# EM Wave Propagation

> Light is E and B regenerating each other. A 1D Yee grid (FDTD) leapfrogs them in time, slows them in glass and blows up past the Courant limit.

- Category: Physics › Electromagnetism
- Level: advanced
- Tags: maxwell, fdtd, yee-grid, courant, waves
- Updated: 2026-11-01
- URL: https://gdcs.coody.app/physics/electromagnetism/em-wave-propagation

## What it is

An electromagnetic wave is a changing electric field that creates a changing magnetic field, which creates the electric field again. In a plane wave E and B are perpendicular to each other and to the direction of travel, in phase, and move at `c = 1/√(με)`.

The **finite-difference time-domain** (FDTD) method solves Maxwell's curl equations directly. Kane Yee's trick is to store E and H on grids offset by half a cell in space and half a step in time, so every derivative is a centred difference. Each step updates H from the spatial change of E, then E from the spatial change of H: a leapfrog.

The scheme is explicit, so it has a stability limit. The **Courant number** `S = cΔt/Δx` must not exceed 1 in 1D (1/√2 in 2D, 1/√3 in 3D); above it, errors grow every step until the grid fills with garbage. Inside a dielectric with permittivity εr the wave slows by `√εr`, its wavelength shrinks, and part of it reflects at the boundary.

The same leapfrog pattern runs water ripples, audio propagation and [sound waves](https://gdcs.coody.app/physics/mechanics/sound-waves) in games; it is the grown-up version of the [fixed timestep](https://gdcs.coody.app/systems/simulation/fixed-timestep) rule that stability depends on step size.

## Key formulas

- **Maxwell's curl equations (1D, Ey and Hz):** $\varepsilon\frac{\partial E_y}{\partial t} = -\frac{\partial H_z}{\partial x}, \qquad \mu\frac{\partial H_z}{\partial t} = -\frac{\partial E_y}{\partial x}$
- **Yee update (normalised: c = 1, Δx = 1):** $H_{i+\frac12}^{n+\frac12} = H_{i+\frac12}^{n-\frac12} - S\,(E_{i+1}^n - E_i^n), \quad E_i^{n+1} = E_i^n - \frac{S}{\varepsilon_r}\,(H_{i+\frac12}^{n+\frac12} - H_{i-\frac12}^{n+\frac12})$
- **Courant stability:** $S = \frac{c\,\Delta t}{\Delta x} \le \frac{1}{\sqrt{d}}$ — d = number of spatial dimensions.
- **Speed and wavelength in a dielectric:** $v = \frac{c}{\sqrt{\varepsilon_r}}, \qquad \lambda = \frac{\lambda_0}{\sqrt{\varepsilon_r}}$
- **Mur absorbing boundary (first order):** $E_0^{n+1} = E_1^{n} + \frac{S-1}{S+1}\left(E_1^{n+1} - E_0^{n}\right)$

## Rules of thumb

- Resolve the shortest wavelength (inside the densest material) with at least 10–20 cells, or numerical dispersion slows short waves.
- Use S ≈ 0.9 of the limit; in 1D, S = 1 is the 'magic time step' where a pulse moves exactly one cell per step with zero dispersion.
- Ramp a sinusoidal source over 1–2 periods; switching it on instantly launches a broadband spike.
- A soft source (add to E) lets reflected waves pass through it; a hard source (overwrite E) reflects them.
- Without an absorbing boundary the grid ends are perfect mirrors and you get standing waves.

## Implementation

One 1D Yee step with a soft source and Mur boundaries:

```ts
export function step(ey: Float64Array, hz: Float64Array, eps: Float64Array,
                     S: number, src: number, value: number, mur: Float64Array) {
  const n = ey.length
  mur[0] = ey[0]; mur[1] = ey[1]; mur[2] = ey[n - 1]; mur[3] = ey[n - 2]
  for (let i = 0; i < n - 1; i++) hz[i] -= S * (ey[i + 1] - ey[i])
  for (let i = 1; i < n - 1; i++) ey[i] -= (S / eps[i]) * (hz[i] - hz[i - 1])
  ey[src] += value // soft source
  const m = (S - 1) / (S + 1)
  ey[0] = mur[1] + m * (ey[1] - mur[0])
  ey[n - 1] = mur[3] + m * (ey[n - 2] - mur[2])
}
```

## Common mistakes

- Storing E and H at the same grid points and times, which loses the second-order accuracy and the stability of the staggered scheme.
- Picking Δt from the frame rate instead of from the Courant limit; a slow frame then makes S > 1 and the sim explodes.
- Checking stability with the slow speed inside a dielectric. The limit is set by the fastest wave on the grid (vacuum).
- Forgetting an absorbing boundary and mistaking the resulting standing wave for physics.

## Interactive visual

Cyan is the electric field Ey (vertical), violet the magnetic field Hz (horizontal), launched from the orange source and absorbed at both ends; the translucent block is a dielectric slab. Raise εr to slow and shorten the wave inside the slab, and push the Courant number past 1 to watch the explicit scheme blow up.

## Prompt

```text
Write a 1D FDTD (Yee) electromagnetic solver in TypeScript for a Three.js visual. Use normalised units (c = 1, Δx = 1) with Float64Array Ey on n nodes, Hz on n−1 half nodes and a per-node relative permittivity array. stepFdtd(grid, S, sourceIndex, source(t)) updates H then E, adds a soft source, applies first-order Mur boundaries at both ends and allocates nothing. Render Ey along y and Hz along z by writing into existing line geometries each frame, clamping non-finite values. Tests: at S = 1 a matched Gaussian pulse moves exactly one cell per step; at S = 0.5 the energy after the pulse exits is < 2% of the start (Mur absorbs); at S = 1.1 energy exceeds 1e6 within 400 steps.
```

## Related

- [Electric Field Lines](https://gdcs.coody.app/physics/electromagnetism/electric-field-lines.md) — Coulomb fields add as vectors; tracing streamlines along the net field draws lines that leave positive charges, end on negative ones and never cross.
- [Sound Waves](https://gdcs.coody.app/physics/mechanics/sound-waves.md) — Sound is a longitudinal pressure wave: frequency sets pitch and wavelength, amplitude sets loudness. The physics behind 3D audio, delay and Doppler.
- [Fixed Timestep](https://gdcs.coody.app/systems/simulation/fixed-timestep.md) — Run physics in constant dt slices from an accumulator, clamp long frames, and interpolate between the last two states for smooth, repeatable motion.

## Further reading

- [OpenStax University Physics Vol. 2 — Plane Electromagnetic Waves](https://openstax.org/books/university-physics-volume-2/pages/16-2-plane-electromagnetic-waves)
- [John B. Schneider — Understanding the FDTD Method (free textbook)](https://eecs.wsu.edu/~schneidj/ufdtd/)

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Source: Game Development Cheat Sheet (https://gdcs.coody.app)