Lesson 6: Waves

Waves carry energy without carrying matter — the physics of sound, light, earthquakes, and Wi-Fi. A-level waves means the wave equation derived, progressive vs stationary waves distinguished mathematically, and interference understood as superposition with path differences. This lesson builds the vocabulary (wavelength, phase, polarisation) and the core mathematics: v = fλ proved, standing waves on strings and in pipes, and the experiments that measure the speed of sound and light. Wave mathematics recurs in quantum physics (Lesson 8) and superposition (Lesson 7).

Describing Waves — v = fλ Derived

Definitions: displacement (from equilibrium), amplitude a (max displacement), wavelength λ (distance per cycle), period T (time per cycle), frequency f = 1/T (cycles per second), phase (position within a cycle, in degrees or radians). Derivation of v = fλ: a wave travels one wavelength λ in one period T, so speed v = λ/T = fλ. This is not a definition but a consequence of what wavelength and period mean — examiners accept this two-line derivation.

Transverse vs longitudinal: transverse oscillates perpendicular to travel (light, strings, water ripples — can be polarised); longitudinal oscillates parallel (sound, slinky springs — compressions and rarefactions). Earthquakes: P-waves (longitudinal, faster, travel through liquid) arrive before S-waves (transverse, slower, blocked by the liquid outer core — proving the core is liquid!). Wavefronts and rays: fronts perpendicular to travel direction; diagrams use them for reflection/refraction.

Worked: a sound wave at 440 Hz (concert A) in air (v = 343 m/s): λ = 343/440 ≈ 0.78 m — comparable to instrument sizes, which is why instruments resonate. A 5G signal at 3.5 GHz: λ = 3×10⁸/3.5×10⁹ ≈ 8.6 cm — antenna sizes follow wavelengths. Red light (λ = 650 nm): f = 3×10⁸/650×10⁻⁹ ≈ 4.6×10¹⁴ Hz. The electromagnetic spectrum spans 20+ orders of magnitude in λ — one equation, v = fλ, organises it all (with v = c for EM waves).

Phase, Polarisation, and the Wave Equation

Phase difference: two points in phase (Δφ = 0) oscillate together; antiphase (Δφ = π) oppositely. Path difference Δx relates to phase: Δφ = 2πΔx/λ — derived: one full wavelength is one full cycle (2π). Points λ/2 apart are antiphase; λ/4 apart in quadrature (π/2). Coherence (constant phase relationship) is required for stable interference — lasers are coherent, light bulbs are not.

Polarisation: transverse waves filtered to one plane. Unpolarised light → Polaroid filter → plane-polarised (intensity halves — the perpendicular component is absorbed). Crossed Polaroids block all light; inserting a third at 45° between them transmits some (Malus's law: I = I₀cos²θ — the middle filter re-polarises). Uses: sunglasses (glare is partially polarised by reflection), LCD screens, 3D cinema glasses, stress analysis (photoelasticity). Longitudinal waves cannot be polarised — polarisation proves light is transverse.

The progressive wave equation: y = a sin(2π(x/λ − t/T)) — displacement at position x, time t. Check: at fixed x, oscillates with period T ✓; at fixed t, sinusoid in x with wavelength λ ✓; wave speed: constant phase requires x/λ − t/T = const, so dx/dt = λ/T = v ✓ — the equation encodes everything. Phase form: y = a sin(kx − ωt) with k = 2π/λ (wave number), ω = 2πf (angular frequency); v = ω/k. Learn to read these: k tells spatial oscillation, ω temporal.

Stationary (Standing) Waves

Stationary waves form when two identical progressive waves superpose in opposite directions (incident + reflected): nodes (zero displacement always) and antinodes (maximum oscillation) at fixed positions — energy is stored, not transmitted. Derivation for a string fixed at both ends: y = a sin(kx−ωt) + a sin(kx+ωt) = 2a sin(kx)cos(ωt) (sum-to-product) — the spatial (sin kx) and temporal (cos ωt) parts separate: every point oscillates in SHM with amplitude 2a|sin kx|. Nodes where sin kx = 0: x = nλ/2.

