---
title: Electromagnetic Waves
module: Maxwell’s Equations and Electromagnetic Waves
moduleNumber: 10
lessonNumber: 2
order: 1002
summary: >
  Once a changing electric flux can drive a magnetic field, the two curl laws feed each
  other: a disturbance in one regenerates the other, and the pair walks off through empty
  space with no medium holding it up. We take the curl of Faraday's law, land on a wave
  equation whose speed is fixed entirely by $\mu_0$ and $\varepsilon_0$, and find that
  $c=1/\sqrt{\mu_0\varepsilon_0}$ falls out of purely electric and magnetic constants.
  The plane-wave solution then fixes the geometry — $\vec E$, $\vec B$, and the
  propagation direction mutually perpendicular, oscillating in phase, with amplitudes
  locked at $E=cB$ — a set of independent predictions any real measurement must meet at once.
topics: [Maxwell’s Equations and Electromagnetic Waves]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 30 — Maxwell’s Equations and Electromagnetic Waves; §§30-2–30-3"
---

## Source-Free Equations

Electromagnetic waves occupy regions where charges and conduction currents need not
be present locally. In vacuum away from sources, Maxwell's equations reduce to

$$
\nabla\mathbin{\cdot}\vec E=0,
\qquad
\nabla\mathbin{\cdot}\vec B=0,
$$

$$
\nabla\mathbin{\times}\vec E
=-\frac{\partial\vec B}{\partial t},
\qquad
\nabla\mathbin{\times}\vec B
=\mu_0\varepsilon_0\frac{\partial\vec E}{\partial t}.
$$

The zero divergence equations constrain the spatial shape of the fields. The curl
equations couple their time variation: a changing magnetic field has an electric curl,
and a changing electric field has a magnetic curl. The source-free equations apply in
the propagation region, not inside an antenna feed, conductor, plasma, or dielectric
whose free charge and current density must be retained explicitly.

$$
% caption: Field geometry of a plane electromagnetic wave in vacuum. The electric
% field, the magnetic field, and the propagation direction are mutually
% perpendicular, and energy travels along the right-handed cross product of the
% electric into the magnetic field.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black,thick] (2.30,1.30)--(6.00,1.30) node[right] {propagation};
\draw[->,acc,very thick] (2.30,1.30)--(2.30,3.15) node[above] {$E$};
\draw[->,black,very thick] (2.30,1.30)--(1.15,0.50) node[below left] {$B$};
\draw[black] (2.30,1.62)--(2.62,1.62)--(2.62,1.30);
\end{tikzpicture}
$$

The wave equation follows by taking the curl of Faraday's law:

$$
\nabla\mathbin{\times}
\left(\nabla\mathbin{\times}\vec E\right)
=-\frac{\partial}{\partial t}
\left(\nabla\mathbin{\times}\vec B\right).
$$

The vector identity

$$
\nabla\mathbin{\times}
\left(\nabla\mathbin{\times}\vec E\right)
=\nabla
\left(\nabla\mathbin{\cdot}\vec E\right)
-\nabla^2\vec E
$$

and the source-free divergence equation remove the first term. Substitution of the
Ampère–Maxwell curl law then gives

$$

\nabla^2\vec E
-\mu_0\varepsilon_0
\frac{\partial^2\vec E}{\partial t^2}=0.

$$

An identical calculation yields

$$

\nabla^2\vec B
-\mu_0\varepsilon_0
\frac{\partial^2\vec B}{\partial t^2}=0.

$$

Both equations have the standard wave-equation form. Comparing their coefficient with
$\nabla^2\psi-(1/v^2)\partial^2\psi/\partial t^2=0$ identifies the
vacuum propagation speed:

$$
c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}.
$$

The result connects electrostatic and magnetic constants to a propagation speed. The
letters $E$ and $B$ in the wave equation denote components or vectors defined over
space and time; one sampled meter value cannot establish the spatial derivatives that
appear in the derivation.

The second-order equation requires initial data. A physically admissible initial
electric field must satisfy $\nabla\mathbin{\cdot}\vec E=0$ in a
source-free volume, and the corresponding magnetic field must satisfy
$\nabla\mathbin{\cdot}\vec B=0$. The two curl equations then set
their initial time rates consistently. Assigning an arbitrary pulse to $E$ and an
unrelated pulse to $B$ generally violates Maxwell's equations even when each pulse
individually resembles a solution of a scalar wave equation.

## Plane-Wave Derivation

A plane wave travelling in the positive $x$ direction can have

$$
\vec E(x,t)=E_y(x,t)\,\hat y,
\qquad
\vec B(x,t)=B_z(x,t)\,\hat z.
$$

With no dependence on $y$ or $z$, the curl equations become

$$
\frac{\partial E_y}{\partial x}
=-\frac{\partial B_z}{\partial t},
\qquad
\frac{\partial B_z}{\partial x}
=-\mu_0\varepsilon_0\frac{\partial E_y}{\partial t}.
$$

The signs depend on the chosen component directions. The physical check is the
right-handed relation $\vec E\mathbin{\times}\vec B$ along the direction of
energy propagation. Reversing the propagation direction reverses one transverse
component when the coordinate basis is held fixed.

