Waves and superposition

Progressive waves, phase difference, intensity, polarisation, standing waves, interference, diffraction gratings, single-slit diffraction and resolution, for H2 Physics.

GCE A-Level H2 Physics 2027
10 lessons

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Before you begin

Waves carry energy without carrying matter, and when two waves meet, their displacements add. The lessons start with a single progressive wave: its graphs, phase difference, intensity and polarisation. They then use the principle of superposition to explain standing waves, two-source interference, diffraction gratings, single-slit diffraction and the limit diffraction puts on resolution.

Be comfortable with: oscillations, because a wave is a chain of oscillating particles described by the same period, frequency, amplitude and phase.

Learning goals

Describe wave models, use wave quantities and interpret wave graphs in space and time.
Relate phase difference to separations in time and position.
Use wave intensity, amplitude and inverse-square relationships with their assumptions.
Explain polarisation and apply Malus’ law to amplitude and intensity.
Apply the principle of superposition to resultant displacement.
Explain standing-wave formation, nodes, antinodes and energy transfer.
Apply boundary conditions to standing waves on stretched strings.
Analyse displacement and pressure patterns in resonant air columns and determine sound wavelength.
Explain single-aperture diffraction and apply first-minimum and Rayleigh criteria.
Explain coherent two-source interference using phase and path difference.
Analyse Young double-slit interference and its small-angle assumptions.
Use diffraction gratings to analyse principal maxima and determine wavelength.

Syllabus statements covered

show an understanding that mechanical waves involve the oscillations of particles within a material medium, such as a string or a fluid, and electromagnetic waves involve the oscillations of electromagnetic fields in space and time
show an understanding of and use the terms displacement, amplitude, period, frequency, phase, phase difference, wavelength and speed
deduce, from the definitions of speed, frequency and wavelength, the equation v = fλ
recall and use the equation v = fλ
analyse and interpret graphical representations of transverse and longitudinal waves with respect to variations in time and position (space)
show an understanding that energy is transferred due to a progressive wave without matter being transferred
recall and use the term intensity as the power transferred (radiated) by a wave per unit area, and the relationship intensity ∝ (amplitude)² for a progressive wave
show an understanding of and apply the concept, that the intensity of a wave from a point source and travelling without loss of energy obeys an inverse square law to solve problems
show an understanding that polarisation is a phenomenon associated with transverse waves
recall and use Malus’ law (intensity ∝ cos²θ) to calculate the amplitude and intensity of a plane-polarised electromagnetic wave after transmission through a polarising filter
explain and use the principle of superposition in simple applications
show an understanding of experiments which demonstrate standing (stationary) waves using microwaves, stretched strings and air columns
explain the formation of a standing (stationary) wave using a graphical method, and identify nodes and antinodes, differentiating between pressure and displacement nodes and antinodes for sound waves
determine the wavelength of sound using standing (stationary) waves
show an understanding of the terms diffraction, interference, coherence, phase difference and path difference
show an understanding of phenomena which demonstrate two-source interference using water waves, sound waves, light and microwaves
show an understanding of the conditions required for two-source interference fringes to be observed
recall and use the equation ax/D = λ to solve problems for double-slit interference, where a is the slit separation and x is the fringe separation
recall and use the equation a sin θ = nλ to solve problems involving the principal maxima of a diffraction grating, where a is the slit separation
describe the use of a diffraction grating to determine the wavelength of light (knowledge of the structure and use of a spectrometer is not required)
show an understanding of phenomena which demonstrate diffraction through a single slit or aperture, or across an edge, such as the diffraction of water waves in a ripple tank with both a wide gap and a narrow gap, or the diffraction of sound waves from loudspeakers or around corners
recall and use the equation b sin θ = λ to solve problems involving the positions of the first minima for diffraction through a single slit of width b
recall and use the Rayleigh criterion θ ≈ λ/b for the resolving power of a single aperture, where b is the width of the aperture

Lessons

Work through them in order.

Core lessons

Practise and check

Topic reference

Quick reference

Idea	Relation	Notes
wave equation	v = fλ	f = 1/T
phase difference	Δφ = (Δ x)/λ × 2π or (Δ t)/T × 2π	Δ x along a wave at one instant; Δ t at one point
intensity	I = P/area, I ∝ A²	same wave type, frequency and medium
point source	I = P/(4π r²)	so I ∝ 1/r² and A ∝ 1/r
unpolarised light through a polariser	I = (1/2)Iᵤ	light leaves polarised along the axis
Malus’ law	I = I₀ cos² θ	I₀ is the polarised intensity arriving
standing wave	node to node = λ/2	node to antinode = λ/4
string fixed at both ends	L = nλ/2	all harmonics
tube closed at one end	L = λ/4,3λ/4,…	odd harmonics; successive resonances λ/2 apart
two-source interference (in phase)	maxima: Δ r = nλ; minima: Δ r = (n + 1/2)λ	Δ r is the path difference
double slit	λ = ax/D	a slit separation, x fringe separation; small angles
diffraction grating	a sin θ = nλ	a slit separation = 1/N; highest order: a/λ rounded down
single slit, first minima	b sin θ = λ	b slit width
Rayleigh criterion	θ ≈ λ/b	in radians; smaller is better

Core knowledge to remember

In a mechanical wave, particles of a medium oscillate; in an electromagnetic wave, electric and magnetic fields oscillate. A progressive wave transfers energy without transferring matter.
Phase difference is the fraction of a cycle by which one oscillation leads another, as an angle. In phase means 0; antiphase means π.
Intensity is the power transferred by a wave per unit area. It is proportional to the square of the amplitude.
Polarisation happens only with transverse waves: a plane-polarised wave oscillates in one direction perpendicular to its direction of travel.
Principle of superposition: when two or more waves meet at a point, the resultant displacement is the sum of the individual displacements.
A standing wave forms when two waves of the same frequency and amplitude travel in opposite directions and superpose. It has nodes and antinodes and transfers no energy. In a sound wave, a displacement node is a pressure antinode.
Coherent sources have a constant phase difference. Path difference is the difference in the distances travelled by two waves to a point.
Diffraction is the spreading of a wave through a gap or around an edge. It is greatest when the gap is about the same size as the wavelength.
The Rayleigh criterion: two sources are just resolved when the central maximum of one diffraction pattern falls on the first minimum of the other.

Common mistakes

Reading a wavelength from a displacement–time graph, or a period from a displacement–distance graph.
Scaling intensity by the amplitude factor instead of its square.
Applying Malus’ law to unpolarised light. One polariser halves unpolarised light.
Taking the distance between neighbouring nodes as a whole wavelength. It is λ/2.
Saying destructive interference always gives zero. It does only when the two amplitudes are equal.
Dividing a distance across N fringes by N instead of N-1.
Using the number of lines per millimetre as the grating spacing, or rounding the highest order up.
Treating b sin θ = λ as a bright fringe. It locates the first dark band of a single slit.