3.3 Waves — AQA 7408 spec coverage

3.3.1.1

Progressive waves

A progressive wave transfers energy without a net transfer of matter; particles of a material medium oscillate about fixed equilibrium positions.
Use $v=f\lambda$ , with $v$ in $\text{m s}^{-1}$ , $f$ in $\text{Hz}$ and $\lambda$ in $\text{m}$ .
For points separated by $x$ along a wave, the phase difference is $\Delta\phi=2\pi x/\lambda$ radians, or $360x/\lambda$ degrees, reduced by whole cycles when appropriate.
Amplitude is the maximum displacement from equilibrium. Do not confuse it with wavelength, which is the shortest distance between points in phase.

Tier 1 · Easy

1 mark

ORIGINAL

Successive crests of a progressive wave are $0.80\,\text{m}$ apart and pass a point every $2.5\,\text{ms}$ . Calculate the wave speed.

Tier 2 · Standard

3 marks

ORIGINAL

Two sensors on a ripple tank are $0.24\,\text{m}$ apart along the direction of a progressive wave of wavelength $0.64\,\text{m}$ . Determine the magnitude of their phase difference in both degrees and radians.

Tier 3 · Hard

5 marks

ORIGINAL

Water waves of frequency $2.4\,\text{Hz}$ and wavelength $0.35\,\text{m}$ travel from P towards Q. Q is $0.22\,\text{m}$ beyond P along the direction of travel. Determine the wave speed, the phase change from P to Q, and the delay between a crest passing P and that crest passing Q.

3.3.1.2

Longitudinal and transverse waves

In a transverse wave, the displacement of particles or fields is perpendicular to the direction of energy transfer; electromagnetic waves and waves on a stretched string are examples.
In a longitudinal wave, particle displacement is parallel to the direction of energy transfer, producing compressions and rarefactions; sound in air is an example.
Only transverse waves can be polarised. Polarisation is therefore evidence that electromagnetic waves are transverse.
A receiving aerial gives the strongest signal when aligned with the transmitted electric-field oscillations. Malus's law is not required for this specification.
All electromagnetic waves travel at $3.00\times10^8\,\text{m s}^{-1}$ in a vacuum; do not use the speed of sound for a radio signal.

Tier 1 · Easy

2 marks

ORIGINAL

State whether a sound wave travelling through air is longitudinal or transverse, and state the direction in which the air molecules oscillate.

Tier 2 · Standard

3 marks

ORIGINAL

A radio transmitter produces a vertically polarised wave. Explain why the signal received by a straight aerial decreases when the aerial is rotated from vertical towards horizontal.

Tier 3 · Hard

5 marks

ORIGINAL

At a distance of $1.20\,\text{km}$ , a detector receives a radio pulse and a sound pulse that were emitted simultaneously. Take the speed of sound as $340\,\text{m s}^{-1}$ . Calculate the arrival-time difference and explain why passing the radio wave through a correctly oriented polariser supports its classification as transverse.

3.3.1.3

Principle of superposition of waves and formation of stationary waves

Superposition means that the resultant displacement at a point is the vector sum of the displacements due to the individual waves.
A stationary wave is formed by two progressive waves of the same frequency travelling in opposite directions; it has nodes of zero amplitude and antinodes of maximum amplitude.
For a string fixed at both ends, adjacent nodes are $\lambda/2$ apart. The $n$ th harmonic has $n$ loops, $n+1$ nodes and frequency $f_n=n(2l)^{-1}\sqrt{T/\mu}$ .
In the first harmonic, $l=\lambda/2$ and $f=(2l)^{-1}\sqrt{T/\mu}$ . A common error is to set $l=\lambda$ .
A stationary wave has no net energy transfer along the pattern; particles within one loop oscillate in phase, while particles in adjacent loops are in antiphase.

Tier 1 · Easy

1 mark

ORIGINAL

State the difference between a node and an antinode in a stationary wave.

Tier 2 · Standard

3 marks

ORIGINAL

A string of length $0.75\,\text{m}$ is fixed at both ends. Its tension is $45\,\text{N}$ and its mass per unit length is $1.8\times10^{-3}\,\text{kg m}^{-1}$ . Determine the frequency of its first harmonic.

Tier 3 · Hard

5 marks

ORIGINAL

A $0.90\,\text{m}$ string fixed at both ends has mass per unit length $2.5\times10^{-3}\,\text{kg m}^{-1}$ . Its third-harmonic stationary wave has frequency $120\,\text{Hz}$ . Determine the string tension and the numbers of nodes and antinodes in this pattern.

