Waves: A-level Physics revision
From the language of progressive waves to interference, diffraction and refraction, this sheet covers the whole of AQA section 3.3 with worked examples, key diagrams and a quick self-check.
Work through it in three passes
Describe waves
Learn the wave quantities, the wave equation, and the difference between transverse and longitudinal waves.
Superpose them
Use the principle of superposition to explain stationary waves, interference and the double-slit experiment.
Bend and spread
Apply diffraction at slits and gratings, and refraction at boundaries including total internal reflection.
- Define amplitude, wavelength, frequency, period, phase and phase difference, and use c = fλ and T = 1/f.
- Distinguish transverse and longitudinal waves and explain polarisation and its applications.
- State the principle of superposition and explain how stationary waves form.
- Find harmonic frequencies on a string and identify nodes and antinodes.
- Explain coherence and path difference, and use the double-slit fringe equation w = λD/s.
- Describe single-slit diffraction and use the grating equation d sinθ = nλ.
- Use refractive index and Snell's law, and find the critical angle for total internal reflection.
- Explain how optical fibres guide light and the causes of signal degradation.
The language of waves
A progressive wave transfers energy from one place to another without transferring matter. The particles of the medium oscillate about fixed equilibrium positions.
A snapshot of a transverse progressive wave. Against distance the repeat length is the wavelength λ; for a single particle against time it is the period T.
| Quantity | Meaning | Unit |
|---|---|---|
| Amplitude A | Maximum displacement from equilibrium. | m |
| Wavelength λ | Distance for one complete cycle. | m |
| Frequency f | Cycles per second; f = 1/T. | Hz |
| Period T | Time for one complete cycle. | s |
| Phase difference | How far two points are through their cycles, in radians or degrees. | rad |
Two points one wavelength apart are in phase (phase difference 2π rad or 360°). Points half a wavelength apart are in antiphase (π rad or 180°). Phase difference = 2π × (path difference)/λ.
Worked example — the wave equation
A sound wave has frequency 256 Hz and travels at 340 m s⁻¹. Find its wavelength and period.
Two ways for a wave to oscillate
In a transverse wave the oscillation is perpendicular to energy transfer; in a longitudinal wave it is parallel, producing compressions and rarefactions.
Transverse
Oscillation is perpendicular to the direction of energy transfer.
Examples: all electromagnetic waves, waves on a string, S-waves, water surface waves.
Longitudinal
Oscillation is parallel to the direction of energy transfer.
Examples: sound, ultrasound, P-waves. Made of compressions and rarefactions.
Polarisation
A plane-polarised wave oscillates in only one plane. Only transverse waves can be polarised, because longitudinal oscillations are always along the direction of travel. Polarisation is therefore evidence that a wave is transverse.
Polaroid sunglasses cut glare (reflected light is partially polarised). Aligning a TV or radio aerial with the transmitter's plane of polarisation maximises the received signal. Rotating one Polaroid filter relative to another reduces the transmitted intensity to zero when they are crossed.
Adding waves together
The principle of superposition states that when two or more waves meet, the total displacement at a point equals the vector sum of the individual displacements.
A stationary (standing) wave forms when two progressive waves of the same frequency and amplitude travel in opposite directions and superpose — for example a wave and its reflection. Energy is stored, not transferred along the wave.
The first three harmonics on a string fixed at both ends. Nodes (N) are always still; antinodes (A) vibrate with maximum amplitude.
| Feature | Progressive wave | Stationary wave |
|---|---|---|
| Energy | Transferred along the wave | Stored, not transferred |
| Amplitude | Same for all points | Varies from zero (nodes) to maximum (antinodes) |
| Phase | Changes along the wave | All points between two nodes are in phase |
| Wavelength | λ | Distance between adjacent nodes = λ/2 |
where L is the string length, T the tension and μ the mass per unit length. The nth harmonic has frequency n f₁.
Worked example — harmonics on a string
A string of length 0.80 m has a first-harmonic frequency of 150 Hz. Find the wavelength of the first harmonic and the frequency of the third harmonic.
The first harmonic has one loop, so L = λ/2.
Coherent waves and the double slit
For a steady interference pattern the sources must be coherent: same frequency and a constant phase difference. Whether interference is constructive or destructive depends on the path difference.
Constructive
Waves arrive in phase and reinforce.
