ → constructive (maximum). Path difference = (n + ½)λ → destructive (minimum), for sources in phase.

Young’s double slits: w = λD/s. Measure across several fringes and divide.

White-light fringes: central white fringe; others are coloured, blue nearer the centre.

Laser safety: do not look along the beam or at reflections; remove shiny surfaces; keep the beam below eye level.

Sound and microwaves: two loudspeakers on one signal generator, or microwaves through two gaps, give maxima and minima explained by path difference.

History of ideas: Newton’s corpuscular theory was long preferred to Huygens’ wave theory; Young’s interference fringes supported the wave theory; the photoelectric effect later showed a particle nature (wave–particle duality).

Required practical 2: interference with Young’s slits and with a diffraction grating to find λ.

3.3.2.2 Diffraction

Single slit: wide, bright central maximum (twice the width of the others), dimmer side maxima. With white light: white centre, side maxima are spectra.

Central maximum wider if λ increases or slit width decreases.

Grating: path difference between adjacent slits = d sin θ, so d sin θ = nλ for maxima. To derive it, draw the right-angled triangle between two adjacent slits and the direction θ, and state that a maximum needs the path difference to be a whole number of wavelengths. d = 1/(lines per metre). Highest order = largest whole number ≤ d/λ (round down). Uses: spectra of stars and gases, measuring wavelengths.

3.3.2.3 Refraction and fibres

• Light bends towards the normal entering a medium of higher n, and slows down.
• TIR needs n₁ > n₂ and angle of incidence > θc.
• Cladding: lower n than the core (allows TIR), protects the core from scratches, stops crossover between fibres.
• Absorption: amplitude falls along the fibre.
• Pulse broadening: pulses lengthen and may overlap.
  • Material dispersion: different wavelengths, different speeds. Fix: monochromatic source.
  • Modal dispersion: different paths, different distances. Fix: very narrow (single-mode) core.
• Step-index fibres only.

Quick self-test

1. A wave has period 0.020 s. What is its frequency?
2. Find the wavelength of a 98 MHz radio wave in a vacuum.
3. Two points on a wave are λ/3 apart. Give the phase difference in degrees and radians.
4. Which of these can be polarised: sound, light, a wave on a string?
5. A stationary wave has wavelength 0.60 m. How far apart are adjacent nodes?
6. A string’s first harmonic is 120 Hz. The tension is multiplied by 2.25. Find the new first harmonic.
7. Find the wavelength of the fifth harmonic on a string of length 1.20 m.
8. Light of wavelength 480 nm passes through slits 0.40 mm apart onto a screen 1.50 m away. Find w.
9. A grating has d = 1.6 µm. What is the highest order seen with 650 nm light?
10. Light enters water (n = 1.33) from air at 50°. Find the angle of refraction.
11. Find the critical angle for a glass–air boundary if the glass has n = 1.60.
12. Why must the cladding have a lower refractive index than the core?

Answers

1. f = 1/0.020 = 50 Hz.
2. λ = 3.00 × 10⁸ / 98 × 10⁶ = 3.06 m.
3. ⅓ cycle = 120° = 2π/3 rad (2.09 rad).
4. Light and a wave on a string (transverse). Sound is longitudinal.
5. λ/2 = 0.30 m.
6. f ∝ √T, √2.25 = 1.5, so f = 180 Hz.
7. λ = 2l/5 = 2.40/5 = 0.48 m.
8. w = (480 × 10⁻⁹ × 1.50)/(0.40 × 10⁻³) = 1.8 × 10⁻³ m (1.8 mm).
9. d/λ = 1.6 × 10⁻⁶ / 650 × 10⁻⁹ = 2.46, so n = 2.
10. sin θ₂ = sin 50°/1.33 = 0.576, θ₂ = 35.2°.
11. sin θc = 1/1.60 = 0.625, θc = 38.7°.
12. Total internal reflection only happens when light meets a boundary with a medium of lower refractive index.

Where marks are usually lost

• Taking node-to-node distance as λ rather than λ/2, or counting nodes instead of the gaps between them.
• Giving phase difference without saying whether it is in degrees or radians.
• Describing a stationary wave without stating that the two waves have the same frequency and travel in opposite directions.
• Saying polarisation “proves light is a wave”: it is evidence that light is transverse.
• Leaving slit separation in mm or wavelength in nm in w = λD/s.
• Using lines per mm as d, or rounding the highest grating order up.
• Defining coherence as “same frequency” only, missing the constant phase difference.
• Mixing up material dispersion (wavelengths) and modal dispersion (paths), or giving the wrong remedy for each.
• Writing sin θc = n₁/n₂, which gives a sine bigger than 1.

The measurements revision notes cover uncertainties for the two required practicals, and the exam preparation guide explains how the papers are set out. Try the free diagnostics to find your weakest topics.

Official syllabus

AQA AS and A-level Physics specification (7407/7408), version 1.4, July 2026, for AS and A-level exams June 2016 onwards, published by AQA. Section 3.3 Waves.