Module 02 — Waves
Duration: 2–3 weeks | Prereq: Module 01
Waves are how energy moves. From sound to light to gravitational ripples in spacetime, the same mathematics describes an enormous range of physical phenomena. This module is a bridge — it connects mechanics to electricity (electromagnetic waves) and to cosmology (redshift, the expanding universe, the cosmic microwave background).
1. What Is a Wave?
A wave is a disturbance that transfers energy through a medium (or through space, in the case of electromagnetic waves) without transferring matter.
The analogy: When a crowd does a Mexican wave, each person bobs up and down but stays in the same seat. Energy travels around the stadium; the people don't.
Types of wave
Transverse waves: oscillations are perpendicular to the direction of travel.
Examples: light, water waves, seismic S-waves, waves on a string.
Longitudinal waves: oscillations are parallel to the direction of travel. The medium compresses and rarifies.
Examples: sound, seismic P-waves, ultrasound.
2. Wave Properties
Key quantities
| Property | Symbol | Definition | Unit |
|---|---|---|---|
| Wavelength | λ (lambda) | Distance between two successive points in phase (e.g., peak to peak) | m |
| Frequency | f | Number of complete waves passing a point per second | Hz (s⁻¹) |
| Period | T | Time for one complete wave | s |
| Amplitude | A | Maximum displacement from equilibrium | m (or same unit as displacement) |
| Wave speed | v or c | Speed at which the wave pattern travels | m/s |
The wave equation
v = fλ
This is one of the most-used equations in physics. Memorise it.
Speed of a wave = its frequency × its wavelength.
Note: For a given medium, wave speed is fixed. If frequency increases, wavelength must decrease (and vice versa). This matters hugely for light and sound.
Worked Example: A sound wave has frequency 440 Hz (concert A) and wavelength 0.773 m. What is its speed?
v = 440 × 0.773 = 340 m/s (approximately the speed of sound in air)
Relationship between T and f: f = 1/T
3. Wave Behaviour
Reflection
A wave bounces off a surface. Angle of incidence = angle of reflection (measured from the normal — the line perpendicular to the surface).
The "normal" convention appears everywhere in optics. Always measure angles from the normal, not the surface.
Refraction
When a wave passes from one medium to another, its speed changes. If it hits the boundary at an angle, it bends — this is refraction.
Snell's Law: n₁ sin θ₁ = n₂ sin θ₂
Where n is the refractive index of each medium, and θ is the angle to the normal.
Refractive index: n = c / v (speed of light in vacuum / speed of light in the medium)
Glass has n ≈ 1.5, meaning light travels at 2/3 of its vacuum speed inside glass.
Why does refraction bend the wave? Think of a car driving from tarmac onto sand at an angle. The wheel that hits sand first slows down while the other wheel is still fast. This causes the car (and the wavefront) to turn.
Total internal reflection
If light in a dense medium hits the boundary at a large angle, it doesn't exit — it reflects internally. This only works going from dense to less dense (e.g., glass to air, not air to glass).
The critical angle θ_c, in general: sin θ_c = n₂ / n₁ (where n₁ > n₂ is the denser medium)
When the second medium is air (n₂ = 1): sin θ_c = 1/n₁, often written as sin θ_c = 1/n.
Above the critical angle → total internal reflection.
Applications: optical fibres (broadband internet, endoscopes), diamonds (the cut maximises total internal reflection for sparkle).
Diffraction
Waves spread out when they pass through a gap or around an obstacle. The spreading is most pronounced when the gap width ≈ the wavelength.
This is why you can hear around a corner (sound has wavelengths of cm to metres, comparable to door gaps) but not see around one (light has wavelengths of ~500 nm, far smaller than typical gaps).
4. Superposition and Interference
The principle of superposition
When two waves meet, the total displacement at any point is the sum of the individual displacements.
Constructive and destructive interference
Constructive interference: waves add together → bigger amplitude. This happens when waves are in phase (peaks align with peaks).
Destructive interference: waves cancel → smaller amplitude (zero if amplitudes are equal). This happens when waves are in antiphase (peaks align with troughs).
Path difference: if two coherent waves travel different distances to reach a point, they arrive with different phases.
- Path difference = nλ (whole number of wavelengths) → constructive interference
- Path difference = (n + ½)λ → destructive interference
Young's double-slit experiment
Thomas Young's 1801 experiment proved that light is a wave by showing that it produces interference patterns.
Two coherent light sources (two slits lit by the same source) produce alternating bright and dark fringes on a screen.
Fringe spacing: w = λD / s
Where w = fringe spacing, λ = wavelength, D = distance to screen, s = slit separation.
This equation lets you measure the wavelength of light using only a ruler.
5. Standing Waves
When a wave reflects back on itself and the two waves interfere, a standing wave forms. Unlike a travelling wave, a standing wave doesn't move — it oscillates in place.
Nodes: points of permanent destructive interference — zero amplitude. These don't move. Antinodes: points of permanent constructive interference — maximum amplitude.
Adjacent node-to-node or antinode-to-antinode distance = λ/2.
Harmonics
A string fixed at both ends supports standing waves where the string length L is a whole number of half-wavelengths: L = nλ/2, so λ = 2L/n.
