Introduction
This chapter connects two ideas that appear separate at first but are deeply related: oscillation (repeated back-and-forth motion at one point) and wave propagation (that oscillation travelling through space). Sound, light, and the electromagnetic spectrum are all waves whose behaviour is built from these foundations, so the chapter ends in optics — the study of light as a wave (and, where relevant, as a stream of photons).
1. Free oscillations, one and two degrees of freedom
A free oscillation occurs when a system, displaced from equilibrium, experiences a restoring force proportional to displacement and oscillates without external driving or damping — the defining example is simple harmonic motion (SHM), x(t) = A cos(ωt + φ), seen in an ideal spring–mass system or small-angle pendulum. The restoring force F = −kx (Hooke’s law for a spring) always points back toward equilibrium, producing sinusoidal motion with angular frequency ω = √(k/m).
A system with two degrees of freedom (for example, two masses connected by springs) supports normal modes — special patterns of motion in which all parts of the system oscillate at the same frequency. Any general motion of the coupled system can be written as a combination (superposition) of these normal modes, a technique that generalises to describe vibrations in molecules, bridges, and crystal lattices (relevant again in Solid State Physics).
2. Forced and damped oscillations, resonance
Real oscillators lose energy to friction or air resistance — this is damping, which causes the amplitude to decay exponentially over time. If an external periodic force is applied (forced oscillation), the system eventually oscillates at the driving frequency rather than its own natural frequency.
Resonance occurs when the driving frequency approaches the system’s natural frequency: the amplitude of oscillation becomes very large, because energy is fed into the system in phase with its own motion on every cycle. Resonance explains why soldiers break step crossing a bridge (to avoid matching the bridge’s natural frequency), why a radio tuner selects one station (an electrical circuit resonating at the station’s frequency), and why a wine glass can shatter at a matching sound frequency.
3. Travelling and standing waves, wave energy, phase and group velocity, sound
A travelling wave, y(x,t) = A sin(kx − ωt), carries energy and momentum through a medium without net transport of the medium’s material. Its phase velocity (v = ω/k) is the speed at which a single crest moves; its group velocity (dω/dk) is the speed at which a pulse or wave packet — and therefore information or energy — actually travels, which can differ from phase velocity in a dispersive medium.
A standing wave forms when two identical waves travel in opposite directions (for example, a wave and its reflection), producing fixed points of zero displacement (nodes) and maximum displacement (antinodes) that do not move along the medium. Standing waves on a string or in an air column explain musical instruments: only certain wavelengths “fit” the boundary conditions of the string or pipe, producing discrete resonant frequencies (the fundamental and its overtones).
Sound is a longitudinal pressure wave requiring a medium; its speed depends on the medium’s stiffness and density (faster in solids than gases, generally).
4. Reflection, refraction, interference, diffraction, polarization
- Reflection and refraction describe what happens when a wave meets a boundary between media: reflection bounces the wave back, refraction bends it as it enters a new medium at an angle (Snell’s law, n₁sinθ₁ = n₂sinθ₂), because the wave’s speed changes with the medium.
- Interference is the superposition of two or more coherent waves, producing constructive interference (crests align, amplitude adds) or destructive interference (crest meets trough, amplitude cancels) — the basis of Young’s double-slit experiment, which was historically decisive evidence for the wave nature of light.
- Diffraction is the bending of waves around obstacles or through apertures, most noticeable when the aperture size is comparable to the wavelength; it sets a fundamental limit on the resolving power of optical instruments.
- Polarization applies only to transverse waves (like light): it describes the orientation of the oscillation direction. Unpolarised light has oscillations in all perpendicular directions; a polarising filter transmits only one orientation, which is why polarised sunglasses reduce glare (reflected light is partially polarised).
5. Interferometers, Newton’s rings, diffraction grating, resolving power, spectrometers
These are practical instruments built from the wave phenomena above. A Michelson interferometer (already met in the mechanics chapter’s discussion of the Michelson–Morley experiment) splits and recombines light beams to detect extremely small path-length differences via interference fringes. Newton’s rings are concentric interference fringes formed between a curved lens and a flat glass plate, used to measure the curvature of lenses or the wavelength of light. A diffraction grating (many closely spaced slits) produces sharp interference maxima at angles depending on wavelength, making it central to spectroscopy — separating light into its component wavelengths far more precisely than a prism. Resolving power describes an instrument’s ability to distinguish two closely spaced wavelengths or two close objects, fundamentally limited by diffraction (the Rayleigh criterion). A spectrometer combines a diffraction grating (or prism) with precise angle measurement to determine the exact wavelengths present in light — essential for identifying elements from their emission spectra.
6. Electromagnetic wave equation, dispersion, coherence, lasers
Maxwell’s equations (met more fully in Electricity and Magnetism) predict that oscillating electric and magnetic fields propagate as a wave at speed c = 1/√(μ₀ε₀), matching the measured speed of light — this was the theoretical confirmation that light itself is an electromagnetic wave.
Dispersion is the variation of a medium’s refractive index with wavelength, causing white light to spread into a spectrum when passed through a prism. Normal dispersion (refractive index decreases with increasing wavelength) is typical in transparent materials; anomalous dispersion occurs near a material’s absorption bands, where this trend reverses.
Coherence describes the extent to which a light source’s waves maintain a constant phase relationship over time and space — essential for producing stable interference patterns. A laser (Light Amplification by Stimulated Emission of Radiation) produces highly coherent light through stimulated emission: a photon of the right energy triggers an excited atom to emit an identical photon (same phase, direction, and wavelength). This only produces net amplification if a population inversion is achieved — more atoms in the excited state than the ground state, normally the reverse of thermal equilibrium — typically created by “pumping” energy into the lasing medium.
Key Takeaways
- Simple harmonic motion is the basic building block of oscillation; coupled systems oscillate in normal modes, and any driven system near its natural frequency shows resonance.
- Waves carry energy without net transport of matter; phase velocity (a point’s speed) can differ from group velocity (a pulse’s, and information’s, speed) in dispersive media.
- Interference, diffraction and polarization are the three signature wave behaviours; diffraction fundamentally limits the resolving power of every optical instrument.
- Interferometers, Newton’s rings, diffraction gratings and spectrometers are practical instruments built directly from interference and diffraction.
- Light is an electromagnetic wave (from Maxwell’s equations); a laser produces coherent light via stimulated emission, which requires an artificially created population inversion.