Waves and Optics
Waves are disturbances that transport energy through space without bulk transport of matter. They can be mechanical (require a medium: sound, water, seismic waves) or electromagnetic (require none: light, radio, X-rays). The study of light, optics, is one of the oldest branches of physics, but its understanding as an electromagnetic wave dates only from Maxwell (1864).
The spatial period of a wave — the distance over which the wave's shape repeats. For a sinusoidal wave it is the distance between successive crests, denoted λ. Wavelength is related to frequency f and wave speed v by v = fλ.
Basic wave parameters
A travelling sinusoidal wave can be written y(x, t) = A sin(kx − ωt + φ), where:
- Amplitude A — maximum displacement.
- Wavelength λ = 2π/k, where k is the wavenumber.
- Frequency f = ω/(2π), where ω is the angular frequency.
- Period T = 1/f.
- Phase velocity v = ω/k = fλ.
Wave intensity is proportional to the square of amplitude: I ∝ A².
Transverse vs longitudinal
- Transverse waves: oscillations are perpendicular to the direction of propagation. Examples: electromagnetic waves, waves on a string.
- Longitudinal waves: oscillations are parallel to the direction of propagation. Example: sound waves in air.
Only transverse waves can be polarised, because longitudinal waves have no transverse component to filter.
Superposition and interference
When two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements (principle of superposition). This produces:
- Constructive interference when waves are in phase (path difference = nλ), giving bright fringes or louder sound.
- Destructive interference when waves are out of phase (path difference = (n + ½)λ), giving dark fringes or silence.
Young's double-slit experiment (1801) demonstrated the wave nature of light. The fringe spacing on a screen at distance D from slits separated by d is:
Δy = λD/d
- Standing waves form when two identical waves travel in opposite directions; they have nodes (zero amplitude) and antinodes (maximum amplitude).
- The fundamental frequency of a string fixed at both ends is f₁ = v/(2L); harmonics are integer multiples.
- Beat frequency between two close frequencies f₁ and f₂ is |f₁ − f₂|.
- Doppler effect changes observed frequency when source or observer moves: f' = f(v ± v_o)/(v ∓ v_s).
Diffraction
Diffraction is the bending of waves around obstacles or through apertures comparable in size to the wavelength. Significant diffraction explains why sound bends around corners while light usually appears to travel in straight lines (its wavelength is far smaller than everyday objects).
For a single slit of width a, the first dark fringe occurs at sin θ = λ/a. For a diffraction grating with N lines per unit length, principal maxima occur at d sin θ = nλ.
Reflection and refraction
Two laws of geometrical optics:
- Law of reflection: angle of incidence = angle of reflection (measured from the normal).
- Snell's law of refraction: n₁ sin θ₁ = n₂ sin θ₂, where n is the refractive index.
Refractive index n = c/v, where v is the speed of light in the medium. n > 1 for all real media (e.g., n = 1.33 for water, n = 1.5 for glass, n ≈ 2.4 for diamond).
Total internal reflection
When light passes from a denser to a less dense medium and the angle of incidence exceeds the critical angle θ_c (sin θ_c = n₂/n₁), it is completely reflected. This is the basis of fibre-optic communication and gives diamond its sparkle.
Lenses and mirrors
The lens equation 1/f = 1/v − 1/u relates focal length f, image distance v and object distance u (sign convention dependent). The lensmaker's equation gives focal length in terms of curvatures and refractive index. Magnification m = v/u = h_image/h_object.
| Optical element | Image (object beyond focal point) |
|---|---|
| Convex lens | Real, inverted, on opposite side |
| Concave lens | Virtual, upright, smaller |
| Concave mirror | Real, inverted (when object beyond C) |
| Convex mirror | Virtual, upright, smaller |
Polarisation
Light is naturally unpolarised: its E-field vibrates in random transverse directions. A polariser transmits only one direction; Malus's law gives the transmitted intensity through a second polariser at angle θ: I = I₀ cos² θ.
Brewster's angle tan θ_B = n₂/n₁ is the angle at which reflected light is completely plane-polarised — used in polarised sunglasses and laser optics.
Remember the visible spectrum mnemonic VIBGYOR (Violet, Indigo, Blue, Green, Yellow, Orange, Red). Violet has the shortest wavelength and is refracted most by a prism; red is refracted least. This dispersion produces the rainbow.
Sound
Sound is a longitudinal pressure wave. Its speed in air at 20 °C is about 343 m/s, in water about 1480 m/s, in steel about 5960 m/s. The decibel scale measures sound intensity logarithmically: β = 10 log₁₀(I/I₀), with I₀ = 10⁻¹² W/m² (the threshold of hearing).
Together, waves and optics underpin modern communications, medical imaging (ultrasound, MRI), laser technology and quantum optics — fields where the wave description of light meets its particle-like (photon) behaviour.