The fundamental frequency (n=1): f₁ = v / 2L
Higher harmonics (overtones): f₂ = 2f₁, f₃ = 3f₁, etc.
This is why guitar strings make musical notes: the length, tension, and density of the string determine which frequencies resonate. Pressing a fret shortens the vibrating length, raising the frequency (higher pitch).
6. The Electromagnetic Spectrum
Light is an electromagnetic wave — oscillating electric and magnetic fields that travel through a vacuum at the speed of light:
c = 3 × 10⁸ m/s (memorise this — it appears everywhere)
All electromagnetic waves travel at c in a vacuum. They differ only in frequency (and therefore wavelength: λ = c/f).
The full spectrum, from longest to shortest wavelength:
| Type | Wavelength range | Frequency range |
|---|---|---|
| Radio | > 0.1 m | < 3 × 10⁹ Hz |
| Microwave | 1 mm – 0.1 m | 3 GHz – 300 GHz |
| Infrared | 700 nm – 1 mm | 300 GHz – 430 THz |
| Visible light | 400 nm – 700 nm | 430 THz – 750 THz |
| Ultraviolet | 10 nm – 400 nm | 750 THz – 30 PHz |
| X-ray | 0.01 nm – 10 nm | 30 PHz – ~10 EHz |
| Gamma ray | < ~0.01 nm | > ~10 EHz |
Visible light is an astonishingly thin slice. The entire visible spectrum spans less than one octave of frequency. The rest of the EM spectrum spans about 25 octaves.
[Connection to Cosmology] The Cosmic Microwave Background — the afterglow of the Big Bang — is microwave radiation. When the universe was young and hot, that radiation was in the visible/UV range. As the universe expanded, the wavelengths were stretched (redshifted) into microwaves. Module 05 picks this up.
7. The Doppler Effect
When a source of waves moves relative to an observer, the observed frequency changes.
- Source moving toward observer → waves bunched up → shorter wavelength → higher frequency (blue-shifted for light)
- Source moving away from observer → waves stretched out → longer wavelength → lower frequency (red-shifted for light)
For sound: f_observed = f_source × (v ± v_observer) / (v ∓ v_source)
Where v is the speed of sound. Sign rule: use + for the numerator when the observer moves toward the source; use − for the denominator when the source moves toward the observer (and the opposite signs for moving away). An easier way to remember: movement that brings source and observer closer together raises frequency; movement that separates them lowers it.
For light (and cosmology): The redshift z = Δλ/λ ≈ v/c (for speeds much less than c)
Why this matters enormously for cosmology: Edwin Hubble observed that virtually all distant galaxies are redshifted — they're moving away from us. The further away a galaxy, the faster it recedes. This was the evidence that the universe is expanding. Module 05 goes into this in detail.
8. Polarisation
Transverse waves can oscillate in any plane perpendicular to their travel direction. Polarisation restricts oscillation to one plane.
Light from the Sun is unpolarised — it oscillates in all planes. A polarising filter only transmits one plane.
Two polarising filters at 90° to each other → no light passes through.
Applications: Polaroid sunglasses (reduce glare from reflected light, which is partially polarised), 3D cinema glasses, LCD screens.
Key point: Only transverse waves can be polarised. Longitudinal waves (like sound) cannot. This is experimental evidence that light is transverse.
9. Self-Check Questions
- A radio wave has frequency 100 MHz. What is its wavelength?
- Light passes from glass (n = 1.5) into air. What is the critical angle?
- In Young's double-slit experiment, the slits are 0.5 mm apart and the screen is 1.5 m away. Green light (λ = 550 nm) is used. What is the fringe spacing?
- A guitar string vibrates at its fundamental frequency of 200 Hz. What is the frequency of the second harmonic?
- A police siren emits sound at 800 Hz. A car is driving away from the police at 30 m/s. What frequency does the driver hear? (speed of sound = 340 m/s)
Answers:
- λ = c/f = 3×10⁸ / 10⁸ = 3 m
- sin θ_c = 1/1.5 = 0.667 → θ_c = sin⁻¹(0.667) = 41.8°
- w = λD/s = (550×10⁻⁹ × 1.5) / (0.5×10⁻³) = 825×10⁻⁹ / 5×10⁻⁴ = 1.65×10⁻³ m = 1.65 mm
- Second harmonic = 2 × fundamental = 400 Hz
- The observer (driver) is moving away from a stationary source (police). Use f = f_source × (v − v_observer) / v = 800 × (340 − 30) / 340 = 800 × 310/340 = 729 Hz
Go Deeper
- Fourier analysis — any wave can be built from combinations of sine waves. This underpins signal processing, music, MRI scanning, and quantum mechanics. The concept is beautiful and surprisingly accessible.
- Gravitational waves — ripples in spacetime itself, detected by LIGO in 2015. These are waves described by the same mathematics you just learned, but the "medium" is the fabric of space. Module 05 touches on this.
- Wave-particle duality — light behaves as both a wave and a particle. Module 04 (Atomic & Quantum) explores this — it overturns everything "obvious" about what waves are.
- Sixty Symbols on YouTube — the episodes on the double-slit experiment, total internal reflection, and the Doppler effect are all excellent.