A monochromatic solution has the form

$$
E_y(x,t)=E_0\cos(kx-\omega t+\phi),
$$

where $k=2\pi/\lambda$, $\omega=2\pi f$, and $\phi$ is a reference phase.
Substitution into the wave equation gives

$$
\omega=ck,
\qquad
c=f\lambda.
$$

The reduced curl equations set the amplitude ratio:

$$
B_0=\frac{E_0}{c}.
$$

Electric and magnetic components of an ideal vacuum plane wave are in phase. Their
zeros, extrema, and sign reversals occur at the same $x,t$ values. A phase
difference between channels in an experiment can result from detector delay,
bandwidth, cable length, a nearby reflecting surface, or a medium with dispersive
response. The phase convention for each instrument belongs in the data record.

$$
% caption: Electric and magnetic fields of an ideal vacuum plane wave sampled at one
% location. The two components have different units and amplitudes, related by
% $E=cB$, yet their zeros and extrema fall at the same instants: the fields oscillate
% in phase.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.80,1.50)--(6.10,1.50) node[right] {time};
\draw[->,black] (0.95,0.45)--(0.95,2.85) node[above] {signal};
\draw[black,dashed] (1.40,0.60)--(1.40,2.55);
\draw[acc,very thick,smooth] plot coordinates {(0.90,1.50)(1.15,2.17)(1.40,2.45)(1.65,2.17)(1.90,1.50)(2.15,0.83)(2.40,0.55)(2.65,0.83)(2.90,1.50)(3.15,2.17)(3.40,2.45)(3.65,2.17)(3.90,1.50)(4.15,0.83)(4.40,0.55)(4.65,0.83)(4.90,1.50)(5.15,2.17)(5.40,2.45)(5.65,2.17)(5.90,1.50)};
\draw[black,very thick,smooth] plot coordinates {(0.90,1.50)(1.15,1.92)(1.40,2.10)(1.65,1.92)(1.90,1.50)(2.15,1.08)(2.40,0.90)(2.65,1.08)(2.90,1.50)(3.15,1.92)(3.40,2.10)(3.65,1.92)(3.90,1.50)(4.15,1.08)(4.40,0.90)(4.65,1.08)(4.90,1.50)(5.15,1.92)(5.40,2.10)(5.65,1.92)(5.90,1.50)};
\node[acc] at (1.40,2.72) {$E$};
\node[black] at (2.40,0.70) {$B$};
\end{tikzpicture}
$$

Phase surfaces satisfy $kx-\omega t+\phi=\text{constant}$. Differentiating a
constant-phase condition gives $\d x/\d t=\omega/k=c$. A crest is therefore a marker
for phase propagation. The relation does not claim that a material object moves at
the electromagnetic wave speed. In vacuum there is no mechanical medium carrying
transverse oscillations; the electric and magnetic fields are the quantities with
spatial structure and time variation.

$$
% caption: Phase fronts of a plane electromagnetic wave. Surfaces of constant phase
% are planes perpendicular to the propagation direction, spaced one wavelength apart;
% each front advances one wavelength in one period, so the phase speed is
% $c=\lambda/T$.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\foreach \x in {1.00,1.80,2.60,3.40,4.20,5.00} {
  \draw[acc,very thick] (\x,0.80)--(\x,2.45);
}
\draw[->,black,very thick] (1.10,2.80)--(5.05,2.80) node[right] {propagation};
\draw[<->,black,thick] (1.00,0.50)--(1.80,0.50);
\node[below] at (1.40,0.50) {one wavelength};
\end{tikzpicture}
$$

### Transversality

The divergence equations impose transversality for a uniform plane wave. Taking
$\vec E=\vec E_0\cos(\vec k\mathbin{\cdot}\vec r-\omega t)$ gives

$$
\vec k\mathbin{\cdot}\vec E_0=0,
\qquad
\vec k\mathbin{\cdot}\vec B_0=0.
$$

Both field amplitudes are perpendicular to the propagation vector. The curl laws
also give

$$
\vec B_0=\frac{1}{\omega}\vec k\mathbin{\times}\vec E_0,
\qquad
\vec E_0=-\frac{c^2}{\omega}\vec k\mathbin{\times}\vec B_0.
$$

These vector relations contain direction, amplitude, and sign information. A proposed
plane-wave field with an electric component parallel to $\vec k$ violates the
source-free divergence condition. A localized antenna near-field can contain
longitudinal components and does not meet the far-field plane-wave approximation.

$$
% caption: Energy transport in a plane wave. The Poynting vector
% $\vec S=\tfrac{1}{\mu_0}\vec E\times\vec B$ points along the propagation direction,
% and its magnitude is the instantaneous power crossing unit area; the electric and
% magnetic fields lie in the plane transverse to that flow.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\fill[acc!10] (2.50,0.98)--(5.60,0.98)--(5.60,1.42)--(2.50,1.42)--cycle;
\draw[->,black,very thick] (2.50,1.20)--(5.95,1.20) node[right] {$S$};
\draw[->,acc,very thick] (2.50,1.20)--(2.50,3.05) node[above] {$E$};
\draw[->,black,very thick] (2.50,1.20)--(1.35,0.50) node[below left] {$B$};
\draw[black] (2.50,1.52)--(2.82,1.52)--(2.82,1.20);
\node[below] at (4.10,0.98) {energy};
\end{tikzpicture}
$$

Transversality is local to the plane-wave model. A finite beam has a small spread of
wave vectors, and its electric field can acquire a small longitudinal component near a
focus or aperture. A waveguide deliberately uses conducting boundaries to sustain
modes with longitudinal field components. The vacuum plane-wave relations provide a
reference limit for measurements made sufficiently far from sources and boundaries.

## Measurements and Spectra

Wave speed can be measured from a pulse delay. Two calibrated sensors separated by
distance $L$ record the same identifiable pulse feature at times $t_1$ and
$t_2$, giving

$$
v_{\rm meas}=\frac{L}{t_2-t_1}.
$$

The distance must be the propagation-path separation between effective sensor
locations, not the distance between instrument housings. Trigger delay, cable delay,
sensor impulse response, and path reflection contribute to the timing uncertainty.
A broad pulse makes a threshold-crossing time sensitive to amplitude; cross-correlation
or a fit to the complete calibrated waveform provides a more stable delay estimate.

