The Electromagnetic Wave Equation

The crown article of E6 ended with a thunderclap: chase Maxwell's two curls into each other and empty space supports self-sustaining waves at $1/\sqrt{\mu_0\varepsilon_0}$ — a number that, confronted with the 1860s measurements of light, remade physics. That was the wave equation as revelation. This chapter — electromagnetic waves, twenty articles of them — needs the wave equation as machinery, and the machinery deserves its own opening treatment: not "does light exist?" (settled) but "what kind of mathematical object governs it, what do its solutions look like in general, what does it fail to contain, and how do sources enter?" Four upgrades over the crown's derivation, each load-bearing for the chapter ahead. First, sources kept: the crown set $\rho = \mathbf{J} = 0$ before deriving; here the curl-of-curl maneuver runs with sources aboard, yielding the inhomogeneous equations whose right-hand sides — gradients of charge density, time-derivatives of current — are a preview of the entire theory of radiation (what shakes, radiates). Second, the general solution: d'Alembert's 1747 result that every one-dimensional solution is exactly two rigid profiles passing through each other at $c$ — arbitrary shapes, no dispersion, superposition exact — the structural facts every later article will spend. Third, the operator: the d'Alembertian $\Box = \nabla^2 - \tfrac{1}{c^2}\partial_t^2$ , whose factorization is d'Alembert's solution and whose Lorentz invariance (E10's business, named here) is why relativity was hiding inside electromagnetism all along. Fourth, the honest deficiency, promoted from the crown's warning box to working technology: the wave equation is a consequence of Maxwell, strictly weaker than Maxwell — and the constraint-completion routine that upgrades a scalar wave solution into a lawful electromagnetic field is the skill this article drills, because the next nineteen articles use it on every wave they touch.

Key takeaways

With sources kept, the curl-of-curl maneuver gives the inhomogeneous wave equations: $\Box\,\mathbf{E} = \tfrac{1}{\varepsilon_0}\nabla\rho + \mu_0\,\partial_t\mathbf{J}$ and $\Box\,\mathbf{B} = -\mu_0\nabla\times\mathbf{J}$ — the source anatomy of all radiation (E9 cashes it).
Vacuum: $\Box\,\mathbf{E} = 0$ , one scalar wave equation per Cartesian component; the crown's derivation, cited not repeated.
d'Alembert's theorem (derived by characteristics): the general 1D solution is $f(z - ct) + g(z + ct)$ — arbitrary profiles, rigid propagation, exact superposition, zero vacuum dispersion. Monochromatic waves are a basis choice (Fourier), not a physical restriction.
The dispersion relation seed: plugging $e^{i(kz-\omega t)}$ gives $\omega = c|k|$ — linear, the "massless" dispersion; every medium article of this chapter is a story about how matter bends this line.
The equation is necessary, not sufficient: solutions must still pass the first-order Maxwell system. The completion routine — solve scalar equations, then impose both divergences and build the partner field from a curl equation — is this chapter's standard operating procedure.
KEY IDEA: Maxwell's equations force every field component to obey $\Box\,\psi =$ (source terms), with $c = 1/\sqrt{\mu_0\varepsilon_0}$ emerging from the constants of statics; the general vacuum solution is superposed rigid profiles at $c$ , constrained back into lawful fields by the first-order system.

The core insight

Maxwell's equations imply that every component of $\mathbf{E}$ and $\mathbf{B}$ obeys the wave equation $\nabla^2\psi - \mu_0\varepsilon_0\,\partial_t^2\psi =$ (source terms), with speed $c = 1/\sqrt{\mu_0\varepsilon_0}$ built from the constants of electrostatics and magnetostatics; its general solutions are superpositions of rigid profiles propagating at $c$ , subject to the first-order Maxwell constraints.

Where this lives today

The vacuum wave equation is the most heavily engineered PDE on Earth. Every antenna-design suite, radar cross-section code, and photonic-crystal simulator is a discretization of (1)–(2) (FDTD's leapfrog grid is Yee's 1966 embedding of the first-order system — chosen over the second-order wave equation precisely to keep the constraints of Derivation B's routine automatically satisfied, a numerical echo of this article's warning). Precision timing civilization runs on d'Alembert rigidity: GPS ranges by pulse arrival at the nanosecond ( $\pm30$ cm), lunar laser ranging at the picosecond ( $\pm$ mm over $7.7\times10^8$ m of round trip), and pulsar timing arrays trust vacuum non-dispersion so completely that the residual interstellar dispersion becomes a plasma survey instrument. At the frontiers, the equation's structural facts are test targets. Linearity: photon–photon scattering (a QED-induced nonlinearity of vacuum, absent classically) is now observed in heavy-ion collisions and hunted with petawatt lasers — measured departures from superposition, exactly where quantum theory predicts effective $E^4$ terms. Linear dispersion: gamma-ray-burst photons spanning six energy decades arrive synchronized to parts in $10^{15}$ after billions of light-years, among the tightest constraints on quantum-gravity-motivated dispersion; and the 2017 neutron-star merger clocked gravitational waves against light across $1.3\times10^8$ ly, agreeing to one part in $10^{15}$ — two different wave equations, one $c$ , as the symmetry article's spacetime reading requires. The operator itself became physics' template: the Klein–Gordon equation is $\Box$ plus a mass term, and every relativistic field theory since has been a decoration of the object this article opened.

