HookNoise-cancelling headphones fight sound with more sound
Switch on noise cancelling and the roar of a jet cabin drops away — not because the sound is being blocked, but because more sound is being added. A microphone on the earcup samples the incoming rumble, a chip inverts the waveform, and the driver plays the inverted copy half a cycle out of step. Crest meets trough, the displacements sum to nearly zero at your eardrum, and the rumble cancels. Paul Lueg patented the idea in the 1930s; Amar Bose resurrected it after a maddeningly loud flight in 1978, shipped aviation headsets in 1989 and the first consumer QuietComfort in 2000. Even the technology's famous weakness is pure wave physics: cancellation works best below about 1 kHz, where sound in air has wavelengths of tens of centimetres — at 1 kHz the wavelength is 34 cm, so a small timing slip barely shifts the phase alignment. At 5 kHz the wavelength is 7 cm and the same slip wrecks it, which is why headphones still let a crying baby through while erasing engine drone.
That one product runs on the central idea of section 3.3: superposition — overlapping waves simply add their displacements. Superposition builds the stationary waves on a guitar string, the fringes in Young's double-slit experiment and the spectra thrown by a diffraction grating; add the geometry of refraction and you have the entire section. Two required practicals live here, and the calculation marks are some of the most bankable on Paper 1 — every formula is short, and every one earns.
ModelFive numbers describe any wave
A progressive wave transfers energy without transferring matter: each particle of the medium (or each point of a field) oscillates about a fixed position while the pattern — and the energy — moves on. Five quantities pin it down. Displacement is where a point is now, relative to equilibrium, with direction. Amplitude is the maximum displacement — not the crest-to-trough distance, a definition AQA tests deliberately. Wavelength \(\lambda\) is the repeat distance. Period \(T\) is the time for one full cycle, and frequency \(f = 1/T\) counts cycles per second. Speed ties them together: \(c = f\lambda\), where \(c\) here means the speed of any wave, not just light.
Phase is the sharpest tool in the box: where in its cycle a point is, measured as an angle — one complete cycle is \(2\pi\) radians. The phase difference between two points a distance \(d\) apart along a wave is \(\Delta\phi = 2\pi d/\lambda\). Points exactly one wavelength apart are in phase (\(2\pi\), equivalent to zero); points half a wavelength apart are in antiphase (\(\pi\) radians), always moving opposite ways. Nearly everything later in the section — interference, stationary waves, the grating equation — is a statement about phase difference dressed in different clothes, so learn to convert path difference into phase difference automatically.
Your Wi-Fi router transmits at 2.4 GHz: \(\lambda = c/f = 3.00\times10^{8} \div 2.4\times10^{9} = 0.125\) m — about the width of your hand, which is why a metal fridge between you and the router casts a serious signal shadow. Now a water wave of wavelength 1.2 m: two corks floating 0.30 m apart have phase difference \(\Delta\phi = 2\pi \times 0.30/1.2 = \pi/2\) rad (90°). When one cork rides a crest, the other passes through equilibrium a quarter-cycle behind. AQA accepts phase difference in degrees or radians — but the data booklet and mark schemes think in radians, so you should too.
ModelTransverse, longitudinal, and the polarisation test
In a transverse wave the oscillation is perpendicular to the direction of energy travel: all electromagnetic waves, water surface ripples, waves on a string. In a longitudinal wave the oscillation is parallel to the travel direction, producing compressions and rarefactions: sound is the exam's standard example. The distinction sounds cosmetic; polarisation makes it physical.
An unpolarised transverse wave oscillates in every plane perpendicular to its direction of travel. A polarising filter passes only one plane, producing plane-polarised waves; a second filter — the analyser — rotated 90° to the first passes almost nothing, and rotating it makes the transmitted intensity cycle bright–dark–bright every 180°. That behaviour is only possible if there are oscillation planes to select between, so polarisation is the evidence that a wave is transverse. Sound cannot be polarised, however you filter it: a longitudinal oscillation has no plane to choose.
AQA names two applications. Polaroid sunglasses: glare reflected off water and wet roads is partially plane-polarised horizontally, so lenses with a vertical transmission axis cut the glare while passing most other light — anglers wear them to see through the surface, not just to look composed. And aerial alignment: TV and FM signals are transmitted plane-polarised, so receiving aerials must be mounted parallel to the transmitting aerial's plane — the reason every rooftop aerial in a town points and tilts the same way, and the reason your picture degrades if a storm twists the mast.