A phase method is effective for a continuous sinusoidal signal. If two sensors
separated by $L$ measure phase difference $\Delta\phi$, then

$$
\Delta\phi=kL+2\pi m,
\qquad
v=\frac{\omega L}{\Delta\phi-2\pi m}.
$$

The integer $m$ counts complete cycles between sensors. It must be determined from
frequency, approximate path length, or a multi-frequency fit. Treating a wrapped
phase reading as the full phase difference can produce a speed error by an integer
factor. Sweeping frequency and fitting unwrapped phase against frequency estimates
group delay and exposes cable-delay offsets as an intercept.

The velocity result belongs with the medium model. Vacuum gives $c$; a homogeneous
linear material gives a characteristic speed $v=1/\sqrt{\mu\varepsilon}$ only when
the stated constitutive parameters represent the frequency range. Conductors, lossy
dielectrics, waveguides, dispersive media, and plasma support more general propagation
relations. The source-free vacuum wave equation provides the reference from which
those deviations are measured.

### Component derivation and sign checks

The vector wave equation can obscure a sign error. A Cartesian component calculation
keeps each derivative visible. Let propagation be along positive $x$, let electric
field point along positive $y$, and let magnetic field point along positive $z$.
The curl of the electric field has only a $z$ component:

$$
\left(\nabla\mathbin{\times}\vec E\right)_z
=\frac{\partial E_y}{\partial x}.
$$

Faraday's law then requires

$$
\frac{\partial E_y}{\partial x}
=-\frac{\partial B_z}{\partial t}.
$$

The curl of $B_z\hat z$ has a $y$ component
$-\partial B_z/\partial x$. Ampère–Maxwell law therefore gives

$$
\frac{\partial B_z}{\partial x}
=-\mu_0\varepsilon_0\frac{\partial E_y}{\partial t}.
$$

Differentiate the first relation with respect to $x$, then replace
$\partial B_z/\partial x$ with the second relation. The result is

$$
\frac{\partial^2E_y}{\partial x^2}
=\mu_0\varepsilon_0
\frac{\partial^2E_y}{\partial t^2}.
$$

The same two equations yield the scalar magnetic wave equation. This component route
is also the fastest way to check a proposed sign convention against a right-handed
coordinate system.

$$
% caption: Snapshot of a linearly polarized plane wave travelling along $x$. The
% electric field points along $y$ and the magnetic field along $z$; the two profiles
% share every zero and extremum, so the fields are transverse and in phase, with
% $E_y=cB_z$.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.70,1.70)--(6.30,1.70) node[right] {$x$};
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\draw[black,very thick,smooth] plot coordinates {(0.90,1.70)(1.15,2.09)(1.40,2.25)(1.65,2.09)(1.90,1.70)(2.15,1.31)(2.40,1.15)(2.65,1.31)(2.90,1.70)(3.15,2.09)(3.40,2.25)(3.65,2.09)(3.90,1.70)(4.15,1.31)(4.40,1.15)(4.65,1.31)(4.90,1.70)(5.15,2.09)(5.40,2.25)(5.65,2.09)(5.90,1.70)};
\draw[->,black,thick] (4.55,3.02)--(5.95,3.02);
\node[above] at (5.25,3.02) {propagation};
\node[acc] at (1.40,2.82) {$E_y$};
\node[black] at (3.40,0.92) {$B_z$};
\end{tikzpicture}
$$

The sign of the amplitude relation follows from a single crest. At a time when
$E_y$ rises with $x$, Faraday's law says that $B_z$ decreases in time. For a
right-moving sinusoid, the crest moves toward larger $x$; the time derivative at a
point on the rising spatial slope is negative. The two signs agree. A left-moving
wave reverses the relation between $E_y$ and $B_z$, so its Poynting direction
changes from positive $x$ to negative $x$.

A right-moving component has

$$
E_y=F(x-ct),
\qquad
B_z=\frac{1}{c}F(x-ct).
$$

A left-moving component has

$$
E_y=G(x+ct),
\qquad
B_z=-\frac{1}{c}G(x+ct).
$$

The sign difference belongs to propagation direction, not to the sign of charge or
to a choice of electric-field unit. Add the two components to represent a general
one-dimensional source-free solution:

$$
E_y(x,t)=F(x-ct)+G(x+ct),
\qquad
B_z(x,t)=\frac{F(x-ct)-G(x+ct)}{c}.
$$

The paired equations show why a standing wave has different energy transport from a
single travelling wave. Equal right- and left-moving components can form a stationary
electric pattern, while their magnetic components have the corresponding opposite
signs. A standing-wave treatment requires boundary conditions; the decomposition already
supplies the field pairing it needs.

An initial-value calculation begins with both $E_y(x,0)$ and $B_z(x,0)$, or
equivalently with $E_y(x,0)$ and $\partial E_y/\partial t(x,0)$. The curl
relation determines the conversion:

$$
\left.\frac{\partial E_y}{\partial t}\right|_{t=0}
=-\frac{1}{\mu_0\varepsilon_0}
\left.\frac{\partial B_z}{\partial x}\right|_{t=0}.
$$

A localized electric pulse with $B_z=0$ at $t=0$ decomposes into equal
counterpropagating components. A pulse launched
by a directional antenna acquires the appropriate accompanying magnetic field through
its source region; once the wave has reached a source-free region, the paired
right-moving relation applies.

The scalar wave equation also permits solutions that violate the vector constraints
when components are assembled carelessly. A numerical or analytic solution must
retain $\nabla\mathbin{\cdot}\vec E=0$ and
$\nabla\mathbin{\cdot}\vec B=0$ at the initial time and during
evolution. Taking divergence of the curl equations shows that their time derivatives
vanish in a source-free region. Constraint errors placed in initial data persist in
the ideal equations; numerical schemes can also create them through inconsistent grid
updates or boundary treatment.