Origins · 1 minJean le Rond d'Alembert and how this was discovered

Jean le Rond d'Alembert

1717–1783 · Paris

The operator wears his name because the solution does. Jean le Rond d'Alembert — foundling (abandoned on the steps of Saint-Jean-le-Rond, whence the name), self-made mathematician, co-editor with Diderot of the Encyclopédie that organized the Enlightenment — derived the wave equation for the vibrating string in 1747, the first partial differential equation of physics, and solved it in the same paper: $f(x - ct) + g(x + ct)$ , the traveling-wave form of this article's Derivation B. The achievement started the eighteenth century's most productive fight: Euler insisted $f$ could be any drawable shape (plucked strings have corners); d'Alembert, demanding analytic functions, resisted; Daniel Bernoulli claimed everything was sums of sines and was disbelieved by both — a three-way dispute about what "function" and "solution" mean that took Fourier's 1807 heat-equation work and a century of analysis (Dirichlet, Riemann) to settle, with all three partly right. His Traité de dynamique (1743) had already recast Newtonian mechanics via what became d'Alembert's principle — the virtual-work formulation the mechanics track builds Lagrangian dynamics from — making him one of two people with a foundational operator in this article and a foundational principle in that track's. That his string equation, constants swapped, turned out to govern light — and that its symmetric $\Box$ would become the invariant kernel of special relativity — is the kind of posthumous compounding interest mathematics pays its founders.

History

1747

— d'Alembert derives and solves the wave equation for the vibrating string: f(x−ct) + g(x+ct)

1750

s — Euler, d'Alembert, and Daniel Bernoulli dispute what counts as a solution — arbitrary shapes vs sums of sines

1807

— Fourier's expansions begin settling the dispute: superposed sines span (almost) everything

1861

— Maxwell finds the same equation hiding in electromagnetism, with c from bench constants (the crown's story)

1887

— Hertz generates and detects the predicted electromagnetic waves at radio frequency

1966

— Yee's FDTD scheme discretizes the first-order system, making the wave equation the most-computed PDE in engineering

2017

— GW170817: gravitational and electromagnetic waves from one merger agree on c to a part in 10^15

The physical picture

Begin with what kind of law a wave equation is, because the chapter will live inside this operator for twenty articles. Laplace's equation — the E2 chapter's home — is an equilibrium law: it relates a function's value to the average of its neighbors and forbids interior extrema; its solutions are as smooth and boring as possible, everything everywhere settling. The wave equation is Laplace's equation with a budget for regret: $\nabla^2\psi = \tfrac{1}{c^2}\partial_t^2\psi$ says curvature-in-space no longer forces settling but accelerates the field toward the average — overshoot, oscillate, and hand the disturbance to your neighbor. Every point is a mass on springs coupled to its neighbors; the continuum limit of that picture (the mechanics track's coupled-oscillator chains) is precisely this equation, and $c$ is the stiffness-to-inertia ratio of the vacuum itself, $1/\sqrt{\mu_0\varepsilon_0}$ — permittivity playing compliance, permeability playing mass density. The mechanical analogy is honest enough to compute with, and its limits (no ether needed — the crown's FAQ) have already been policed.

Figure

The wave equation's three structural facts: rigid profiles at c, exact superposition, and the light-cone geometry of the d'Alembertian

