Everything begins
with a wave.

We use a periodic, zero-mean signal with nine possible harmonics. A 512-point discrete Fourier transform measures aₙ = (2/N)Σ fⱼ cos(2πnj/N) and bₙ = (2/N)Σ fⱼ sin(2πnj/N). Orthogonality separates the frequencies. For our sine-plus-phase convention, aₙ = Aₙ sin φₙ and bₙ = Aₙ cos φₙ. Energy is the period average of f², proportional to Σ Aₙ². The dashed curve keeps only frequencies at or below the selected cutoff.

Square and sawtooth presets are truncated Fourier series, not exact discontinuous waves. Amplitudes are dimensionless; the fundamental frequency is 1/T.

The equations here use SI units in vacuum. E is electric field, B magnetic field, Q enclosed charge, I enclosed conduction current, and Φ the flux through a surface. ε₀ and μ₀ are the vacuum constants. A closed-surface integral is marked ∯; a closed-loop integral is marked ∮.

The electric view is a cross-section of an isolated point charge. The magnetic view shows closed field lines around a long straight wire; the rectangular surface section has as much inward as outward magnetic flux. Faraday’s view assumes a prescribed, spatially uniform B(t) within a disk of radius R = 1 m. At its rim, Eθ = −(R/2)dB/dt. Ampère–Maxwell’s view looks along the axis of an ideal circular capacitor gap with uniform E(t); I = 0 in the gap and Bθ = (μ₀ε₀R/2)dE/dt at the rim. Both changing-field examples are local, quasistatic idealizations that neglect edge effects; they are not full radiation solutions. Their physical cycle is 1 second, animated more slowly. Field-line spacing and arrows are illustrative.

This is a linearly polarized plane wave in vacuum. It shows field values along a line, not a material ribbon moving through space. Cyan is E and coral is cB, so the drawn amplitudes match even though E and B have different units. Time is slowed enormously, and the axes are diagrammatic. k = 2π/λ, ω = 2πf, c = 299,792,458 m/s. The photon readout gives energy per photon, not total beam intensity; this classical field picture does not visualize an individual photon.

The transmitted field amplitude is E₀ cos θ. Intensity is proportional to the time average of the field squared, giving I = I₀ cos² θ. I₀ here is the intensity of an already linearly polarized input beam; the one-half factor for initially unpolarized light is not part of this example. Directions are measured counterclockwise from the horizontal and polarization axes repeat every 180°.

These are dimensionless conjugate coordinates x and k with a fixed mean wave number k₀ = 5. The plotted Gaussian densities each integrate to 1. Δ denotes standard deviation, and the shaded bands mark ±1Δ. The real-space density is √(2/π) Δk exp(−2Δk²x²); the complex amplitude is proportional to exp(−Δk²x²) exp(ik₀x). The distributions use independent vertical scales, labeled on each plot. This is a snapshot at t = 0, with no time evolution or measurement collapse simulated.

All states satisfy Δx Δp ≥ ℏ/2. This unchirped Gaussian reaches equality. The relation comes from the wavefunction’s Fourier structure, rather than an imperfection in a measuring device.