### Spectral content and dispersion

A finite pulse contains a range of spatial frequencies. Represent one component as a
superposition of Fourier terms with wave number $k$. In vacuum every term satisfies
$\omega=ck$, so phase velocity and group velocity both equal $c$. A compact
pulse retains its shape in the ideal one-dimensional vacuum model because the
dispersion relation is linear.

$$
% caption: Spatial-frequency content of a localized pulse. A spatially broad pulse is
% built from a narrow band of wave numbers, whereas a short pulse needs a wide band;
% in vacuum every component obeys $\omega=ck$, so all of them travel at the same speed
% and the pulse keeps its shape.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.70,0.60)--(5.95,0.60) node[right] {wave number};
\draw[->,black] (0.70,0.60)--(0.70,2.90) node[above] {amplitude};
\draw[acc,very thick,smooth] plot coordinates {(0.95,0.62)(1.25,0.92)(1.55,2.20)(1.85,0.92)(2.15,0.62)};
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\node[acc] at (1.55,2.48) {broad pulse};
\node[black] at (4.20,2.06) {short pulse};
\end{tikzpicture}
$$

A homogeneous material with scalar, frequency-independent $\varepsilon$ and
$\mu$ has

$$
v=\frac{1}{\sqrt{\mu\varepsilon}},
\qquad
k=\omega\sqrt{\mu\varepsilon}.
$$

The material description belongs to the region in which the wave propagates. Relative
permittivity and permeability measured at low frequency may not describe a fast
pulse. A dielectric can have a frequency-dependent polarization response; a conductor
can introduce attenuation and phase shift; a magnetic material can have resonant
permeability. The simple speed formula requires a specified frequency band and a
homogeneous, linear response.

When $\omega(k)$ is nonlinear, phase velocity
$v_{\rm p}=\omega/k$ and group velocity $v_{\rm g}=\d\omega/\d k$ differ. A
narrow-band envelope travels near $v_{\rm g}$, while individual crests travel near
$v_{\rm p}$. A time-of-flight pulse measurement responds to envelope or feature
delay; a sinusoidal phase measurement responds to phase delay. Reporting only one
speed without naming the measurement observable loses that distinction.

$$
% caption: Dispersion relations. In vacuum $\omega=ck$ is a straight line through the
% origin, so phase and group speeds both equal $c$. In a dispersive material
% $\omega(k)$ curves: the chord from the origin gives the phase speed and the local
% tangent gives the group speed, and the two differ.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.70,0.60)--(5.95,0.60) node[right] {wave number};
\draw[->,black] (0.70,0.60)--(0.70,3.05) node[above] {frequency};
\draw[acc,very thick] (0.90,0.72)--(5.40,2.92);
\draw[black,very thick,smooth] plot coordinates {(0.90,0.72)(1.60,1.18)(2.30,1.56)(3.00,1.86)(3.70,2.10)(4.40,2.26)(5.10,2.36)};
\node[acc] at (2.05,2.55) {vacuum};
\node[black] at (4.75,1.98) {material};
\end{tikzpicture}
$$

Pulse broadening provides a direct dispersive diagnostic. Send pulses with two
different bandwidths through the same path, correct the sensor response, and compare
their arrival-time and width changes. A path-independent timing offset suggests a
shared trigger or cable delay. A bandwidth-dependent shape change indicates material
or waveguide dispersion. The analysis should retain the input waveform; a received
pulse width alone cannot separate source bandwidth from propagation broadening.

## Numerical and Boundary Models

The one-dimensional wave equation offers a controlled numerical check on the analytic
relations. Sample $E_y$ at positions $x_j=j\Delta x$ and times
$t_n=n\Delta t$. A centered second-difference update is

$$
E_j^{n+1}
=2E_j^n-E_j^{n-1}
+S^2\left(E_{j+1}^n-2E_j^n+E_{j-1}^n\right),
\qquad
S=\frac{c\Delta t}{\Delta x}.
$$

The dimensionless number $S$ compares numerical information travel during one time
step with one spatial cell. In one dimension the centered scheme remains stable for
$S\leq1$. Values above one permit a disturbance to jump farther than one cell per
update, and round-off or truncation error grows instead of remaining bounded. A stable
calculation can still be inaccurate: stability prevents runaway error, while spatial
and temporal resolution control phase and amplitude error.

An electromagnetic finite-difference calculation benefits from staggering electric
and magnetic samples. In the one-dimensional Yee arrangement, electric values occupy
integer positions and integer time levels, while magnetic values occupy half-position
and half-time levels. Faraday's law updates magnetic values from neighboring electric
values; Ampère–Maxwell law updates electric values from neighboring magnetic values.
The staggered layout preserves the local curl structure more directly than an
independent scalar-wave update for each component.

$$
B_{j+1/2}^{\,n+1/2}
=B_{j+1/2}^{\,n-1/2}
-\frac{\Delta t}{\Delta x}
\left(E_{j+1}^{\,n}-E_j^{\,n}\right),
$$

$$
E_j^{\,n+1}
=E_j^{\,n}
-c^2\frac{\Delta t}{\Delta x}
\left(B_{j+1/2}^{\,n+1/2}-B_{j-1/2}^{\,n+1/2}\right).
$$

The apparent asymmetry reflects the different units of $E$ and $B$. The updates
are paired by the same vacuum constant $c^2$. Initialize both component arrays from
a right-moving or left-moving paired state. Initializing a sampled electric pulse
without the matching magnetic array launches an unintended mixture of directions and
can be mistaken for a boundary reflection.

Grid dispersion changes the relation between numerical frequency and wave number.
For the centered scalar update,

$$
\sin^2\left(\frac{\omega_{\rm num}\Delta t}{2}\right)
=S^2\sin^2\left(\frac{k\Delta x}{2}\right).
$$

Small $k\Delta x$ reproduces $\omega_{\rm num}\simeq ck$. Near the grid limit,
the sine terms curve the relation and short wavelengths travel at the wrong numerical
phase speed. A pulse then broadens even in vacuum, purely as a discretization artifact.
Increase the number of cells per shortest wavelength until phase delay, amplitude, and
pulse width converge within the stated accuracy target. Refining time step alone
cannot repair spatial aliasing.