Harmonics on a string of length L: L = nλ/2, so f_n = nv/2L — the harmonic series (n = 1 fundamental, n = 2, 3 overtones). A guitar string (L = 0.65 m, v = 400 m/s): f₁ = 400/(2×0.65) ≈ 308 Hz; pressing the 12th fret halves L, doubling f (octave). Pipes: closed at one end — node at closed end, antinode at open: L = (2n−1)λ/4, only odd harmonics (clarinet's hollow sound); open–open: L = nλ/2 (flute). End correction: antinode slightly beyond the open end (~0.6r) — mention in practicals.

Worked — tension from frequency: a guitar string of length 0.64 m and mass per unit length μ = 1.1 g/m sounds its fundamental at 196 Hz (the G string). Wave speed on the string: v = 2Lf₁ = 2(0.64)(196) ≈ 251 m/s. Tension: from v = √(T/μ), T = μv² = 1.1×10⁻³ × 251² ≈ 69 N. Guitar strings really are under car-jack tensions — which is why the neck is reinforced, and why f rises if you tighten the peg (T up → v up → f up at fixed L).

Required practical — stationary waves on a string: vibrator driving a string over a pulley with hanging masses (tension T = mg). Vary T or L, find resonance (maximum amplitude), measure λ from loop lengths. Verify v = √(T/μ) by plotting f² vs T or v² vs T: from f = (1/2L)√(T/μ), f² = T/(4L²μ) — gradient gives μ. Melde's experiment in modern form; nodes measured with a paper rider dancing at antinodes.

Measuring Wave Speeds — Sound and Light

Speed of sound: resonance tube — water in a tall cylinder, tuning fork over the top; lower the water until resonance (loudest): first resonance at L = λ/4 (closed pipe), so λ = 4L (+ end correction). With f = 512 Hz fork and L = 0.165 m: λ = 0.66 m, v = fλ ≈ 338 m/s. Repeat with different forks; plot L vs 1/f: gradient = v/4. Temperature correction: v ≈ 331 + 0.6T(°C) m/s — state the room temperature. Echo method: clap 50 m from a wall, time 20 echoes, v = 2×50×20/t — crude but instructive.

Speed of light: too fast for stopwatches — Fizeau's toothed wheel (1849) chopped light into pulses; modern: modulated laser + phase measurement, or the chocolate-and-microwave standing-wave experiment. Worked: in a microwave oven (f = 2.45 GHz from the rating plate), melted spots on a chocolate bar are 6.1 cm apart. Adjacent antinodes of a stationary wave are λ/2 apart, so λ = 0.122 m and c = fλ = 2.45×10⁹ × 0.122 ≈ 2.99×10⁸ m/s — a fundamental constant measured in a kitchen. Uncertainty: the spot spacing is readable to ±0.2 cm, a ±3.3% uncertainty, so c = (2.99 ± 0.10)×10⁸ m/s — consistent with 3.00×10⁸. A-level expects knowledge that c = 3.00×10⁸ m/s in vacuum and how interferometry measures it precisely.

Refraction and wave behaviour at boundaries: v changes → λ changes (f fixed by the source!) → direction changes (Snell's law n₁sinθ₁ = n₂sinθ₂, with n = c/v). Water waves slowing in shallow water bend toward the normal — the same physics as lenses. Total internal reflection (θ > critical angle, sin c = n₂/n₁): fibre optics, endoscopes, broadband — the applications justify the theory.

Doppler Effect and Exam Technique

Doppler: moving source compresses wavefronts ahead (higher f) and stretches behind (lower f). Formula: f' = f·v/(v ∓ v_s) (source moving toward/away); for moving observer: f' = f(v ± v_o)/v. Worked: ambulance siren 700 Hz, approaching at 25 m/s (v = 343 m/s): f' = 700×343/(343−25) ≈ 755 Hz; receding: 700×343/368 ≈ 652 Hz. The pitch drop as it passes is ~100 Hz — unmistakable. Redshift: galaxies' light stretched by cosmic expansion — v = H₀d (Hubble's law, Lesson 18) — the Doppler effect measuring the universe.

