Wave Packets

Bandwidth and packet extent

A single harmonic component extends across all space and repeats for all time. A localized disturbance has a leading region, a central region, and a trailing region. Its mathematical description therefore combines a range of wave numbers. A right-moving, narrow-band disturbance has the complex signal

The measured displacement, voltage, pressure variation, or another real signal is $y(x,t)$. The complex form keeps the amplitude and phase of every harmonic component together. The spectral amplitude $\tilde{\psi }(k)$ states how strongly each wave number contributes at $t=0$; the dispersion relation $\omega (k)$ sets the subsequent phase accumulation. A narrow spectral band centered near $k_0>0$ gives a recognizable carrier pattern inside an envelope. A broad band gives a shorter disturbance with fewer carrier oscillations.

The integral carries a physical qualification. Linear superposition must describe the medium over the amplitudes in use. A taut string at small transverse slope, small-amplitude sound, a linear electrical line, and a weak optical pulse satisfy that condition within the required accuracy. Large amplitudes can make the propagation law amplitude-dependent. Then the spectral components exchange energy and the simple fixed relation $\omega (k)$ no longer predicts the whole packet.

At an initial time, the transform pair is

The transform pair converts a measured spatial trace into wave-number content and a proposed spectrum into an expected pulse shape. The two descriptions contain the same information when the spatial record is long enough and adequately sampled. A finite record approximates the integrals, a limitation developed later in the lesson.

The expression $k$ denotes angular wave number, measured in radians per metre. Ordinary spatial frequency ${\nu }_x$ measures cycles per metre, so

Keeping angular and ordinary frequency separate avoids a common factor-of-$2\pi$ error in packet calculations. A graph labeled in hertz requires $f$ on its horizontal axis; a graph labeled in radians per second requires $\omega$. The same distinction applies to wave number. Spectrometers and fast Fourier transform software often report either convention, so the axis label must be read before a slope is converted into a speed.

Carrier and envelope

A spectrum concentrated near $k_0$ can be written in the form

The slowly varying complex amplitude $A$ is the envelope. Its magnitude identifies the region where the packet is appreciable. Its argument can carry a gradual phase variation in addition to the rapid carrier oscillation. The separation applies when the envelope length is several carrier wavelengths and the spectrum is narrow relative to $k_0$. A packet containing only one or two cycles has no clean visual division between carrier and envelope, even though the Fourier integral remains valid.

Envelope width requires an operational definition. A laboratory report may use full width at half maximum, the interval above a threshold, a root-mean-square width, or the time between two selected landmarks. These definitions yield different numbers for the same pulse. A report needs to name the definition and retain it for every data set. Root-mean-square widths have compact Gaussian Fourier relations and propagate directly through uncertainty calculations.

Bandwidth, duration, and spatial extent.

A short pulse must occupy a broad frequency interval. The order-of-magnitude relation is

The symbols $\Delta f$, $\Delta t$, $\Delta k$, and $\Delta x$ do not carry one universal definition. The numerical constant depends on whether width means a standard deviation, a half-maximum span, a first-null span, or a chosen signal gate. The reciprocal relationship survives those choices. Halving the duration roughly doubles the frequency range needed to synthesize the pulse. In a nondispersive medium with speed $c$, $\Delta x\sim c\Delta t$ and $\Delta k\sim 2\pi \Delta f/c$.

A Gaussian packet is an exact benchmark. Let its initial complex signal be

Its spectral amplitude is another Gaussian centered at $k_0$,

With ${\sigma }_x$ and ${\sigma }_k$ interpreted as standard deviations of the corresponding intensity-like distributions, the product is

The equality applies to this Gaussian convention. Calculations based on full width at half maximum require the Gaussian conversion $\mathrm{FWHM}=2\sqrt{2\ln 2}\sigma$ for the relevant quantity. Applying one width convention on one side of the relation and another convention on the other side creates a misleading bandwidth estimate.