Now the honest architecture of what this article does and does not re-derive. The vacuum wave equation is the crown's Derivation A — the curl-of-curl maneuver, the sign-stability audit, the confrontation of $1/\sqrt{\mu_0\varepsilon_0}$ with Weber–Kohlrausch and Fizeau: all of it stands, none of it is repeated, and a reader arriving here without it should read Maxwell's Equations in Vacuum first. What that derivation deliberately discarded — by setting $\rho = \mathbf{J} = 0$ at the door — is where this article's first derivation lives: keep the sources and the same maneuver produces wave equations with drive terms, and the anatomy of those terms is a table of contents for the radiation chapter. A static charge's $\nabla\rho$ does not radiate (it is cancelled by the static solution); a steady current's $\partial_t\mathbf{J} = 0$ does not radiate; but shake a charge — make $\partial_t\mathbf{J} \neq 0$ — and the right-hand side switches on, driving waves. Everything E9 will compute (dipole radiation, Larmor's formula, antennas) is the response of $\Box$ to these drives. The second thing the crown left undone is the solution theory, and it starts with a fact so strong it feels illegal: in one dimension, d'Alembert wrote down every solution the equation has — not a family, not a basis: all of them. Two arbitrary profiles, one moving each way at $c$ , shape frozen. The rigidity is vacuum's linear dispersion ( $\omega = ck$ for every $k$ : all Fourier components march in step — the "massless" dispersion relation whose bending, by media, boundaries, and plasmas, is essentially the plot of the rest of this chapter). The arbitrariness is why monochromatic waves — the next article's subject — are a choice of basis rather than a physical restriction: any pulse is a Fourier superposition of them, and superposition is exact because the equation is linear. And the two-way structure is the operator's factorization showing through: $\Box$ splits into left-movers and right-movers, information riding the characteristics $z \mp ct$ — the light-cone geometry that E10 will recognize as Lorentz invariance in plain sight.

The equations, with sources aboard

The objects this chapter will solve, in their full inhomogeneous form (Derivation A):

\nabla^2\mathbf{E} - \mu_0\varepsilon_0\,\frac{\partial^2\mathbf{E}}{\partial t^2} = \frac{1}{\varepsilon_0}\,\nabla\rho + \mu_0\,\frac{\partial\mathbf{J}}{\partial t}, \qquad (1)

where: the left side is the d'Alembertian acting on each Cartesian component of $\mathbf{E}$ (sign convention: $\Box \equiv \nabla^2 - \tfrac{1}{c^2}\partial_t^2$ ); the right side is the drive — and its two terms are worth reading slowly, because they are the whole of radiation theory in precis. $\nabla\rho$ : spatial variation of charge density drives $\mathbf{E}$ (near a static charge this term is exactly balanced by the static Coulomb solution — no waves); $\mu_0\,\partial_t\mathbf{J}$ : accelerating charge drives $\mathbf{E}$ dynamically — this is the term that antennas, synchrotrons, and every glowing atom exercise, and the E9 chapter's Jefimenko/dipole machinery is its systematic solution. The magnetic twin:

\nabla^2\mathbf{B} - \mu_0\varepsilon_0\,\frac{\partial^2\mathbf{B}}{\partial t^2} = -\,\mu_0\,\nabla\times\mathbf{J}, \qquad (2)

where: only the current's curl drives $\mathbf{B}$ — no $\rho$ term (no magnetic charge to gradient, equation II of the crown's plaque doing its silent work), and no $\partial_t$ on the source (the time derivatives redistributed differently through the maneuver). Set $\rho = \mathbf{J} = 0$ and both collapse to the crown's vacuum equations, $\Box\mathbf{E} = 0$ , $\Box\mathbf{B} = 0$ — six scalar copies of

\frac{\partial^2\psi}{\partial z^2} - \frac{1}{c^2}\frac{\partial^2\psi}{\partial t^2} = 0 \qquad\text{(one spatial dimension)}, \qquad (3)

where: $\psi$ stands for any component and $c = 1/\sqrt{\mu_0\varepsilon_0} = 2.998\times10^8$ m/s — the constants of bench electricity assembling the speed of light, the structural fact this chapter inherits from the crown without re-arguing its history. Feeding (3) the trial $\psi = e^{i(kz - \omega t)}$ gives the dispersion relation

\omega = c\,|k|, \qquad (4)

where: linearity in $k$ is the deep content: phase velocity $\omega/k$ and group velocity $d\omega/dk$ coincide at $c$ for every wavelength — no spreading, no chirping, profiles rigid (worked example 1 uses this operationally). Media will curve this line ( $\omega(k)$ through $n(\omega)$ — articles 0590 and 0595), waveguides will gap it, plasmas will cut it off; vacuum's straight line is the reference against which all of that is measured.

Figure

The drive anatomy: which sources switch the wave equation on — statics absorbed, steady currents silent, acceleration radiates

Derivation AThe sourced wave equations — what shakes, radiates

The goal is equations (1) and (2): run the crown's curl-of-curl maneuver without discarding $\rho$ and $\mathbf{J}$ , and read the anatomy of the source terms. The strategy is identical to the vacuum case — curl of one curl equation, expand, substitute the other — with the bookkeeping of source terms as the new content. Watch two things: where each source term originates (Gauss's law versus Ampère's current), and why the drives take these particular derivative structures — the answers are the selection rules of radiation.