Destructive
Waves arrive in antiphase and cancel.
Young's double-slit experiment. Light is diffracted at each slit, overlaps, and forms equally spaced fringes on the screen.
where w is the fringe spacing, D the slit-to-screen distance and s the slit separation. The equation is valid when D ≫ s.
Never look directly into a laser beam or its reflection. Lasers are used because the light is monochromatic and coherent, giving sharp fringes.
Worked example — fringe spacing
Laser light of wavelength 650 nm passes through slits 0.30 mm apart. The screen is 2.4 m away. Find the fringe spacing.
Waves spreading through gaps
When a wave passes through a gap or around an obstacle it diffracts (spreads out). Spreading is greatest when the gap is about the same size as the wavelength.
Single-slit diffraction with monochromatic light: a wide, bright central maximum twice the width of the dimmer side maxima.
- A narrower slit gives a wider central maximum.
- A longer wavelength gives a wider central maximum.
- With white light the central maximum is white; the side maxima are spectra with blue nearest the centre.
The diffraction grating
A grating has many equally spaced slits, giving sharp, bright maxima. For slit spacing d:
A diffraction grating splits a monochromatic beam into a zero-order beam and symmetric higher orders.
Worked example — grating angle
Light of wavelength 590 nm is shone on a grating with 300 lines per mm. Find the angle of the first-order maximum.
Light crossing a boundary
The refractive index of a medium is n = c / cₛ, where cₛ is the speed of light in the medium. A higher n means a slower, optically denser medium.
Left: a ray bends towards the normal entering a denser medium. Right: above the critical angle the ray is totally internally reflected.
When light travels from a denser to a less dense medium it bends away from the normal. Beyond the critical angle θc it is totally internally reflected:
A step-index fibre has a high-index core surrounded by lower-index cladding, so light is guided by repeated total internal reflection. The cladding protects the core, keeps it separated to prevent crossover, and increases the critical angle so only near-axial rays travel.
An optical fibre guides light along the core by total internal reflection at the core–cladding boundary.
Signal degradation: modal dispersion (rays taking different path lengths) and material dispersion (different wavelengths travelling at different speeds) both cause pulse broadening; absorption reduces the amplitude. These limit the rate and distance over which data can be sent.
Worked example — critical angle
A glass–air boundary has a glass refractive index of 1.52. Find the critical angle.
Transverse or longitudinal?
For each wave, select whether it is transverse or longitudinal.
Mistakes to avoid
Stationary waves do not transfer energy
Unlike progressive waves, a stationary wave stores energy. No net energy passes a node.
Node spacing is half a wavelength
Adjacent nodes (or adjacent antinodes) are λ/2 apart, not λ. A node and the next antinode are λ/4 apart.
Coherence needs constant phase
Two sources can have the same frequency but still not be coherent if their phase difference changes with time.
Only transverse waves polarise
Sound (longitudinal) cannot be polarised. If a wave can be polarised, it must be transverse.
Self-check
1. A wave has period 0.025 s. State its frequency.
f = 1/T = 1/0.025 = 40 Hz
2. Why must the two sources in a double-slit experiment be coherent?
So the phase difference between them is constant, giving a stable (non-shifting) interference pattern of fringes.
3. Light of wavelength 500 nm hits a grating with d = 2.0 × 10⁻⁶ m. Find the second-order angle.
sinθ = nλ/d = (2 × 500 × 10⁻⁹)/(2.0 × 10⁻⁶) = 0.50, so θ = 30°
4. State two differences between a stationary and a progressive wave.
A stationary wave stores energy and has points of varying amplitude (nodes and antinodes); a progressive wave transfers energy and every point has the same amplitude.
Before the exam
- Define the wave quantities and use c = fλ and T = 1/f.
- Relate phase difference to path difference.
- Distinguish transverse and longitudinal waves; explain polarisation and its uses.
- State the principle of superposition and describe how stationary waves form.
- Identify nodes and antinodes and find harmonic frequencies.
- Explain coherence and apply w = λD/s.
- Describe single-slit diffraction and use d sinθ = nλ for a grating.
- Use n = c/cₛ and n₁ sinθ₁ = n₂ sinθ₂.
- Find the critical angle and explain total internal reflection in optical fibres.
- Describe modal and material dispersion as causes of signal degradation.