The duration-bandwidth relation has practical consequences for pulse design. A timing experiment benefits from a short envelope because nearby returns remain separated in time. The same short envelope spreads across more frequencies, which can encounter frequency-dependent loss or dispersion. A long narrow-band packet measures phase and a single carrier frequency well, yet it gives poor localization of an event in time. Required timing and frequency resolution set the acceptable compromise.

Finite-duration records introduce a related issue in data analysis. Multiplying an unbounded signal by a finite time gate broadens its spectrum through convolution with the spectrum of that gate. A sharp rectangular cut gives long spectral side-lobes. A tapered gate lowers side-lobes but widens the main spectral lobe. Gate selection belongs in the measurement method. Report the gate, its duration, and whether the measured amplitude or power spectrum has been displayed.

Phase, group velocity, and dispersion

One harmonic component has the phase

A particular crest is a constant-phase marker. Setting $\mathbf{d}\varphi =0$ gives

The subscript p denotes phase. This speed follows an individual carrier crest or any other fixed value of phase. It can be measured by tracking a repeated zero crossing with the same phase direction, provided the crest identity remains clear between sensors. The envelope speed can differ from the phase speed.

Group velocity emerges from two neighboring components. Let $k_1=\bar{k}-\Delta k/2$, $k_2=\bar{k}+\Delta k/2$, and define their corresponding angular frequencies in the same way. Addition of the two cosine waves gives

The first cosine gives a slowly varying envelope when the differences are small. Its maxima travel at $\Delta \omega /\Delta k$. As the spectral separation tends to zero, the packet velocity becomes

The subscript g denotes group. The derivative is the tangent slope of the dispersion curve; $\omega /k$ is the origin-to-point slope. They coincide when the relation is locally a straight line through the origin.

The two-wave derivation gives intuition, while the derivative requires a narrow enough spectrum that one tangent describes the occupied part of $\omega (k)$. A broad packet samples substantial curvature and higher derivatives. Its peak, centroid, leading threshold crossing, and energy-weighted center can follow different trajectories as the waveform distorts.

Phase and group tracking answer different experimental questions. A phase-delay measurement compares the phase of a narrow-band sinusoid at two positions. It gives a delay modulo whole cycles and needs an independent estimate of the cycle count. An envelope-delay measurement sends a localized packet and records its arrival at two positions. It gives a broad-band transit time, often without cycle ambiguity. A packet can have a phase marker that overtakes its envelope or falls behind it without any material feature moving at either speed.

Group velocity often approximates the speed of energy transport in a lossless, linear, single-mode system. That statement has conditions. Strong attenuation, gain, multiple propagating modes, sharp spectral filtering, boundaries, and a very broad spectrum can separate an energy centroid from the envelope maximum. The earliest causal change at a receiver also depends on source turn-on and the high-frequency behavior of the medium. A laboratory report should state which landmark was timed and call the result a peak speed, centroid speed, correlation speed, or phase speed as appropriate.

Dispersion relations.

The dispersion relation specifies the allowed angular frequency for each wave number. It may come from a differential equation, a lattice model, a measured phase delay, or a two-dimensional transform of a recorded pulse. The curve is a property of the mode and the state of the medium. Tension, depth, temperature, material structure, mean flow, and boundary geometry can alter it. A graph of frequency against wave number therefore carries more physical information than a single quoted wave speed.

An ideal nondispersive medium with a right-moving branch has

Substitution into the packet integral gives

Every part of the initial profile shifts by the same distance $ct$. Carrier crests, envelope peak, width, and a threshold crossing all keep their relative positions. The result uses the entire straight-line relation over the occupied bandwidth. A dispersion curve that is approximately straight close to $k_0$ preserves shape only over the corresponding small spectral interval.

The phrase nondispersive applies to a bandwidth and a measurement tolerance. A steel cable, for example, may appear nondispersive in a low-frequency experiment where tension controls the restoring force. Bending stiffness becomes important at shorter wavelength and produces curvature in the relation. A pulse can therefore retain shape over a metre-scale path and broaden substantially over a shorter wavelength range in the same physical specimen. The appropriate question is whether variation in $\mathbf{d}\omega /\mathbf{d}k$ across the actual spectral band is small enough for the desired measurement accuracy.