Step 1. Curl of Faraday, exactly as before: $\nabla\times(\nabla\times\mathbf{E}) = -\partial_t(\nabla\times\mathbf{B})$ . Expand the left with the E0 identity, but now Gauss's law keeps its source: $\nabla(\nabla\cdot\mathbf{E}) = \nabla(\rho/\varepsilon_0)$ :

\frac{1}{\varepsilon_0}\nabla\rho - \nabla^2\mathbf{E} = -\frac{\partial}{\partial t}\left(\nabla\times\mathbf{B}\right).

Step 2. Feed the right side the full Ampère–Maxwell law, $\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\,\partial_t\mathbf{E}$ :

\frac{1}{\varepsilon_0}\nabla\rho - \nabla^2\mathbf{E} = -\mu_0\frac{\partial\mathbf{J}}{\partial t} - \mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2},

and rearrange into (1). Why the drives have these shapes: $\mathbf{E}$ 's sources are charge (through Gauss — hence a gradient: only spatially varying $\rho$ can curve field lines into waves) and changing current (through Ampère–Maxwell's displacement partnership — hence one time derivative: steady currents make static $\mathbf{B}$ , not waves). The radiation selection rule falls out before any radiation theory exists: static anything → no drive; uniformly drifting charge → transformable-away drive (E10's boost argument); acceleration → radiation. Larmor's formula is this observation, quantified (E9).

Step 3. The magnetic twin. Curl of Ampère–Maxwell; expand with $\nabla\cdot\mathbf{B} = 0$ (no source — the monopole absence acting); substitute Faraday:

-\nabla^2\mathbf{B} = \mu_0\nabla\times\mathbf{J} + \mu_0\varepsilon_0\,\partial_t(\nabla\times\mathbf{E}) = \mu_0\nabla\times\mathbf{J} - \mu_0\varepsilon_0\,\frac{\partial^2\mathbf{B}}{\partial t^2},

which is (2). Why $\mathbf{B}$ 's drive is a curl: magnetic fields have no scalar source to gradient; the only way matter drives them is through the rotational structure of its currents. A current with $\nabla\times\mathbf{J} = 0$ everywhere (a uniform current-filled universe, say) drives no $\mathbf{B}$ -wave — consistent with symmetry.

Step 4. Read the system's architecture. Six scalar inhomogeneous wave equations — but not independent: the same fields must also satisfy the four first-order Maxwell equations, which correlate the components (transversality, the $E$ – $B$ lock) and correlate the sources (take the divergence of (1)'s right side and continuity appears — charge conservation embedded in the drive structure, the E7 chapter's opening law showing up as an integrability condition). This redundancy is why the practical solution theory of radiation (E9) abandons the field wave equations for the potential form, $\Box A^\mu = -\mu_0 J^\mu$ in Lorenz gauge — one cleaner equation, sources undifferentiated — deferred by name. For this chapter's vacuum waves, the redundancy is instead handled by the completion routine of Derivation B. Note: the vacuum limit of these equations collapses to the crown's $\Box\mathbf{E}=0$ , $\Box\mathbf{B}=0$ as expected; taking the divergence of the E-drive ( $\varepsilon_0^{-1}\nabla\rho + \mu_0\partial_t\mathbf{J}$ ) and applying the continuity equation $\partial_t\rho + \nabla\cdot\mathbf{J} = 0$ confirms the drive terms are mutually consistent — charge conservation is embedded as an integrability condition of the sourced system.

Go deeper · Derivation B · 2 mind'Alembert's general solution — and the completion routine that makes it electromagnetic

Derivation Bd'Alembert's general solution — and the completion routine that makes it electromagnetic

The goal is two-fold: prove that every solution of the 1D wave equation (3) is $f(z-ct) + g(z+ct)$ with $f$ , $g$ arbitrary twice-differentiable profiles — the strongest solution theorem in classical field theory — and then upgrade a scalar solution into a lawful electromagnetic field via the first-order constraints, the routine this chapter will run on every wave it meets. The strategy for the first part is the method of characteristics: change to the coordinates the operator itself prefers and watch it factorize. For the second, Maxwell's own equations do the constraining. Watch the operator factorize — the calculation is three lines and explains the light-cone picture outright.

Step 1. Characteristic coordinates. Define $u = z - ct$ , $v = z + ct$ . Chain rule: $\partial_z = \partial_u + \partial_v$ , $\partial_t = -c\,\partial_u + c\,\partial_v$ , so

\partial_z^2 - \frac{1}{c^2}\partial_t^2 = (\partial_u + \partial_v)^2 - (\partial_v - \partial_u)^2 = 4\,\partial_u\partial_v.

The wave equation becomes $\partial_u\partial_v\,\psi = 0$ — the operator has factorized into one derivative along each characteristic direction.

