Electromagnetic Waves

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

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.

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

The vector identity

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

An identical calculation yields

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:

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 $\nabla \cdot \vec{E}=0$ in a source-free volume, and the corresponding magnetic field must satisfy $\nabla \cdot \vec{B}=0$. 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

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

The signs depend on the chosen component directions. The physical check is the right-handed relation $\vec{E}\times \vec{B}$ 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

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

The reduced curl equations set the amplitude ratio:

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.

Phase surfaces satisfy $kx-\omega t+\varphi =\text{constant}$. Differentiating a constant-phase condition gives $\mathbf{d}x/\mathbf{d}t=\omega /k=c$. 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.

Transversality

The divergence equations impose transversality for a uniform plane wave. Taking $\vec{E}={\vec{E}}_0\cos (\vec{k}\cdot \vec{r}-\omega t)$ gives

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

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.

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

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 $\Delta \varphi$, then

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 $v=1/\sqrt{\mu \epsilon }$ 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:

Faraday's law then requires

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

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

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.

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

A left-moving component has

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:

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:

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 $\nabla \cdot \vec{E}=0$ and $\nabla \cdot \vec{B}=0$ 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.

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

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 $v_{\mathrm{p}}=\omega /k$ and group velocity $v_{\mathrm{g}}=\mathbf{d}\omega /\mathbf{d}k$ differ. A narrow-band envelope travels near $v_{\mathrm{g}}$, while individual crests travel near $v_{\mathrm{p}}$. 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.

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

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 $S\le 1$. 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.

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,

Small $k\Delta x$ reproduces ${\omega }_{\mathrm{num}}\simeq ck$. 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

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

where ${\tilde{q}}_i$ is the physical quantity coupled to the sensor, $H_i$ is the calibrated complex response, and ${\tilde{n}}_i$ 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 ${\tau }_{i,\mathrm{ch}}$ denote the channel delay. A raw two-channel phase difference contains

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

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 $s_2(t)$:

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 $1/\sqrt{\mu \epsilon }$.

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:

Taking a curl with Faraday's law gives

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

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,

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.

Loss is weak when $\sigma \ll \omega \epsilon$. The phase constant then remains close to $\omega \sqrt{\mu \epsilon }$, while the amplitude decreases slowly with distance. A strong conductor has $\sigma \gg \omega \epsilon$; fields penetrate a short skin depth,

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

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 $1/\sqrt{{\mu }_0{\epsilon }_0}$ 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.

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.

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:

The intercept ${\tau }_{\mathrm{off}}$ 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

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.