Deep-water gravity waves provide a familiar dispersive example. For wavelength small compared with water depth, the relation is

Longer waves have smaller $k$ and travel faster than shorter waves in this model. The crest pattern moves twice as fast as the envelope. A person watching a train of water ripples can therefore see crests emerge at the rear of a group, cross its middle, and fade near the front while the group itself travels at half the crest speed. The result belongs to deep water; finite depth changes the relation and approaches the nondispersive shallow-water result at sufficiently long wavelength.

A discrete mass-and-spring chain shows how an underlying length scale creates dispersion. Equal masses $m$ separated by distance $a$ and joined by springs of constant $K$ have, for the first Brillouin zone,

For $0<ka<\pi$,

At long wavelength, $\sin (ka/2)\approx ka/2$ and the chain behaves like a continuous string with speed $a\sqrt{K/m}$. Near the zone edge, neighboring masses move almost out of phase, the dispersion curve flattens, and the group speed falls toward zero. This model also explains a numerical hazard: a spatial grid is a discrete chain in computational form, even when the intended physical medium is continuous.

Measurements of a dispersion relation usually sample a finite range. One method drives a narrow-band source at several frequencies, measures phase shift between two positions, resolves the integer-cycle ambiguity, and infers $k(f)$. A second method launches a broad packet, records $y(x,t)$ on a sensor line, and locates spectral power in the $(k,\omega )$ plane. The first method has high frequency resolution and weak timing localization. The second separates several occupied branches at once but needs careful spatial calibration, sampling, and windowing.

Curvature produces broadening and chirp.

The local shape of the dispersion curve controls the first changes in a narrow packet. Put $q=k-k_0$ and expand about the central wave number:

The constant term advances the carrier phase. The linear term translates the envelope at $v_{\mathrm{g}}$. The quadratic term gives different parts of the spectrum a phase shift that grows as $q^{2}t$. Its effect is visible once the relative phase across the occupied bandwidth reaches a substantial fraction of a cycle. The cubic term and higher terms become relevant for a broad spectrum, a long path, or a region where curvature changes rapidly.

The expansion is a local model. It applies when the occupied interval $k_0-\Delta k/2$ to $k_0+\Delta k/2$ lies within one smooth branch of the dispersion relation. A packet whose spectrum crosses a cutoff, a mode crossing, or a strongly attenuated band needs the full measured relation. Fitting a quadratic across a kink may give a numerical coefficient while hiding the physical change of mode.

The Gaussian initial packet used earlier has an exact width formula when the quadratic term is retained:

The result requires ${\sigma }_{x0}$ to be the standard deviation of the intensity-like envelope and that the initial envelope has no chirp. Dimensional checks expose the structure. Since ${\omega }_2$ has units of metres squared per second, ${\omega }_{2}t/{\sigma }_{x0}^2$ is dimensionless. At $t=0$, the width is ${\sigma }_{x0}$. At large $|t|$, the width grows approximately as $|{\omega }_2|t/(2{\sigma }_{x0})$. A narrower initial packet has a broader spectrum and therefore broadens more rapidly under the same curvature.

The quadratic phase also produces chirp. The local wave number is the spatial derivative of the total phase,

At a fixed time, changing crest spacing across the envelope indicates that $k_{\mathrm{local}}$ differs from one side of the packet to the other. A positive frequency-versus-time slope at a fixed sensor is called an up-chirp; a negative slope is a down-chirp. The sign depends on the sign of ${\omega }_2$ and on the coordinate and Fourier conventions. A report should give the measured trace and the convention before attaching an up- or down-chirp label.