MechanismStationary waves: interference with yourself
The principle of superposition: where two or more waves meet, the resultant displacement at each point is the vector sum of the individual displacements — after which the waves pass through each other completely unchanged. Send a wave down a stretched string and let it reflect from a fixed end, and the string carries two waves of the same frequency and wavelength travelling in opposite directions. Their superposition is a stationary (standing) wave: a pattern that stores energy in place instead of transporting it.
At the nodes, the two waves arrive permanently in antiphase, so the displacement is always zero. At the antinodes, they arrive in phase and the string swings with maximum amplitude. Adjacent nodes sit half a wavelength apart. The contrasts with a progressive wave are exam gold: on a stationary wave the amplitude depends on position (zero at nodes, maximum at antinodes) rather than being the same everywhere, and all points between two adjacent nodes oscillate in phase, with a phase jump of \(\pi\) as you cross a node — whereas along a progressive wave the phase varies smoothly with position.
A string fixed at both ends can only sustain patterns whose nodes land on the ends, so only certain wavelengths fit. The simplest — one loop, a single antinode — is the first harmonic: string length \(l = \lambda/2\), giving \(f_1 = \tfrac{1}{2l}\sqrt{T/\mu}\), where \(T\) is the tension and \(\mu\) the mass per unit length. Higher harmonics stack extra loops at integer multiples of \(f_1\). The formula is a guitar in algebra: shorter (fret the string), tighter (tuning peg) and lighter (thinner string) all raise the pitch.
A 0.600 m length of string with mass per unit length 1.2 g m⁻¹ is tensioned by a hanging 0.50 kg mass: \(T = mg = 0.50 \times 9.81 = 4.9\) N. First harmonic: \(f_1 = \tfrac{1}{2 \times 0.600}\sqrt{4.9 \div 1.2\times10^{-3}} = 0.833 \times 63.9 \approx 53\) Hz. Now interrogate the formula. Double the hanging mass: \(f \propto \sqrt{T}\), so \(f_1\) rises by \(\sqrt{2}\) to about 75 Hz. Halve the length: \(f \propto 1/l\), so it doubles to about 107 Hz. Swap in a string of four times the mass per unit length: \(f \propto 1/\sqrt{\mu}\), so it halves. Each knob's effect drops straight out of the equation — and 'predict and justify the change' is exactly how AQA phrases it.
DataRequired practical 1: making the string equation confess
The apparatus is a signal generator driving a small vibration generator attached to a string that runs over a pulley to a mass hanger: the hanging weight sets the tension \(T = mg\), a metre rule measures the vibrating length \(l\), and \(\mu\) comes from weighing a known length of the same string. Sweep the driving frequency until the string resonates in a single loop — the first harmonic — and record that frequency. The design is one independent variable at a time: vary \(l\) with \(T\) and \(\mu\) fixed, then \(T\) with \(l\) and \(\mu\) fixed, then \(\mu\) by switching strings, with the dependent variable always the first-harmonic frequency.
The analysis linearises the equation. Testing length: plot \(f_1\) against \(1/l\) and expect a straight line through the origin. Testing tension: plot \(f_1^2\) against \(T\) — squaring kills the square root — and the gradient hands you \(1/(4l^2\mu)\), an indirect measurement of \(\mu\) you can check against the balance.
The uncertainties are where the marks hide. The dominant one is judging peak resonance: amplitude changes slowly near the maximum, so approach it from below and from above and take the mid-frequency of the range that looks maximal — and quote that range as the uncertainty in \(f_1\). The node at the pulley is not a perfect point, smearing \(l\) by a few millimetres. \(\mu\) is a small mass divided by a length, so weigh several metres of string rather than a short offcut, or the balance's resolution dominates. Slotted masses can be percent off their stamped values — weigh them. And the signal generator's dial can carry a systematic calibration error: checking its output against an oscilloscope or a calibrated frequency meter is the improvement examiners most often reward.