The time step also limits what a recorded numerical trace can represent. A pulse with
significant spectral content above the Nyquist frequency of the time grid aliases into
lower frequencies. Monitor the source waveform and the propagated waveform in the
same spectral plot. A clean-looking trace can contain phase error that appears only
when compared at two separated locations. Store raw arrays, grid spacing, time step,
boundary prescription, source injection rule, and the number of update steps; a
rendered animation has little diagnostic value without those records.

Boundary treatment must match the physical problem. A perfect-conductor end imposes
zero tangential electric component and generates a reflected wave. A simple
zero-gradient array edge can create a reflection even when the intended model is open
space. Absorbing layers gradually damp a wave before it reaches the outer computational
edge, but their parameters must be tested by launching a pulse toward the layer and
measuring the returned amplitude. The reflected fraction belongs in a numerical
uncertainty budget.

Numerical validation uses several independent checks. In a uniform source-free
section, compare the measured grid delay with the analytic $L/c$ delay. Check the
ratio of electric and magnetic amplitudes against $c$. Verify that discrete
divergence remains near round-off for a plane-wave initial state. Repeat with smaller
$\Delta x$ and $\Delta t$, then compare waveforms after interpolation at common
physical coordinates. Agreement of one crest position is weaker evidence than
agreement of delay, amplitude ratio, energy flow, and residual boundary return over
the full record.

### Source regions and the plane-wave limit

The source-free equations begin outside the region that drives the wave. A transmitting
antenna, a current loop, or a voltage-fed plate contains charge density and current
density. Maxwell's curl law there has the full source term

$$
\nabla\mathbin{\times}\vec B
=\mu_0\vec J
+\mu_0\varepsilon_0\frac{\partial\vec E}{\partial t}.
$$

The source produces fields with spatial variation set by its geometry and by
wavelength. Far from a compact source, the radiative part can approach a transverse
plane or spherical wave over a limited observation area. Close to the source, electric
and magnetic amplitudes can have different distance dependences and the simple
$B=E/c$ relation need not hold. A measurement location should be identified as
source-near, transition, or radiation-region geometry before the plane-wave formulas
are applied.

Finite conductors impose boundary conditions on the electromagnetic field. At an
ideal conducting surface, tangential electric field is zero in the conductor frame.
The reflected field combines with the incident field to meet that condition. A real
metal has finite conductivity and skin depth; its tangential field and associated
loss depend on frequency and surface quality. Waveguides use conducting boundaries
intentionally, so their modes have component structures different from free-space
plane waves.

The plane-wave limit also assumes the observation aperture is small relative to the
radius of curvature of a spherical front. A distant point source produces spherical
surfaces. Over a small detector array, adjacent normal directions are nearly parallel
and amplitude variation can be small, giving an effective plane wave. A large array,
a near detector, or an aperture edge samples curvature and requires a wavefront model
with transverse position dependence.

An integral Maxwell-law check is valuable near an interface or finite conductor.
Select a loop whose sides lie on opposite sides of the boundary and shrink its height
while retaining its tangential span. The circulation equation relates the tangential
field jump to any surface current. Select a pillbox straddling the same boundary to
relate normal electric flux to surface charge. The required source terms depend on
the actual conductor or dielectric model. Applying vacuum source-free equations
through an electrode boundary removes the surface charge and current that
create the emitted wave.

The geometric boundary and the computational boundary are distinct. A metal plate
inside a simulation represents a physical surface condition. The outer edge of a
simulation domain represents a truncation of space and needs an absorbing or
analytically matched treatment. Confusing the two can turn a numerical reflection into
an apparent physical standing wave. Store a diagram showing both boundaries, the
source location, the observation interval, and the coordinate direction used for
component signs.

### Calibrated phase and time-of-flight

Wave-speed measurements compare records from two locations, so the acquisition chain
forms part of the physical model. A receiving antenna, photodiode, pickup loop, or
electric-field probe has a response function. Cables add delay and frequency-dependent
loss. Digitizer channels have gain, phase, and clock errors. The recorded signals
cannot be treated as direct samples of one field component until the sensing geometry
and transfer functions have been stated.

Write the frequency-domain record from channel $i$ as

$$
\widetilde s_i(\omega)
=H_i(\omega)\widetilde q_i(\omega)
+\widetilde n_i(\omega),
$$

where $\widetilde q_i$ is the physical quantity coupled to the sensor,
$H_i$ is the calibrated complex response, and $\widetilde n_i$ represents
instrument noise. Antenna orientation, effective height, impedance loading, and
frequency response relate terminal voltage to $E_y$. A pickup loop responds to
time-varying magnetic flux, so its voltage represents a frequency-weighted magnetic
signal.

Calibrate channel delay with a known common input or a reference path whose physical
propagation delay is negligible compared with the desired uncertainty. Let
$\tau_{i,\rm ch}$ denote the channel delay. A raw two-channel phase difference
contains

$$
\Delta\phi_{\rm raw}(\omega)
=\Delta\phi_{\rm phys}(\omega)
-\omega\left(\tau_{2,\rm ch}-\tau_{1,\rm ch}\right)
+\Delta\phi_{\rm gain}(\omega).
$$

The correction must preserve sign. Swapping channels reverses the physical delay and
the channel-delay term. A cable replacement can change both phase and amplitude; a
calibration performed with a different cable route does not transfer automatically to
the measurement geometry.

