Waves

Waves transfer energy without transferring matter. Understanding wave behaviour — including reflection, refraction, diffraction, interference, and the wave-particle duality — is essential for explaining phenomena from sound to light to quantum mechanics.

Topics Covered

Wave Properties

• Progressive waves — transverse (displacement perpendicular to propagation: light, electromagnetic) vs. longitudinal (displacement parallel to propagation: sound)
• Wave terms — amplitude, wavelength $\lambda$, frequency $f$, period $T = \frac{1}{f}$, wave speed $v = f\lambda$
• Phase and phase difference — in phase ($\Delta\phi = 0$ or $2\pi$), antiphase ($\Delta\phi = \pi$); path difference $\Delta x = \frac{\lambda \Delta\phi}{2\pi}$
• Electromagnetic spectrum — radio, microwave, infrared, visible, ultraviolet, X-ray, gamma; all travel at $c = 3.0 \times 10^8\,\text{m/s}$ in a vacuum

Superposition and Interference

• Principle of superposition — resultant displacement is the sum of individual displacements
• Constructive interference — path difference $= n\lambda$; amplitudes add
• Destructive interference — path difference $= (n + \frac{1}{2})\lambda$; amplitudes cancel
• Two-source interference — Young’s double slit: fringe spacing $\Delta y = \frac{\lambda D}{s}$; coherent sources required
• Diffraction gratings — $d\sin\theta = n\lambda$; calculating wavelength or grating spacing; resolving power
• Stationary (standing) waves — formed by superposition of two progressive waves travelling in opposite directions; nodes (zero amplitude) and antinodes (maximum amplitude); fundamental frequency and harmonics

Refraction and Total Internal Reflection

• Snell’s law — $n_1 \sin\theta_1 = n_2 \sin\theta_2$; refractive index $n = \frac{c}{v}$
• Total internal reflection (TIR) — occurs when $\theta > \theta_c$ and light travels from a more dense to less dense medium; critical angle $\sin\theta_c = \frac{n_2}{n_1}$ ($n_1 > n_2$)
• Optical fibres — TIR in the core; cladding with lower refractive index; applications in communications and medicine; modal and material dispersion
• Lenses — converging and diverging; focal length, principal focus, magnification $m = \frac{v}{u}$; the lens equation $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

Polarisation

Polarisation is evidence for the transverse nature of electromagnetic waves. Only transverse waves can be polarised — longitudinal waves (sound) cannot.

• Polarisation by reflection — light reflected from a non-metallic surface is partially polarised; the reflected light vibrates in one plane
• Polarising filters — transmit only one plane of vibration; rotating the filter varies the transmitted intensity from maximum to zero (when crossed at 90°)
• Malus’s law — $I = I_0 \cos^2\theta$, where $\theta$ is the angle between the polariser and the analyser
• Applications — sunglasses (reduce glare from reflective surfaces), LCD screens, stress analysis in engineering

Stationary vs. Progressive Waves

Property	Progressive Wave	Stationary Wave
Energy transfer	Yes	No (energy stored)
Amplitude	Constant for all points	Varies: zero at nodes, maximum at antinodes
Phase	Changes along the wave	All points between nodes are in phase
Wavelength	Distance between consecutive identical points	Twice the distance between adjacent nodes

Study Tips

1. Draw wave diagrams — sketch displacement-distance and displacement-time graphs for transverse and longitudinal waves. Label amplitude, wavelength, and period.
2. Understand coherence — interference requires coherent sources (constant phase relationship). In exams, always mention this when describing interference experiments.
3. Practise fringe spacing calculations — Young’s double slit and diffraction grating problems are standard exam fare. Know the derivations, not just the formulas.
4. Stationary vs. progressive waves — be able to compare them: stationary waves store energy, progressive waves transfer energy; stationary waves have nodes, progressive waves do not.
5. For TIR problems — always check two conditions: (1) light travels from more dense to less dense medium, AND (2) angle of incidence exceeds the critical angle.

How to Use These Notes

Follow the sidebar order. Each page provides physical principles, derivations, worked examples with diagrams, and exam-style problems. Start with wave properties, then superposition and interference, then refraction and TIR.