amazon.co.ukA-Level Physics waves and optics revision: wave properties, superposition, standing waves, Young's double slits, diffraction gratings, refraction, Snell's law and total internal reflection with derivations and required practicals - AQA 7408 3.3, OCR A H556 4.3, Edexcel 9PH0 Topic 5.
A progressive wave transfers energy without transferring matter (AQA 7408 3.3.1; OCR A H556 4.3.1; Edexcel 9PH0 Topic 5). Key quantities: amplitude A (maximum displacement - sets intensity, I ∝ A^2), wavelength λ, frequency f, period T = 1/f, and the wave speed v = fλ. Transverse waves oscillate perpendicular to the direction of travel (all EM waves, waves on strings); longitudinal waves oscillate parallel (sound: compressions and rarefactions). EM waves travel at c = 3.00 × 10^8 m/s in vacuum and slower v = c/n in media.
Phase difference derivation: a point on a wave one full wavelength further along has completed exactly one extra cycle, i.e. a phase difference of 2π radians. Points separated by Δx have phase difference Δφ = 2πΔx/λ. So points λ/2 apart are in antiphase (Δφ = π), and points λ apart are in phase. In time, a point completes 2π radians in one period, so Δφ = 2πΔt/T for a time lag Δt. Displacement of a point: y = A sin(2πft) for a wave starting at zero displacement - be able to sketch and read displacement-time and displacement-position graphs and convert between them.
Polarisation (AQA 7408 3.3.2): only transverse waves can be plane-polarised because their oscillations occur in a choice of perpendicular directions; longitudinal waves oscillate along one line only, so there is nothing to filter - the standard proof that light is transverse. Two polarising filters at 90 degrees give extinction (crossed polaroids); at angle θ, Malus's law I = I_0 cos^2 θ gives the transmitted intensity. Applications: polaroid sunglasses cut glare reflected off horizontal surfaces (partially polarised light), and stress analysis in plastics.
A wave of frequency 50 Hz travels at 25 m/s. Find the phase difference between two points 0.25 m apart.
Line 1: λ = v/f = 25/50 = 0.50 m (M1 A1).
Line 2: Δφ = 2πΔx/λ = 2π × 0.25/0.50 = π radians = 180 degrees (M1 A1) - the points are in antiphase.
Principle of superposition: when two or more waves meet at a point, the resultant displacement is the (vector) sum of the individual displacements (AQA 7408 3.3.3). Constructive interference occurs when waves arrive in phase (path difference nλ); destructive when in antiphase (path difference (n + ½)λ) - for coherent sources. Coherence - constant phase difference and equal frequency - is the condition for a stable pattern; two separate lamps are not coherent, which is why Young used a single source split in two.
Standing (stationary) waves (AQA 7408 3.3.4; OCR A H556 4.3.3): form when two identical progressive waves travelling in opposite directions superpose - typically a wave and its reflection. The pattern has fixed nodes (zero amplitude, destructive superposition) and antinodes (maximum amplitude) with adjacent nodes λ/2 apart; energy is not transferred along the wave, it is stored between nodes. On a string fixed at both ends, both ends must be nodes.
Harmonics derivation (string length L): the first harmonic (fundamental) fits half a wavelength between the two end nodes: λ_1 = 2L, so f_1 = v/2L where v = √(T/μ) is the wave speed on the string under tension T with mass per unit length μ. The second harmonic has an extra node at the centre: λ_2 = L, f_2 = 2f_1; in general the nth harmonic has n antinodes (n - 1 internal nodes), λ_n = 2L/n and f_n = n f_1 - a whole-number (harmonic) series. Same logic for tubes: closed at one end, only odd harmonics fit (λ = 4L, 4L/3, 4L/5...); open at both ends, all harmonics (like the string).
A string is stretched over a pulley with masses (setting the tension) and driven by a vibration generator at the far end. Adjust the frequency until clear loops appear: the driver end and pulley are (near-)nodes. Measure the length L of n loops - each loop is half a wavelength, so λ = 2L/n. Plot f against 1/λ: gradient = v, the wave speed on the string. Then v = √(T/μ) lets you verify the effect of tension (doubling T multiplies v by √2) or of the string's mass per unit length. Errors: judging the exact frequencies of resonance (amplitude changes continuously - use the frequency giving maximum amplitude at fixed driving amplitude), the end near the driver is not a perfect node (antinode correction), and parallax in measuring L. Uncertainty in f comes from the signal generator display resolution.
Two-source interference (AQA 7408 3.3.5; Edexcel 9PH0 Topic 5): Young's double slits split one laser (or a lamp with a colour filter and single slit) into two coherent sources. Where the path difference is nλ, waves arrive in phase - a bright fringe; where it is (n + ½)λ - a dark fringe. The result is equally spaced alternating fringes.
Fringe spacing derivation (w = λD/s): for slits separated by s and a screen at distance D, the nth bright fringe occurs where the path difference equals nλ. For D much larger than s, the geometry gives path difference ≈ s sin θ ≈ s tan θ ≈ sx/D for small angles (the small-angle approximation sin θ ≈ tan θ ≈ θ is the key step). Setting the path difference to nλ: sx_n/D = nλ, so the nth bright fringe sits at x_n = nλD/s from the centre. Fringe spacing w = x_(n+1) - x_n = λD/s. Hence: longer wavelength (red fringes wider than blue), larger screen distance, and closer slits all widen the pattern - and measuring w across many fringes (say 10, then divide by 10) reduces percentage uncertainty.