Continuous-wave phase wraps every $2\pi$. A phase meter may report $+170^\circ$
at one frequency and $-175^\circ$ at a nearby frequency even when physical delay
changes smoothly. Unwrapping adds or subtracts whole cycles to form a continuous
phase sequence. The correct branch follows a plausible delay range and is checked by
measurements at multiple separations. A single frequency leaves an integer-cycle
ambiguity whenever the path contains more than one wavelength.

Fit the unwrapped phase to

$$
\Delta\phi_{\rm phys}(\omega)=\phi_0-\omega\tau_{\rm prop},
\qquad
\tau_{\rm prop}=\frac{L}{v_{\rm p}}
$$

for a nondispersive phase measurement. The sign of the slope follows the chosen
Fourier convention and the order of channels. Determine the sign from a known cable
delay before assigning a negative fitted slope to a physical advance. Weighted linear
regression is appropriate when phase uncertainty changes with signal amplitude.
Retain covariance between slope and intercept; channel-reference uncertainty affects
the intercept and can correlate with a narrow-band slope estimate.

Time-domain pulse analysis uses a different observable. Cross-correlate calibrated
records $s_1(t)$ and $s_2(t)$:

$$
C_{12}(\tau)=\int s_1(t)s_2(t+\tau)\,\d t.
$$

The lag at the correlation maximum estimates delay when the waveform is preserved
along the path. Window the record around the pulse, remove a measured baseline, and
inspect the correlation peak width. A broad or double peak can indicate multipath
arrival, waveform dispersion, or an unresolved trigger error. A maximum at one
sample interval should not be reported with sub-sample precision unless an
interpolation or likelihood model has been stated.

Path length has its own uncertainty. The relevant distance runs between effective
sensing planes along the propagation direction. Antenna phase centers can move with
frequency. A sensor surface tilted by angle $\theta$ has an effective delay that
depends on the incident wavefront and aperture size. Record mechanical coordinates,
orientation, and any correction from physical housing marks to electrical reference
planes. Reversing the apparatus direction or exchanging sensor positions provides a
test for a fixed channel delay: physical propagation delay reverses sign under the
channel order, while a common geometry error may persist.

Amplitude and phase quality gates protect the speed fit. Reject records with clipped
channels, low signal-to-noise ratio, unresolved interference, or gain changes between
calibration and acquisition. Estimate delay on repeated independent records rather
than many overlapping windows from one trace. Separate random timing scatter from
shared clock scale, channel-delay calibration, baseline geometry, and medium-property
uncertainty. A report containing raw records, response corrections, unwrapping rule,
fit residuals, and the full uncertainty model permits the measured propagation speed
to be independently checked.

Time-of-flight and phase methods can be reconciled in vacuum. A pulse delay gives
$L/c$, and an unwrapped sinusoidal phase slope gives the same delay. In a
dispersive material, their disagreement can be physical: pulse-envelope delay follows
group propagation while a carrier phase slope follows phase propagation. The data
analysis should name the waveform feature, frequency interval, and medium model before
comparing either result with $1/\sqrt{\mu\varepsilon}$.

## Loss and Dispersion

Conductivity changes the source-free material wave equation. In a homogeneous linear
medium with $\vec J=\sigma\vec E$, Ampère–Maxwell law contains the conduction
term as well as the displacement term:

$$
\nabla\mathbin{\times}\vec B
=\mu\sigma\vec E
+\mu\varepsilon\frac{\partial\vec E}{\partial t}.
$$

Taking a curl with Faraday's law gives

$$
\nabla^2\vec E
-\mu\varepsilon\frac{\partial^2\vec E}{\partial t^2}
-\mu\sigma\frac{\partial\vec E}{\partial t}=0.
$$

The last term represents conversion of electromagnetic energy into material heating.
An identical form applies to $\vec B$ under the same homogeneous-material
assumptions. A conductor is therefore not represented by the vacuum wave equation
with a slower numerical speed inserted by hand. Its current density changes the
differential equation and introduces attenuation.

At a single frequency, write the field as the real part of

$$
\widetilde E(z,t)=E_0e^{\mathrm i(kz-\omega t)},
\qquad
k=\beta+\mathrm i\alpha.
$$

The factor $e^{\mathrm i kz}$ contains oscillation $e^{\mathrm i\beta z}$ and
amplitude decay $e^{-\alpha z}$. The real positive quantities $\beta$ and
$\alpha$ are phase and attenuation constants. With the stated time convention,

$$
k^2=\omega^2\mu\varepsilon+\mathrm i\omega\mu\sigma.
$$

The sign convention must remain fixed from equation to data analysis. Using
$e^{+\mathrm i\omega t}$ changes the sign of the imaginary term and the chosen
complex propagation constant. Measured attenuation remains positive in either
convention; a reported negative attenuation usually marks a reference-direction or
calibration error.

$$
% caption: A plane wave in a conducting medium. The carrier keeps a fixed spatial
% period while its amplitude decays as $e^{-\alpha z}$; the dashed envelope shows the
% attenuation, and a single amplitude reading cannot separate the loss constant
% $\alpha$ from the phase constant $\beta$.
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Loss is weak when $\sigma\ll\omega\varepsilon$. The phase constant then remains
close to $\omega\sqrt{\mu\varepsilon}$, while the amplitude decreases slowly with
distance. A strong conductor has $\sigma\gg\omega\varepsilon$; fields penetrate a
short skin depth,

$$
\delta=\sqrt{\frac{2}{\omega\mu\sigma}}.
$$

The skin-depth expression applies to a good conductor under its stated conditions.
It describes the decay of a time-harmonic field within the material, not attenuation
of a free-space wave traveling beside a wire. Frequency, permeability, temperature,
surface roughness, and geometry can change a practical conductor response.