MechanismCoherence, path difference and Young's double slit
Two wave sources interfere stably only if they are coherent: same frequency, constant phase difference. Then geometry decides everything through path difference — the extra distance one wave travels to reach a point. A whole number of wavelengths, \(n\lambda\): the waves arrive in phase, superpose constructively, bright fringe. An odd half-integer, \((n+\tfrac{1}{2})\lambda\): antiphase, destructive, dark fringe. In 1801 Thomas Young pushed sunlight through a single slit (to create one coherent source) and then a double slit, saw the alternating fringes, and handed physics the decisive evidence that light behaves as a wave — against a century of loyalty to Newton's particle theory.
A laser makes the experiment almost unfairly easy because its light is monochromatic and coherent across the beam: aim it through slits a fraction of a millimetre apart and fringes appear metres away. (Standard lab discipline applies: never look into the beam or its specular reflections, keep it below eye level, terminate it on a matt screen.) The fringe spacing is \(w = \lambda D/s\), with \(s\) the slit separation and \(D\) the slit–screen distance. Because \(D/s\) can be several thousand, the geometry works as a lever that stretches a sub-micrometre wavelength into millimetres you can measure with a ruler — that is the entire cleverness of the arrangement. Swap the laser for white light and the pattern signs its name: a white central fringe (zero path difference for every wavelength), flanked by fringes smeared into spectra with the blue edge nearer the centre — blue's shorter \(\lambda\) means narrower spacing — fading after a few orders as the colours overlap.
A helium–neon laser (\(\lambda = 633\) nm) illuminates slits 0.400 mm apart, with a screen 2.50 m away: \(w = \lambda D/s = (633\times10^{-9} \times 2.50) \div 4.00\times10^{-4} = 4.0\times10^{-3}\) m — 4.0 mm fringes. In required practical 2 you run this backwards, \(\lambda = ws/D\), and the craft is all in \(w\): one fringe measured with a millimetre rule carries over 10% uncertainty, but measuring across ten spacings (eleven bright fringes) gives about 40 mm, cutting the percentage tenfold. Then audit the other two: \(s\) is tiny, so take the manufacturer's value or measure it with a travelling microscope; \(D\) is huge, so a ±5 mm tape error barely registers. Ranking the three uncertainties — and saying which one limits the experiment — is precisely the AO3 question AQA attaches to this practical.
DataDiffraction, and the grating that measures light itself
Diffraction is the spreading of a wave as it passes through a gap or past an edge, strongest when the gap width is comparable to the wavelength — the reason you hear a conversation through a doorway before you see the speakers. A single slit throws a characteristic pattern: a broad, bright central maximum, dark minima either side, then weak secondary maxima. Two control knobs matter: narrow the slit and the central maximum gets wider but dimmer; increase the wavelength and it widens too, so red spreads more than blue, and white light gives a white centre with colour-fringed edges.
A diffraction grating multiplies the double slit into thousands of slits per millimetre. Constructive interference now survives only where every slit's contribution arrives in phase, which happens at sharply defined angles: \(d\sin\theta = n\lambda\), where \(d\) is the slit spacing and \(n\) the order. Between those angles, the many contributions cancel almost completely — so instead of soft fringes you get bright, needle-sharp beams separated by darkness. That sharpness plus the large angles makes the grating the precision half of required practical 2: measure the distance between the \(+n\) and \(-n\) spots on the screen, halve it, divide by the grating–screen distance for \(\tan\theta\) — large angles mean small percentage uncertainties, which is exactly why the grating beats the double slit for measuring \(\lambda\). It is also working spectroscopy: each wavelength peaks at its own angle, so a grating fans starlight into a spectrum, and the dark absorption lines identify which elements the light passed through — how astronomers read a star's chemistry without leaving the ground.
A green laser pointer (\(\lambda = 532\) nm) hits a grating ruled at 600 lines per millimetre. First convert the ruling: \(d = 1\times10^{-3} \div 600 = 1.67\times10^{-6}\) m. First order: \(\sin\theta_1 = \lambda/d = 0.319\), so \(\theta_1 = 18.6°\). Highest visible order: \(n \le d/\lambda = 3.13\), so \(n = 3\) — you see seven beams in total, the zero order plus three each side. And check the spacing: \(\sin\theta_3 = 3 \times 0.319 = 0.958\) gives \(\theta_3 = 73.3°\) — nowhere near \(3 \times 18.6°\). Orders stretch apart as \(n\) climbs because \(\sin\theta\), not \(\theta\), is what grows linearly. Multiplying the first-order angle by \(n\) is the classic dropped mark.