Group delay is a frequency-domain route to the same quantity. A wave component passing through a uniform length $L$ acquires phase $\Phi (\omega )=k(\omega )L$. Differentiating gives

A phase-delay instrument can estimate the slope of $\Phi (\omega )$ across a narrow band, then infer group delay without launching a visibly short pulse. Phase must be unwrapped continuously before differentiation. A missing cycle adds $2\pi$ to one point and creates a false large slope. Multiple sensor spacings help identify and remove that ambiguity.

Dispersion and attenuation leave different signatures. Pure phase dispersion preserves the magnitude of each spectral component while changing its phase. Frequency-dependent attenuation changes the magnitude and can alter packet width even if the phase relation is nearly linear. A practical propagation model uses a complex transfer factor,

where $\alpha (k)$ represents amplitude decay. Fitting broadening with a phase-only model when high-frequency components are being removed can return an incorrect curvature. Record spectra at more than one propagation distance whenever the medium has appreciable loss.

Packet-speed measurement and sampling

A pulse-speed result starts with a definition of the measured event. The largest sample value, the first threshold crossing, the envelope maximum, a fitted Gaussian center, the signal centroid, and the peak of a cross-correlation all answer slightly different questions. Their values converge for a clean, symmetric packet that changes little over the sensor spacing. They diverge when the packet broadens, develops a tail, overlaps a reflection, or carries strong noise. Naming a method after the calculation is preferable to calling every result a generic wave speed.

Two fixed sensors separated by a surveyed distance $d$ provide a basic pulse-delay measurement. If the same packet landmark appears at times $t_1$ and $t_2$, then

The distance must be the propagation-path distance between the effective sensing points. A microphone diaphragm, an accelerometer mounting point, and a camera pixel each have a physical location; a cable label or a ruler position on a bench can be offset from it. A sensor pair placed too close together yields a small delay that is dominated by timing uncertainty. A very large separation gives a clean delay but allows more distortion, attenuation, and reflections. A pilot recording can locate a spacing that balances those effects.

Peak timing uses the time coordinate at the largest envelope value. It works well when the pulse has a single dominant lobe and adequate signal-to-noise ratio. An unfiltered raw carrier can shift its largest instantaneous value by a fraction of a cycle as phase changes, so an envelope or a fitted template is often more stable. Threshold timing suits a sharp leading edge but depends on threshold level, gain, baseline drift, and high-frequency noise. A fixed fraction of each packet's own peak reduces gain sensitivity, yet it shifts when the leading-edge shape changes with propagation.

A centroid method reduces dependence on one noisy sample. After subtracting a baseline and selecting a gate that contains the packet,

where $E(t)$ is an envelope or another nonnegative amplitude measure. Squaring is appropriate when the chosen quantity is proportional to amplitude and the desired weight is energy-like. Other physical signals may require a calibrated energy flux instead. The integration gate must include comparable fractions of the packet at both sensors. Cutting off a growing tail at only one sensor biases the centroid toward the retained side.

Cross-correlation is effective when the waveform is recognizable but noisy. For baseline-corrected records $s_1$ and $s_2$, define

If sensor 2 receives a delayed copy, $s_2(t)=s_1(t-{\tau }_d)$, the maximum occurs at $\tau ={\tau }_d$ with this convention. The correlation speed is

Distorted or reflected traces produce shifted or multiple correlation peaks. A large secondary peak can indicate an echo, a periodic carrier ambiguity, or two propagation modes. Display the correlation curve, its search interval, and the two original gated records. The maximum should be fitted or interpolated between samples when sub-sample timing matters.

Phase-delay measurements need a different reduction. Isolate a narrow spectral line, compute the complex phase at each sensor, and form the unwrapped difference $\Delta \varphi (f)$. A right-moving component over a path of length $d$ has

The integer $N$ counts full cycles between sensors. It is an experimental unknown unless the spacing is short compared with wavelength or a swept-frequency record tracks the phase continuously. Sampling a single frequency at widely separated sensors can produce several equally plausible phase speeds. A broadband packet arrival or a third sensor adds independent information that resolves the branch.