Exam technique: (1) always write v = fλ with substitution before numbers; (2) for phase/path problems, convert to fractions of λ first (Δφ = 2πΔx/λ) — geometry before arithmetic; (3) stationary-wave questions: draw the mode shape, mark nodes/antinodes, then apply L = nλ/2 or (2n−1)λ/4 as appropriate — wrong end-conditions are the top error; (4) polarisation proves transverse — state it when asked 'what does this observation tell us about the wave?'; (5) practicals: name the graph, its gradient's meaning, and the control variables.

Common mistakes

• Node–antinode spacing: adjacent node and antinode are λ/4 apart; adjacent nodes are λ/2 apart. Resonant lengths follow from the end conditions, not from memory of 'λ/4' alone.
• Closed-pipe harmonics: only odd harmonics exist (f, 3f, 5f…) — there is no second harmonic in a stopped pipe.
• Phase difference units: Δφ = 2πΔx/λ gives radians; if the question wants degrees or fractions of a cycle, convert at the end.
• Amplitude vs intensity: intensity ∝ amplitude² — doubling amplitude quadruples intensity (and halving intensity at a Polaroid is an amplitude effect, not 'half the waves stopped').
• Doppler signs: write the physical picture (approaching → higher f) before selecting ∓; a formula sign chosen blindly is a coin toss.

Data Analysis — Two Resonances Beat the End Correction

The resonance-tube measurement has a hidden systematic error: the antinode sits slightly beyond the open end (end correction c ≈ 0.6 × tube radius), so a single resonance gives λ = 4(L + c), not 4L — using 4L alone underestimates v. The professional fix is to use TWO resonances of the same fork. For a 512 Hz fork a student finds the first resonance at L₁ = 0.165 m and the second at L₂ = 0.495 m. Successive resonances are λ/2 apart, and the end correction cancels in the difference: λ = 2(L₂ − L₁) = 2(0.330) = 0.660 m, so v = fλ = 512 × 0.660 ≈ 338 m/s. Uncertainty: each length is readable to ±0.002 m, so the difference carries ±0.004 m (±1.2%) and v = 338 ± 4 m/s — and the accepted value at 20 °C (343 m/s) sits just outside that range, prompting the right evaluation discussion: the small remaining shortfall points to the temperature correction (v ≈ 331 + 0.6T) or fork calibration, not to random scatter. Eliminating a systematic error by differencing is a Paper 5 technique worth collecting: the same trick appears in pendulum timing (time 20 swings, not 1) and in potentiometer null measurements.

Exam Experience

(a) Unpolarised light passes through a single Polaroid filter. State what happens to the intensity and explain what this, together with crossed-filter behaviour, shows about light. [2] (b) In a microwave oven (f = 2.45 GHz), melted spots on a bar of chocolate are 6.1 ± 0.2 cm apart. Determine the speed of light with its uncertainty. [4] (c) A guitar string (L = 0.64 m, μ = 1.1 g/m) sounds its fundamental at 196 Hz. Calculate the tension. [3]

Mark scheme (a) Intensity halves [1]; only the component in one plane is transmitted; that a wave can be polarised at all shows it is TRANSVERSE (longitudinal waves have no transverse plane to select). [1] (b) Spots are adjacent antinodes: λ/2 = 6.1 cm → λ = 0.122 m [1]; c = fλ = 2.45×10⁹ × 0.122 = 2.99×10⁸ m/s [2]; uncertainty 0.2/6.1 ≈ 3.3% → c = (2.99 ± 0.10)×10⁸ m/s, consistent with 3.00×10⁸. [1] (c) v = 2Lf₁ = 2 × 0.64 × 196 = 251 m/s [1]; T = μv² = 1.1×10⁻³ × 251² ≈ 69 N. [2]