ModelRefraction, total internal reflection and the fibre under your street
Light slows down inside transparent materials, and the refractive index \(n = c/c_s\) records by how much: glass at \(n \approx 1.5\) carries light at about \(2\times10^{8}\) m s⁻¹. Crossing a boundary, the change of speed swings the direction: Snell's law, \(n_1\sin\theta_1 = n_2\sin\theta_2\), with every angle measured from the normal, never the surface. Into a denser medium (higher \(n\)) light bends towards the normal; out of it, away.
That second case has a cliff edge. Leaving a dense medium, the refracted ray bends further and further from the normal until, at the critical angle \(\sin\theta_c = n_2/n_1\), it grazes along the boundary itself. Beyond \(\theta_c\) no refracted ray can exist and the boundary reflects everything: total internal reflection, which needs both conditions — travelling from higher \(n\) to lower \(n\), and incidence beyond the critical angle.
An optical fibre weaponises this: a hair-thin glass core sheathed in cladding of slightly lower refractive index, so light entering steeply enough ricochets along by TIR for kilometres. The cladding is not packaging — it is physics. It guarantees a controlled n-step at the core boundary even where fibres touch, flex or get scratched (a scratch on a bare core would change the boundary and leak light), and it stops signal crossing between neighbouring fibres in a bundle. Two effects degrade the signal. Absorption: the glass soaks up energy, shrinking amplitude until repeaters must re-boost it. Pulse broadening: modal dispersion — rays bouncing at different angles travel different total path lengths and arrive spread out — plus material dispersion, where different wavelengths travel at slightly different speeds. Broadened pulses smear into their neighbours and cap the data rate, so engineers fight back with monomode cores so narrow only the straight-through path fits, and near-monochromatic laser sources.
A fibre has core \(n_1 = 1.55\) and cladding \(n_2 = 1.45\): \(\sin\theta_c = 1.45/1.55 = 0.935\), so \(\theta_c = 69.3°\). A ray striking the core wall at 80° from the normal exceeds the critical angle, reflects totally, and does so at every subsequent bounce — losing essentially nothing to transmission for kilometre after kilometre. Strip the cladding and use bare glass in air: \(\sin\theta_c = 1.00/1.55\) gives \(\theta_c = 40.2°\) — TIR becomes easier, yet the fibre gets worse, because every fingerprint, water droplet or touching neighbour now changes the boundary and bleeds signal. The cladding buys control, not a bigger critical angle — an evaluation point that separates band-A answers on fibre questions.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
The calculation marks in this section are lost to housekeeping, not physics. Check the calculator is in degrees for Snell's-law and grating angles but think in radians for phase. Convert a grating's lines-per-millimetre to a spacing in metres before anything else: d = 10⁻³ ÷ N. In \(w = \lambda D/s\), keep s (slit separation) and w (fringe separation) straight — swapping them is the classic error. And for highest order, compute \(d/\lambda\) and take the whole-number floor, checking \(\sin\theta \le 1\); an answer of 3.13 means three orders, not 'order 3.13'.
The 6-mark levels question here is almost always 'explain how the stationary wave forms', and the mark scheme is a checklist you can pre-load: two waves, same frequency and wavelength, similar amplitude, travelling in opposite directions (reflection provides the second), superposition, nodes where the waves are permanently in antiphase giving zero displacement, antinodes where they arrive in phase. Six ingredients, six sentences. For required-practical questions, pair every error with its remedy — resonance judged by eye, so approach the peak from both sides and take the mid-frequency; one fringe too small to measure, so measure across ten and divide; node at the pulley ill-defined, so quote the extra uncertainty in l. AQA's AO3 marks sit in the pairing, not in naming the error alone.
Diagram and definition marks are the quiet third of this section. Draw angles from the normal and label the normal explicitly; mark nodes and antinodes on any stationary-wave sketch; define amplitude as maximum displacement and phase difference in radians. When asked for evidence, match it precisely: polarisation shows light is transverse; interference and diffraction show it is a wave; a white central fringe flanked by spectra is the double-slit signature of white light. Vague evidence sentences ('it proves waves exist') score nothing — name the property the observation pins down.