Step 2. Integrate twice. $\partial_v\psi$ is independent of $u$ : $\partial_v\psi = G(v)$ . Integrate in $v$ : $\psi = g(v) + f(u)$ with $g' = G$ and $f$ the integration "constant" (a function of $u$ alone). Hence

\boxed{\ \psi(z,t) = f(z - ct) + g(z + ct)\ }

— and nothing else. Why this is exhaustive where most PDE solution formulas are not: the change of variables is invertible and the double integration introduced no restrictions; every twice-differentiable solution has exactly this form, with $f$ and $g$ read off the initial profile and its initial velocity (d'Alembert's formula). Arbitrary shapes propagate rigidly: plug in any $f$ — Gaussian, square pulse, your signature — and the equation transports it unchanged at $c$ . Vacuum has no preferred wavelength to treat differently; that is (4)'s linearity, restated.

Step 3. Superposition and the basis reading. Linearity makes any sum of solutions a solution — including sums over the monochromatic family $e^{i(kz \mp \omega t)}$ , which by Fourier's theorem spans the $f$ and $g$ of Step 2. Monochromatic plane waves (next article) are thus complete scaffolding, not a physical assumption; every theorem proved for them extends to arbitrary pulses by superposition. In three dimensions the count enlarges (plane waves in all directions; spherical waves $\psi = \tfrac{1}{r}f(r - ct)$ — checked by direct substitution, the $1/r$ being the energy-conservation dilution this chapter's Poynting articles priced) but the structural facts — rigid transport at $c$ , exact superposition, Fourier completeness — carry over intact.

Step 4. The completion routine — from scalar solutions to electromagnetic fields. A candidate $\mathbf{E} = E_0\,f(z - ct)\,\hat{\mathbf{x}}$ solves (3) componentwise; now impose Maxwell in order. Gauss: $\nabla\cdot\mathbf{E} = \partial_x E_x = 0$ ✓ (profile depends only on $z$ — but note a $z$ -polarized candidate $E_z = f(z-ct)$ would fail here unless constant: longitudinal waves killed by the divergence, the crown's impostor warning now a checklist line). Faraday, to build the partner: $\nabla\times\mathbf{E} = -\partial_t\mathbf{B}$ gives $\partial_t B_y = -\partial_z E_x = -E_0 f'$ , integrating to $\mathbf{B} = \tfrac{E_0}{c}f(z - ct)\,\hat{\mathbf{y}}$ (the static integration constant set aside). Ampère–Maxwell, as audit: $\nabla\times\mathbf{B} = \mu_0\varepsilon_0\,\partial_t\mathbf{E}$ — substitute and both sides equal $-\tfrac{E_0}{c}f'\,\hat{\mathbf{x}}\cdot...$ — equality holds identically ✓. Gauss-for-B: ✓ trivially. The lawful wave: transverse, $\mathbf{B} = \tfrac1c\,\hat{\mathbf{z}}\times\mathbf{E}$ , $E = cB$ , $\mathbf{E}\times\mathbf{B}$ forward — every crown-promised property, now derived by routine rather than quoted. This four-check sequence — divergences to kill impostors, one curl to build the partner, the other curl to audit — is the standard operating procedure for every wave configuration in the nineteen articles ahead. Note: the spherical scalar wave $\psi = f(r-ct)/r$ can be confirmed by direct substitution into the radial Laplacian; the completion routine's constraints — transversality, $E = cB$ , and the $\mathbf{E}$ – $\mathbf{B}$ phase lock — hold for an arbitrary profile $f$ , not only for the monochromatic case, and the longitudinal-impostor rejection at Gauss's law applies equally generally.

Limiting cases

Case	Result	Meaning
$\rho = \mathbf{J} = 0$	$\Box\mathbf{E} = 0$ , $\Box\mathbf{B} = 0$	the crown's vacuum equations, recovered
static sources	drives cancelled by static solutions	Coulomb/Biot–Savart carry the load; no waves
steady currents	$\partial_t\mathbf{J} = 0$ : no E-drive	DC circuits do not glow
accelerating charge	$\partial_t\mathbf{J} \neq 0$ : drive on	radiation; E9's entire subject in one term
trial $e^{i(kz-\omega t)}$	$\omega = c\vert k\vert$	linear dispersion; no spreading, group = phase = $c$
1D general solution	$f(z-ct) + g(z+ct)$ , arbitrary profiles	d'Alembert: complete, rigid, superposable
spherical symmetry	$\psi = \tfrac1r f(r - ct)$	$1/r$ amplitude: energy conservation over growing spheres
longitudinal candidate	killed by $\nabla\cdot\mathbf{E} = 0$	completion routine's first checkpoint
in media (preview)	$c \to c/n(\omega)$ : dispersion line bends	articles 0590/0595 own the machinery

Common trapSolving the wave equation is the beginning of the check, never the end — run the completion routine every time