Sensor response belongs in the method section. A sensor can add phase delay, attenuate high frequencies, saturate, or respond to a spatial average rather than a point. Identical channels cancel some systematic delay in a difference measurement; different channels require calibration. Record the trigger source, sample clock relationship, sensor orientation, mounting condition, source drive waveform, and the time interval used for analysis. These details determine whether a repeated measurement has the same physical meaning.

Sampling limits the packet information in a record.

A digitizer records a sequence with interval $\Delta t$ and sampling rate $f_{\mathrm{s}}=1/\Delta t$. The highest recoverable temporal frequency is the Nyquist frequency

Every appreciable component in the analogue signal must lie below $f_{\mathrm{Ny}}$ after the analogue front end has done its work. Components above that boundary fold into the recorded band as aliases. A 900 Hz tone sampled at 1.0 kHz produces the same sample sequence as a 100 Hz tone with a suitable phase. Once the samples have been stored, later processing cannot recover which original tone was present. An analogue low-pass stage before digitization removes frequencies that the sampler cannot represent.

The sample interval also sets timing granularity. Packet delays can lie between sample times; a smooth local fit or an interpolated cross-correlation peak estimates a fractional-sample delay. The attainable precision depends on bandwidth, noise, clock jitter, and waveform shape. Repeated trials estimate an empirical delay spread that includes many of these effects.

For $N$ equally spaced samples, the discrete Fourier transform represents frequency bins separated by

Longer acquisition time narrows the bin spacing and separates nearby stable tones. Rapidly changing packets can leave the record window, while long records can include reflections, source drift, or unrelated events. Gate selection must therefore match the physical question: use a clean early window for a direct-path speed, or model later arrivals when they are part of the experiment.

Zero padding appends samples with value zero before a transform. It draws the discrete transform on a denser set of displayed frequency locations, which makes a peak easier to interpolate or plot. Record duration and the main-lobe width of the chosen gate remain unchanged, so resolving power remains that of the acquired measurement. A report should distinguish the original count of acquired samples from the transform length after padding.

Spatial sampling has the same structure. A camera line scan, a row of probes, or a numerical grid has spacing $\Delta x$. The highest unaliased wave number is

The shortest wavelength represented without spatial aliasing is ${\lambda }_{\min }=2\Delta x$. Accurate propagation needs more than two points per wavelength because phase and amplitude estimates remain poorly resolved at that limit. Ten or more points per shortest important wavelength is a common engineering target when gradients, dispersion, or nonlinear post-processing matter; convergence checks should set the actual requirement.

A line of $N_x$ spatial samples spanning $L=N_x\Delta x$ has wave-number bin spacing

A long spatial aperture distinguishes nearby wave numbers. A short aperture represents a localized packet in a small region but smears its spectrum. Temporal and spatial design must be considered together when the goal is a dispersion curve: the packet must remain inside the recorded space-time window, the sensors must resolve its carrier, and the window must be large enough to distinguish its spectral band from nearby modes.

The two-dimensional transform of a sensor array produces $S(k,\omega )$. A right-moving branch and a left-moving branch occupy different signed regions of this plane under a stated transform convention. The sign convention is easy to reverse by changing the exponential in software. Validate it with a known traveling test signal before interpreting a measured ridge as a forward or backward mode.

Numerical propagation requires an additional audit because a stable calculation can still have the wrong dispersion relation. The centered finite-difference update for the one-dimensional wave equation is

Substitution of a trial solution proportional to $\exp [i(jk\Delta x-n\omega \Delta t)]$ into this scheme gives

Stability in one dimension requires $r\le 1$. At $r<1$, the numerical phase and group velocities fall below $c$ for wave numbers near the grid limit. The computed packet broadens even when the physical relation is nondispersive. Reducing both steps while holding the physical packet fixed tests whether observed broadening is a medium effect or a grid effect.