The ratio of electric and magnetic complex amplitudes is the intrinsic impedance of
the medium. In vacuum it reduces to the real constant

$$
Z_0=\sqrt{\frac{\mu_0}{\varepsilon_0}}.
$$

In a lossy material the ratio has magnitude and phase. Electric and magnetic sensor
records can therefore differ in phase even in a single traveling mode without an
instrument fault. Sensor-transfer calibration remains necessary because an apparent
phase difference can also come from cables, antenna loading, or a pickup-loop
derivative response.

Geometric spreading can mimic material attenuation. A spherical wave from a localized
source has amplitude that decreases approximately as $1/r$ in a lossless far zone,
whereas a guided plane-like mode can maintain nearly constant cross-sectional
amplitude until material loss dominates. Divide out the modeled geometric factor
before fitting an attenuation coefficient. A measurement path that changes both
distance from the source and detector orientation confounds propagation loss with
antenna-pattern variation.

A frequency sweep separates several mechanisms. Ohmic attenuation often grows with
frequency through skin effect. Dielectric loss can have a different frequency
dependence set by polarization relaxation. Waveguide cutoff changes the phase relation
near a modal threshold. Fit a physical model only over a range in which the
propagation mode, source coupling, and sensor response remain identified. A single
straight line on a logarithmic amplitude graph does not establish a universal loss
law beyond that range.

## Verification and Worked Survey

The electromagnetic wave equation is a model with several independent predictions.
In a uniform vacuum plane-wave region, the phase speed is $c$, field components are
transverse, electric and magnetic amplitudes have ratio $c$, and the component
phases are aligned. A single delay measurement tests only one of those predictions.
Testing several quantities makes a model failure diagnosable: an amplitude-ratio
error can arise from sensor calibration, a direction error can arise from probe
orientation, and a position-dependent phase error can arise from reflection or
dispersion.

Field-component orientation should be checked by rotating a directional sensor about
the propagation axis. A linear electric probe has a response proportional to the
projection of the electric field onto its sensing axis. The response should follow a
cosine pattern in a plane-wave region after background subtraction. A magnetic loop
has an analogous normal-axis dependence. Angular scans identify cross-polarized
pickup, mount scattering, and sensor-axis misalignment before amplitude-ratio data are
used in a Maxwell-equation comparison.

Spatial residuals carry physical information. A sinusoidal residual versus distance
can indicate reflection from a boundary, because incident and returned waves interfere.
A monotonic residual with frequency can indicate dispersion, channel-delay error, or
an evolving source pattern. A discontinuity after a connector or material change
points to an interface or calibration transition. Plot signed residuals against
distance, frequency, orientation, and source power. An RMS number without coordinates
cannot distinguish these mechanisms.

Repeatability must be structured. Record multiple independent source acquisitions,
then repeat the complete setup after reconnecting cables or repositioning sensors.
The first group estimates short-term noise; the second group probes configuration
repeatability. Temperature can alter cable delay, oscillator frequency, material
permittivity, and sensor gain. Source power can alter amplifier compression or
material response. Record these conditions alongside waveforms so that a later
residual pattern can be traced to an identifiable variable.

Dimensional and limiting checks catch many calculation errors. The vacuum speed from
$1/\sqrt{\mu_0\varepsilon_0}$ must have dimensions of velocity. The plane-wave
amplitude ratio has units $E/B$, also velocity. In the lossless-material limit,
$\sigma$ tends to zero and the damping term in the wave equation vanishes. In the
low-frequency conductor limit, skin depth grows and a thin conductor can approach a
uniform current distribution. These checks should be applied to each algebraic result
before numerical values are substituted.

The final statement of a wave-speed result names the source, propagation medium,
frequency interval or pulse bandwidth, source-to-sensor geometry, field component
measured, calibration reference, delay-extraction method, uncertainty components, and
validity conditions for the plane-wave model. It also lists excluded effects such as
near-source coupling, interface reflections, material loss, waveguide modes, and
dispersion. That scope does not weaken the result; it specifies the physical regime in
which the measured quantity tests the Maxwell wave equation.

> **Worked example (two-antenna wave-speed survey).** Two matched receiving antennas have
> calibrated phase centers separated by $L=(4.000\pm0.005)\ \mathrm m$. After channel-delay
> correction, cross-correlation of repeated pulse records gives a propagation delay
> $\tau=(13.40\pm0.05)\ \mathrm{ns}$. The speed estimate is
>
> $$
> v=\frac{L}{\tau}
> =2.985\times10^8\ \mathrm{m\,s^{-1}}.
> $$
>
> Treating baseline and delay as independent, first-order propagation of uncertainty gives
>
> $$
> \left(\frac{u_v}{v}\right)^2
> =\left(\frac{u_L}{L}\right)^2
> +\left(\frac{u_\tau}{\tau}\right)^2,
> $$
>
> a relative standard uncertainty of $3.9\times10^{-3}$, so
> $u_v\approx1.2\times10^6\ \mathrm{m\,s^{-1}}$. The value sits close to $c$, but agreement
> alone does not validate the plane-wave model: cross-polarized response, reflections, or an
> amplitude-calibration error can each survive this single scalar check.

The survey therefore records field orientation, channel response, source and return
geometry, background level, and residual timing structure in addition to the scalar delay.
A shared clock calibration can correlate
delay estimates from every repetition. A tape-scale calibration can correlate
baseline estimates across every sensor separation. Those common terms remain after
averaging repeated traces and should be reported separately from record-to-record
scatter.

$$
% caption: Geometry of the two-antenna wave-speed survey. A pulse leaves the source,
% and its arrival is timed at two receivers whose phase centres are separated by the
% baseline $L$; a movable absorber tests whether a reflected path contributes a
% second arrival.
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\node[below] at (3.25,1.05) {sensor A};
\node[below] at (5.35,1.05) {sensor B};
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$$

An uncertainty budget groups each term by mechanism. Baseline placement includes
the physical survey, antenna phase-center uncertainty, and wave-normal alignment.
Timing includes digitizer sample interval, interpolation method, channel-delay
calibration, trigger jitter, and correlation-peak ambiguity. Multipath includes
reflections from floor, walls, fixtures, and the source mount. Medium uncertainty
includes humidity or pressure only when the path is in a material whose propagation
constant depends appreciably on those conditions. Sensor response includes angular
misalignment, polarization leakage, and amplifier compression.

