General wave theory independent of medium: the wave equation, superposition, interference, Doppler effect, Huygens' principle. A crosscutting tree referenced by EM (light), quantum (matter waves), and mechanics (sound, seismic).
wave-phenomena
Wave equation
Linear PDE governing the propagation of a disturbance ψ at speed v: ∂²ψ/∂t² = v²∇²ψ.
Superposition principle
For linear systems, the response to a sum of inputs equals the sum of individual responses. Underlies interference and diffraction.
Doppler effect
Observed frequency of a wave depends on the relative velocity of source and observer.
Huygens' principle
Every point on a wavefront acts as a source of spherical wavelets; the new wavefront is the envelope of these wavelets.
Electromagnetic wave propagation
In vacuum, Maxwell's equations yield transverse electromagnetic waves travelling at c.
Dispersion relation
Functional relation ω(k) between angular frequency and wavenumber for a medium/mode. Distinguishes non-dispersive (ω ∝ k) from dispersive…
Phase velocity
Speed v_p = ω/k at which a single-frequency wave's crests propagate. Can exceed c in a medium (vacuum dispersion), but carries no…
Group velocity
Speed v_g = ∂ω/∂k at which a wave packet (envelope) — hence energy/information — propagates. Subluminal in causal media.
Shock wave
Discontinuous jump in density/pressure/velocity arising from nonlinear wave steepening when characteristics cross (e.g., supersonic flow,…
Soliton
Localised, shape-preserving traveling-wave solution of a nonlinear dispersive PDE (KdV, sine-Gordon, NLS). Balances dispersion against…
Solitons (KdV, NLS)
Localized wavepackets preserving shape via nonlinearity-dispersion balance. KdV for shallow water, NLS for optical fibers. Inverse…
Wave-packet dispersion
Gaussian packet spreads as σ(t) = σ₀√(1 + (t/τ)²); group velocity ≠ phase velocity when dispersion present. Pulse broadening in optical…
Metamaterials & negative-index
Engineered subwavelength structures with effective ε<0 and µ<0 → negative refractive index. Veselago 1968 prediction; Smith-Schultz-Shelby…
Coherence (spatial + temporal)
Temporal (Δν·Δt ≈ 1) + spatial (Van Cittert-Zernike) coherence govern interference visibility. Stellar interferometry, optical coherence…
Dispersion relation ω(k)
Relates frequency to wavenumber for given medium; group velocity v_g = dω/dk vs phase velocity v_p = ω/k; governs pulse propagation.
Phase vs group velocity
v_p = ω/k transports wave crests; v_g = dω/dk transports envelope/energy; anomalous when dn/dω<0; can differ in sign.
WKB approximation for waves
ψ ≈ A(x) e^(iφ(x)) with slowly-varying A, k(x); connection at classical turning points via Airy functions; caustics amplify amplitude.
Fermat's principle (stationary optical path)
Light paths extremize optical path length ∫n ds; derives Snell's law, reflection, curved-mirror focus; geometric optics limit.
Snell–Descartes law and TIR
n₁ sin θ₁ = n₂ sin θ₂; total internal reflection at θ_c = sin⁻¹(n₂/n₁); evanescent wave in forbidden medium (frustrated TIR).
Babinet's principle
Diffraction by complementary screens sum to unobstructed field; U₁ + U₂ = U_free; exchange of opaque/clear regions.
Bragg's law
2d sin θ = nλ for constructive reflection from parallel crystal planes; X-ray crystallography, neutron diffraction; 1915 Nobel.
Standing waves and normal modes
Superposition of counter-propagating waves; nodes/antinodes fixed in space; cavity modes quantize frequency via boundary conditions.
Gaussian pulse dispersion
Narrow pulse broadens: σ(z) = σ₀√(1+(z/z_d)²) where z_d ~ σ₀²/|β₂|; group velocity dispersion in optical fibers.
Korteweg–de Vries soliton
η(x,t) = A sech²((x-ct)/L) balances dispersion vs nonlinearity; integrable via inverse scattering; tsunami-like in shallow water.
Wave impedance and matching
Z = √(μ/ε) for EM, √(ρB) for acoustic; reflection r = (Z₂-Z₁)/(Z₂+Z₁); quarter-wave antireflection coatings.
Classical Doppler effect
f' = f (c±v_obs)/(c∓v_src); different for longitudinal vs transverse; foundation for radar, sonar, LIDAR.
Shock waves & Rankine–Hugoniot
Discontinuous jumps in ρ, v, P across shock; [ρv]=0, [P+ρv²]=0, [h+v²/2]=0; Mach cone sin μ = 1/M.
Evanescent waves
Exponentially-decaying non-propagating fields beyond TIR boundary or cut-off waveguides; near-field microscopy, frustrated TIR coupling.
Kramers–Kronig relations
Re χ(ω) and Im χ(ω) of causal linear response connected by Hilbert transform; imposes n(ω) and α(ω) consistency.
Wavefront aberrations (Zernike)
Aberration W(ρ,θ) = Σc_nm Z_nm expanded in orthonormal Zernike polynomials; adaptive-optics correction in astronomy, ophthalmology.