The crown flagged it; this chapter operationalizes it: $\Box\mathbf{E} = 0$ admits solutions Maxwell forbids, because the curl-of-curl maneuver discards information (it is one-way: Maxwell ⟹ wave equation, never conversely). The failure modes are specific and worth memorizing as a checklist of impostors. Longitudinal waves ( $\mathbf{E}$ along propagation): solve $\Box$ beautifully, die at Gauss's law. Unpartnered fields (an $\mathbf{E}$ -wave with $\mathbf{B} = 0$ , or with an uncorrelated $\mathbf{B}$ ): die at Faraday — the partner field is constructed, not chosen, and its amplitude ( $E = cB$ ), direction ( $\hat{\mathbf{k}}\times\hat{\mathbf{E}}$ ), and phase (locked, in step) are all outputs. Mismatched profiles ( $\mathbf{E}$ and $\mathbf{B}$ with different shapes $f \neq \tilde f$ ): die at the Ampère audit. The routine that catches all three — divergences first, build the partner by one curl, audit with the other — costs five minutes and is mandatory whenever a wave is written down by hand, guessed, or inherited from a scalar-wave intuition (acoustics and quantum mechanics train reflexes that electromagnetism's transversality punishes). One more scope note: this article's rigid-profile theorems are vacuum statements. The moment matter enters — even dilute matter — dispersion bends (4), profiles spread, and group and phase velocities part company; carrying "shapes propagate rigidly" into a fiber or a plasma is a classic borrowed-theorem error, and the media articles will replace it carefully.

Worked examples

1 · Radar ranging on the dispersion relation (confidence builder)

An air-traffic radar emits a $1.0\ \mu$ s pulse at $2.8$ GHz. The echo from an aircraft returns $t = 0.62$ ms later. Find the range, the pulse's length in flight, its bandwidth, and state precisely which property of equation (3) the whole scheme relies on.

Step 1. Range: the pulse travels out and back at $c$ regardless of its shape (d'Alembert rigidity): $R = \tfrac12 c\,t = 0.5\times2.998\times10^8\times6.2\times10^{-4} = \mathbf{93\ km}$ .

Step 2. Pulse length in flight: $\ell = c\tau = 2.998\times10^8\times10^{-6} = \mathbf{300\ m}$ — a 300-metre slab of field sliding through the sky, shape frozen.

Step 3. Bandwidth: a $1\ \mu$ s envelope carries Fourier content $\Delta f \sim 1/\tau = \mathbf{1\ MHz}$ around the 2.8 GHz carrier — thousands of monochromatic components (next article's basis) superposed.

Step 4. The property being trusted: linearity of the dispersion relation (4). All $\sim10^3$ Fourier components travel at exactly the same speed, so the pulse neither spreads nor chirps over 186 km of flight, and "echo time" is unambiguous at the nanosecond level ( $\pm1$ ns → $\pm15$ cm of range). In a dispersive medium the components would drift apart and the pulse would smear — which is precisely what happens to radio pulses crossing the interstellar plasma, where the arrival-time smear across frequencies is inverted by astronomers to measure the electron column (dispersion measure of pulsars: the failure of vacuum rigidity, put to work). Vacuum's straight line $\omega = ck$ is the reason ranging, GPS timing, and lidar are trivial in principle: the wave equation forgets nothing.

2 · Building the lawful wave behind an FM signal (exam level)

An FM broadcast wave at $f = 100$ MHz propagates along $+\hat{\mathbf{z}}$ with measured electric amplitude $E_0 = 5.0$ mV/m along $\hat{\mathbf{x}}$ at a receiver. Construct the full lawful field, audit it, and extract the operational numbers.

Step 1. Kinematics from (4): $\omega = 2\pi f = 6.28\times10^8$ rad/s; $k = \omega/c = 2.10$ rad/m; $\lambda = 2\pi/k = \mathbf{3.0\ m}$ . Candidate: $\mathbf{E} = E_0\cos(kz - \omega t)\,\hat{\mathbf{x}}$ .

Step 2. Completion routine. Gauss: $\partial_xE_x = 0$ ✓. Faraday builds the partner: $\partial_tB_y = -\partial_zE_x \Rightarrow \mathbf{B} = \tfrac{E_0}{c}\cos(kz-\omega t)\,\hat{\mathbf{y}}$ , amplitude $B_0 = 5.0\times10^{-3}/2.998\times10^8 = \mathbf{1.7\times10^{-11}\ T}$ — seventeen picotesla, a million times below Earth's field, yet cleanly detected because it oscillates at $10^8$ Hz where nothing geomagnetic competes.