Each term should have a sensitivity coefficient. A baseline error enters
$v=L/\tau$ with coefficient $1/\tau$; a delay error enters with coefficient
$-L/\tau^2$. An absorber test estimates multipath by changing the reflected path
while holding source, antennas, and time base fixed. A source-power test estimates
compression by changing signal level while preserving pulse shape. A rotation test
estimates polarization leakage by comparing co-polar and cross-polar response.
Combining all effects into one unexplained standard deviation removes the physical
information needed to improve the experiment.

The delay extraction should be repeated at several baselines. A linear regression of
delay against baseline estimates inverse propagation speed from the slope:

$$
\tau(L)=\tau_{\rm off}+\frac{L}{v}.
$$

The intercept $\tau_{\rm off}$ represents residual fixed channel delay. A nonzero
intercept does not by itself prove a calibration failure; it may quantify a remaining
electrical reference-plane offset. The slope is the physical quantity of interest
provided that source geometry, sensor response, and propagation mode stay unchanged
as baseline is varied.

A multi-baseline survey can also detect reflections. A direct wave delay changes linearly
with distance. A reflection from a fixed wall changes with a different path length and
can appear as a secondary correlation maximum whose separation varies with antenna
position. Use time gates only after preserving the full trace. Deleting the secondary
arrival without documenting its magnitude can bias a phase or delay estimate and mask
a breakdown of the free-space model.

Compare electric and magnetic records only after the sensors have been referenced to
the same physical location. A separated E probe and B loop sample different phases
of a short-wavelength wave. Place them within a small fraction of wavelength, correct
their distinct transfer functions, and calculate the local complex ratio. A result
close to $E/B=c$ with phase near zero supports the vacuum plane-wave relation. A
deviation can be assigned only after cross-axis response, placement, source-near
fields, reflections, and channel calibration have been constrained.

The survey should include a null configuration. Terminate the source or rotate each
sensor to its nominal null axis and record background. A background trace may contain
local oscillator leakage, digitizer pickup, ambient transmitters, or static offset.
Subtracting a mean value without preserving noise spectrum can create artificial
correlation peaks. Report the background amplitude and its spectral lines alongside
the source-on data.

The worked survey demonstrates a general reporting standard. Preserve raw waveforms,
calibration files, coordinate survey, source settings, channel map, processing
scripts, time gates, phase branches, regression residuals, and uncertainty
covariance. State the medium, frequency range, wave polarization, propagation
geometry, and distance from sources and boundaries. A future reader can then test
whether a disagreement with $c$ came from physical propagation, source geometry,
or the data-reduction chain.

### Integral evidence for the local equations

The differential curl equations describe a limit of circulation over a shrinking
surface. Laboratory probes have finite area, so they measure spatial averages rather
than a derivative at a mathematical point. A small pickup loop of area $A_\ell$
oriented with normal $\hat n$ measures induced voltage related to

$$
V_\ell=-\frac{\d}{\d t}\int_{A_\ell}\vec B\mathbin{\cdot}\hat n\,\d A.
$$

The loop response approaches a local magnetic component only when the field changes
little across the loop and the calibration accounts for the frequency derivative.
An electric probe has its own finite sensing length and loading impedance. Mapping a
curl from measured data requires a spatial stencil whose separation, orientation, and
uncertainty are stated explicitly.

An integral Faraday test uses a closed measurement contour. Measure the electric
circulation along the contour by calibrated potential differences or segmented
electric probes, then compare it with the time rate of magnetic flux through the
bounded surface. A small contour improves the local approximation but reduces signal
and increases sensitivity to lead placement. A large contour increases signal while
sampling spatial variation and possible source currents. Select a contour size from
the wavelength, field-gradient scale, sensor noise floor, and desired uncertainty.

Ampère–Maxwell evidence follows the complementary route. A loop around the propagation
axis measures magnetic circulation, while a spanning surface samples changing electric
flux. In an ideal plane wave, a circular loop has symmetries that can simplify the
integral, but a finite beam or nearby source breaks those symmetries. Use numerical
integration of the measured component map when a single-radius shortcut lacks support.
The surface normal and loop direction must follow one right-hand convention through
every sign in the comparison.

Finite-difference derivative estimates amplify spatial noise. If adjacent samples have
uncertainty $u_E$ and separation $\Delta x$, a centered derivative has an
uncertainty scale proportional to $u_E/\Delta x$. Reducing $\Delta x$ improves
spatial resolution while worsening noise amplification. A family of maps taken at
several spacings identifies a plateau in which derivative estimates are stable within
the combined uncertainty. This physical sampling tradeoff resembles the temporal
bandwidth tradeoff in pulse-delay estimation.

The local wave equation is best checked by consistency across several measurements:
spatial curvature from a component map, temporal curvature from a synchronized time
record, transverse orientation from a rotation scan, and propagation delay from a
multi-baseline survey. No one measurement reconstructs every derivative and component.
Agreement across those observables ties the reported speed to the Maxwell curl laws
and to the source-free assumptions used in the derivation.

The full conclusion should distinguish a measured propagation result from a universal
claim. A vacuum plane-wave experiment can establish consistency with
$\omega=ck$, $B_0=E_0/c$, and transverse geometry over a stated frequency,
baseline, and uncertainty interval. Material loss, finite apertures, conducting
boundaries, source-near fields, and numerical truncation require their own wave
models. Keeping those regimes separate preserves the content of the wave equation
instead of turning it into a generic label for any changing electric and magnetic
signal.