Measuring λ this way (AQA required practical 2; Edexcel CP8): measure s with a vernier/travelling microscope (the dominant uncertainty), D with a tape, w across as many fringes as possible. Percentage uncertainty in λ = Δw/w + Δs/s + ΔD/D. White light gives a white central fringe with coloured edges - each wavelength produces its own spacing, overlapping away from the centre.
Light of wavelength 500 nm passes through double slits 0.20 mm apart. Fringes are observed on a screen 2.5 m away.
Line 1: w = λD/s (M1: equation, coherent symbols).
Line 2: w = (500 × 10^-9 × 2.5)/(0.20 × 10^-3) = 6.25 × 10^-3 m = 6.25 mm (A1: unit conversion).
Line 3: Across 10 fringes the spacing would be 62.5 mm, measurable to better than 1% with a mm rule - the practical evaluation point (A1).
Diffraction is the spreading of waves through gaps or around obstacles - significant when the gap width is comparable to λ (AQA 7408 3.3.6). A single slit of width a gives a central bright maximum twice as wide as the side maxima, with the first minimum at a sin θ = λ (one full wavelength of path difference across the slit cancels in pairs - Huygens construction in words: every point across the slit acts as a source, and pairs of sources λ/2 apart cancel in the direction where their path difference is λ/2). Narrower slit → wider pattern; longer wavelength → wider pattern.
Diffraction grating: many equally spaced slits (spacing d = 1/N for N lines per metre) give sharp, bright maxima because many beams reinforce only at exact angles. The condition for the nth order maximum: d sin θ = nλ. The number of visible orders is the largest n with sin θ = nλ/d ≤ 1 - integer division of d/λ, and the 90-degree case (sin θ = 1) is not observable. Gratings resolve spectra (monochromatic light: one line per order; white light: a spectrum in each order with red at larger angles because λ is larger).
Applications and standard questions: atomic spectra analysis, CD/DVD surface structure, X-ray crystallography (Bragg condition). For the grating experiment measure the angle θ to each order with a spectrometer and plot sin θ against n - a straight line of gradient λ/d - to average random error and check for systematic zero error.
Light of wavelength 640 nm is incident normally on a grating with 500 lines per mm. Find the angle of the second-order maximum and the number of visible orders.
Line 1: d = 1/500 mm = 2.0 × 10^-6 m (M1: slit spacing from lines per mm).
Line 2: d sin θ = nλ, so sin θ = (2 × 640 × 10^-9)/2.0 × 10^-6 = 0.64 (M1 A1).
Line 3: θ = arcsin(0.64) = 39.8 degrees (A1).
Line 4: Maximum order: n ≤ d/λ = 2.0 × 10^-6/6.40 × 10^-7 = 3.1, so n = 3 (integer part) (A1).
Refraction is the change of direction when a wave crosses a boundary between media at an angle, caused by the change of speed (AQA 7408 3.3.7; OCR A H556 4.3). The refractive index of a medium is n = c/v (speed in vacuum over speed in the medium; n ≥ 1, air ≈ 1.00, water ≈ 1.33, glass ≈ 1.5). Across a boundary, Snell's law: n_1 sin θ_1 = n_2 sin θ_2, with angles measured from the normal. Derivation from wavefronts: wavefronts must be continuous across the boundary, so the frequency is unchanged and the wavelength changes (λ = v/f = c/(nf)) - the wavefront geometry (shared hypotenuse) gives sin θ_1/sin θ_2 = v_1/v_2 = n_2/n_1.
Total internal reflection (TIR) requires two conditions: travel from a denser to a less dense medium (n_1 > n_2) and incidence above the critical angle. Derivation: at the critical angle the refracted ray travels along the boundary (θ_2 = 90 degrees). Snell's law: n_1 sin θ_c = n_2 sin 90 = n_2, so sin θ_c = n_2/n_1. For glass-air (n = 1.5): θ_c = arcsin(1/1.5) = 41.8 degrees. Below θ_c, partial reflection plus refraction; above, complete reflection - TIR is how optical fibres and prisms (periscopes, binoculars) work.
Optical fibres: a high-refractive-index core surrounded by lower-index cladding keeps light in the core by TIR. Signal degradation: modal dispersion (rays at different angles take different path lengths, spreading pulses - fixed by single-mode/monomode fibres) and material/spectral dispersion (different wavelengths travel at different speeds - fixed by monochromatic light). Absorption/scattering also attenuates. The cladding also prevents light leaking between adjacent fibres (cross-talk) and protects the core surface.
A glass-air interface has n_glass = 1.52. A fibre of length 100 m has core refractive index 1.50.
Line 1: sin θ_c = 1.00/1.52 = 0.658, θ_c = 41.1 degrees (M1 A1).
Line 2: Light along the fibre axis: v = c/n = 3.00 × 10^8/1.50 = 2.00 × 10^8 m/s (A1).
Line 3: t = 100/2.00 × 10^8 = 5.0 × 10^-7 s (A1); rays at larger angles take longer - modal dispersion spreads the pulse (A1: physics of the effect).