Step 3. Ampère audit: $[\nabla\times\mathbf{B}]_x = -\partial_zB_y = \tfrac{E_0k}{c}\sin(kz-\omega t)$ versus $\mu_0\varepsilon_0\partial_tE_x = \tfrac{\omega E_0}{c^2}\sin(kz-\omega t)$ — equal iff $k = \omega/c$ ✓: the audit re-derives the dispersion relation, which is the sign the routine is closing consistently rather than vacuously.

Step 4. Operational payoff (with the E7 chapter's ledgers, cited): intensity $\langle S\rangle = \tfrac12 E_0B_0/\mu_0 = \tfrac12\times5.0\times10^{-3}\times1.67\times10^{-11}/1.257\times10^{-6} = \mathbf{3.3\times10^{-8}\ W/m^2}$ — 33 nW/m², a comfortable FM signal (receivers work down to $\sim$ pW/m²). Note what the completion routine delivered that the scalar wave equation alone never could: the direction, magnitude, and phase of $\mathbf{B}$ ; the transversality; and the energy flow — the difference between "a function that waves" and "an electromagnetic wave."

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.2 · Jackson, Classical Electrodynamics, 3rd ed., §7.1 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9 · Zangwill, Modern Electrodynamics, §16.2 The curl-of-curl derivation yields wave equations with coefficient $\mu_0\varepsilon_0$ ; numerically, $1/\sqrt{\mu_0\varepsilon_0}\approx2.998\times10^8$ m/s using CODATA values $\varepsilon_0=8.854\times10^{-12}$ F/m, $\mu_0=1.257\times10^{-6}$ H/m, consistent with the exact definitional value $c=299\,792\,458$ m/s. The vacuum derivation and the historical confrontation of $c$ are treated in Maxwell's Equations in Vacuum (the E6 crown) and cited, not repeated here; this article derives the sourced equations (drive terms $\varepsilon_0^{-1}\nabla\rho + \mu_0\partial_t\mathbf{J}$ and $-\mu_0\nabla\times\mathbf{J}$ ), d'Alembert's general solution by characteristics, and the first-order completion routine. Key worked numbers: radar $R = ct/2 = 93$ km at $t = 0.62$ ms, $\ell = 300$ m, $\Delta f \sim 1$ MHz; FM example $k = 2.10$ rad/m, $B_0 = E_0/c = 1.67\times10^{-11}$ T, $\langle S\rangle = E_0^2/2\mu_0c = 3.3\times10^{-8}$ W/m².

Builds on: Maxwell's Equations in Vacuum · Vector Calculus Identities · Partial Differential Equations

Leads to: Monochromatic Plane Waves in Vacuum · Complex Notation for Electromagnetic Waves · Electromagnetic Waves in Linear Dielectric Media · Energy Density and the Poynting Vector

Why does each Cartesian component get its own scalar wave equation, but not each spherical or cylindrical component?

Because the curl-of-curl identity's clean split $\nabla\times(\nabla\times\mathbf{E}) = \nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E}$ hides a subtlety: the vector Laplacian $\nabla^2\mathbf{E}$ equals "the scalar Laplacian applied to each component" only in Cartesian coordinates, where the basis vectors are constant. Spherical and cylindrical unit vectors rotate from point to point, so differentiating a component drags basis-vector derivatives along, coupling the components: the wave equation for $E_r$ involves $E_\theta$ and $E_\phi$ too. Practical consequences: plane-wave problems decompose trivially (this chapter's bread), while spherical radiation problems (E9) either work component-wise in Cartesian, use scalar potentials/Debye potentials, or organize by vector spherical harmonics — machinery invented precisely to tame the coupling. A pointed special case worth knowing: the scalar spherical wave $\psi = f(r - ct)/r$ is exact, but there is no purely radial vector wave $\mathbf{E} = \psi\,\hat{\boldsymbol{r}}$ — it would violate transversality (and Gauss), which is why even the simplest antenna's far field has the angular structure it does rather than radiating uniformly.

If the sourced equations drive E with the gradient of rho, why doesn't a static point charge radiate — its gradient is enormous?

Because the equation's response to a static drive is a static solution, and "radiation" means the time-dependent remainder. Decompose: for time-independent $\rho$ , the ansatz $\mathbf{E}(\mathbf{r})$ static reduces (1) to $\nabla^2\mathbf{E} = \varepsilon_0^{-1}\nabla\rho$ — which is exactly the equation the Coulomb field satisfies (take the gradient of Poisson's equation from E1). The static drive is fully absorbed by the static field; nothing is left over to propagate, and the wave content is zero. The general structure: solutions of an inhomogeneous linear equation are (particular solution matching the source) + (homogeneous solutions = free waves), and radiation is sourced only by the part of the drive that no quasi-static particular solution can track — which, chasing the time derivatives, means accelerating charge. This is also why "uniformly moving charge doesn't radiate" needs one more ingredient (a boosted Coulomb field is the tracking solution — E10 makes it exact), and why the radiation chapter's formulas all carry accelerations. The drive terms in (1) are necessary plumbing; net radiation is about what the plumbing cannot statically accommodate.