3.3.2.1

Interference

Coherent sources have a constant phase difference and the same frequency. Coherence is required for a stable interference pattern.
Constructive interference occurs for path difference $n\lambda$ ; destructive interference occurs for path difference $(n+\tfrac12)\lambda$ when the sources are in phase.
For Young's double slits at small angles, fringe spacing is $w=\lambda D/s$ , where $D$ is slit-to-screen distance and $s$ is slit separation.
Measure across several fringe spacings and divide to reduce percentage uncertainty. Keep $\lambda$ , $D$ , $s$ and $w$ in consistent units.
A laser supplies monochromatic coherent light, but laser beams must never be viewed directly or through an optical instrument.

Tier 1 · Easy

2 marks

ORIGINAL

State the two conditions that two sources must satisfy to be coherent.

Tier 2 · Standard

3 marks

ORIGINAL

Light of wavelength $600\,\text{nm}$ illuminates two slits separated by $0.40\,\text{mm}$ . A screen is $2.4\,\text{m}$ from the slits. Calculate the fringe spacing.

Tier 3 · Hard

5 marks

ORIGINAL

In a Young double-slit experiment, the distance from the first to the ninth bright fringe is $28.8\,\text{mm}$ . The screen is $1.80\,\text{m}$ from the slits and the wavelength is $520\,\text{nm}$ . Determine the slit separation. Explain the appearance near the centre when the laser is replaced by white light.

3.3.2.2

Diffraction

Diffraction is the spreading of a wave after it passes through an aperture or around an obstacle; it is most pronounced when the aperture is comparable with the wavelength.
For a single slit, decreasing slit width or increasing wavelength increases the angular width of the central maximum.
For a plane transmission grating at normal incidence, maxima satisfy $d\sin\theta=n\lambda$ , where $d$ is grating spacing and $n$ is the order.
Convert a line density $N$ into spacing with $d=1/N$ in SI units, and reject an order if the calculation requires $\sin\theta>1$ .
A monochromatic single-slit pattern has a broad bright central maximum and weaker side maxima; white light gives a white centre with coloured edges.

Tier 1 · Easy

1 mark

ORIGINAL

State how the width of the central diffraction maximum changes when monochromatic light passes through a narrower single slit.

Tier 2 · Standard

3 marks

ORIGINAL

A diffraction grating has $400$ lines per $\text{mm}$ . Light of wavelength $589\,\text{nm}$ is incident normally. Calculate the angle of the first-order maximum.

Tier 3 · Hard

5 marks

ORIGINAL

A plane transmission grating has $600$ lines per $\text{mm}$ . It is illuminated normally with light of wavelength $480\,\text{nm}$ . Determine the highest observable order and the angle of this order. Explain why the next order cannot occur.

3.3.2.3

Refraction at a plane surface

The refractive index is $n=c/v$ . At a boundary, Snell's law is $n_1\sin\theta_1=n_2\sin\theta_2$ , with angles measured from the normal.
Total internal reflection occurs only when light travels from higher to lower refractive index and the incidence angle exceeds the critical angle, where $\sin c=n_2/n_1$ .
A step-index fibre has a higher-index core and lower-index cladding, allowing total internal reflection while protecting the boundary and reducing signal leakage between fibres.
Modal dispersion arises because rays follow paths of different lengths; material dispersion arises because refractive index and speed depend on wavelength.
Dispersion and absorption broaden or weaken pulses, limiting bandwidth and repeater spacing. A common error is to use angles to the surface rather than to the normal.

Tier 1 · Easy

2 marks

ORIGINAL

Light travels through a transparent material at $2.00\times10^8\,\text{m s}^{-1}$ . Calculate its refractive index.

Tier 2 · Standard

3 marks

ORIGINAL

A ray in glass of refractive index $1.52$ meets a glass-air boundary at an incidence angle of $44.0^{\circ}$ . Determine whether total internal reflection occurs.

Tier 3 · Hard

6 marks

ORIGINAL

A step-index optical fibre has core refractive index $1.50$ and cladding refractive index $1.47$ . A ray in the core meets the boundary at $80.0^{\circ}$ to the normal. Show that the ray is guided. Explain how material dispersion and modal dispersion broaden a light pulse in this fibre.