Is d'Alembert's two-profile completeness special to one dimension?

Yes — and the failure of its naive 3D generalization is real physics. In 1D, the solution manifold is exactly two function's worth; in 3D, plane waves travel in a two-sphere of directions and the general solution is an integral over them (Fourier synthesis) — no finite list of rigid profiles suffices. Two genuinely new 3D phenomena follow. Huygens' principle in the strong sense: in three dimensions (and every odd dimension), a sharp pulse remains sharp — the field at a point depends on the source only on the light cone, not inside it — so a lightning flash is heard once, not as an endless afterglow. Remarkably this fails in even dimensions: a 2D "flash" (think of a pond wave) has a trailing wake, and flatland optics would smear every image. Focusing and diffraction: 3D solutions can concentrate and interfere in ways 1D profiles cannot, which is the entire subject of physical optics downstream. What survives in every dimension: rigidity of each plane-wave component (linear dispersion), exact superposition, and propagation confined to speed $c$ . Those three, not the two-profile formula, are the structural facts the chapter spends.

The wave equation for E involves second derivatives, but Maxwell's equations are first-order. Which is more fundamental for initial conditions?

The first-order system, decisively — and counting initial data shows why. The six-component wave equations naively want twelve functions of position as initial data ( $\mathbf{E}$ , $\mathbf{B}$ , and their time derivatives). Maxwell wants six ( $\mathbf{E}$ and $\mathbf{B}$ now), minus two constraints (the divergences), and it determines the time derivatives itself — Faraday hands you $\partial_t\mathbf{B}$ , Ampère–Maxwell hands you $\partial_t\mathbf{E}$ . The mismatch is exactly the impostor freedom the warning box polices: wave-equation initial data can launch longitudinal modes and unpartnered fields that Maxwell would never evolve into being. The professional formulation (shared with numerical relativity, which faces the same structure): treat the divergences as constraints on initial data, evolve with the curls, and prove constraint propagation — data satisfying the constraints at $t = 0$ satisfies them forever (the crown's self-policing consistency). This is why FDTD codes evolve the first-order Yee system and why "solve the wave equation, then constrain" (Derivation B's routine) is a hand-calculation convenience resting on the first-order system's authority, never a replacement for it.

What happens to this article's theorems in a medium — which survive, which die?

A useful triage, since the rest of the chapter lives in media. Survives intact: linearity/superposition (for linear media — intense fields in nonlinear crystals are E11-adjacent territory), and the completion-routine discipline (first-order Maxwell-in-matter constraints still rule; only the constitutive details change). Survives modified: the wave equation itself — in transparent media it becomes $\Box_n\mathbf{E} = 0$ with $c \to c/n$ , and for monochromatic waves that is the whole story at fixed frequency. Dies: profile rigidity and the linear dispersion relation. Real media have $n(\omega)$ — resonances make dispersion inevitable (a causal medium cannot have constant $n$ ; the Kramers–Kronig relations of E8's absorption article make that a theorem) — so group and phase velocities split, pulses spread and chirp, and d'Alembert's arbitrary-profile transport is replaced by envelope equations and group-velocity analysis. The triage explains the chapter's architecture: monochromatic waves first (0583–0589, where fixed frequency hides dispersion), then media and dispersion (0590+, where the vacuum theorems are carefully renegotiated). Carry the vacuum reflexes into a fiber and the fiber will re-educate you; this article's rigidity theorems are the baseline, and their controlled failure is the syllabus.

Why present the field wave equations at all if the radiation chapter will switch to potentials?

Three reasons of practice, one of principle. Practice: the field equations are constraint-transparent (you can see transversality and the $E$ – $B$ lock being imposed — the completion routine — where the potential formulation buries them in gauge choices); they are what FDTD-style numerics actually evolve; and for the source-free waves of this chapter they are strictly simpler — no gauge to fix, no potentials to differentiate at the end. The potential form $\Box A^\mu = -\mu_0J^\mu$ (Lorenz gauge) wins exactly when sources dominate the problem: its drive is the undifferentiated current (compare (1)'s gradient-and-derivative anatomy), it is four equations rather than six-plus-constraints, and Green's-function technology applies cleanly — which is why E9 adopts it for retarded potentials and radiation. Principle: seeing both formulations of one physics — fields with constraints versus potentials with gauge freedom — is the first serious encounter with a trade-off that structures modern field theory (manifest physicality versus manifest covariance), and the E10 chapter's covariant repackaging will assume you have felt both sides. This article is the field-side foundation; the potential side gets its own opening two chapters hence.