The case $r=1$ in this one-dimensional centered scheme is special. On the resolvable branch, ${\omega }_{\mathrm{num}}=ck$. Boundary reflections, source discretization error, floating-point accumulation, and material-model error still require separate checks. In two or three dimensions, directional grid effects reappear even when a one-dimensional Courant ratio is chosen carefully. A numerical report needs the grid spacing, time step, boundary treatment, source bandwidth, and a refinement comparison.

Uncertainty, validation, and record preparation

Every pulse-speed measurement contains at least a distance uncertainty and a timing uncertainty. For independent small standard uncertainties, a speed $v=d/\Delta t$ has the approximate relative uncertainty

The expression applies to random or independently estimated components. Systematic biases require a separate treatment. A shared clock offset can cancel in a same-clock delay measurement, while unequal channel delays remain as a fixed bias. A tape-measure scale error affects every sensor spacing in the same direction. A report should list these contributions separately from trial-to-trial scatter and explain how each was bounded or calibrated.

Common timing contributions include

• digitizer sample interval and clock jitter;
• interpolation or template-fit uncertainty;
• sensor response and unequal cable delays;
• source trigger stability;
• baseline noise and the selected analysis gate;
• waveform distortion between the two sensing positions.

Distance uncertainty can include ruler calibration, sensor aperture, curved path geometry, thermal expansion of a guide, and the location of an effective sensor center. A one-dimensional model assumes one path. A broad source or a receiver with large area averages over a family of paths and can move the measured envelope peak even with perfect clocks.

Validation and failure checks

A strong packet model predicts more than one number. It predicts the arrival delay, carrier phase, width change, chirp sign, and spectral evolution over several distances. Hold one record aside when fitting a dispersion relation, then use the fit to predict that unseen record. Agreement across several packet widths is more informative than agreement with a single peak time.

Residual structure points toward a physical cause. A repeated delayed copy suggests a reflection. Two arrival ridges in a space-time transform suggest multiple modes. A shift that depends on amplitude suggests nonlinearity. A systematic high-frequency loss suggests attenuation or sensor response. A change that disappears when the grid is refined identifies numerical dispersion. The diagnostic should return to the raw records; an attractive fitted curve alone cannot separate these mechanisms.

An analysis record should retain enough information for another person to repeat the reduction. Include the source waveform and amplitude, medium state, geometry, sensor positions, sample rate, analogue bandwidth, clock method, gate edges, baseline treatment, transform convention, window type, peak or correlation rule, and uncertainty method. Save raw traces as well as processed figures. These metadata convert a speed estimate into an auditable experimental result.

Preparing and interpreting a packet record.

The source waveform matters as much as the propagation path. A burst generator can produce a nominally narrow-band carrier with a sharp turn-on, a tapered envelope, an initial chirp, or amplitude drift from one trial to the next. Each choice changes the launched spectrum. Characterize the source close to its emission point before using a distant record to infer a medium property. Otherwise a source chirp or a source-spectrum notch can be mistaken for propagation-induced dispersion.

A frequency-domain description of a linear source-path-sensor chain has the form

The factors represent source conversion, propagation, sensor response, and additive noise. A transfer-function correction may be appropriate when those factors have been independently calibrated. Division by a very small response factor amplifies noise and should be constrained to a band with adequate signal strength. Correction cannot recover information outside the acquisition bandwidth or beyond the sampling limit.

An envelope extraction procedure needs an explicit bandwidth choice. For a narrow-band carrier, form a complex quadrature representation of the recorded signal and take its magnitude. Broad, one-lobed pulses can be treated more transparently by direct smoothing of a calibrated energy-like measure. Both methods can shift peaks near the edge of a short gate. Apply the same procedure at every sensor and inspect a raw trace alongside its envelope before treating a fitted maximum as a physical packet center.

Packet overlap limits what a delay estimator can resolve. Two arrivals that are well separated relative to their envelope width produce two distinct correlation features. As their separation falls below the width, a single peak drifts between them. Increasing bandwidth can sharpen the envelope and improve timing separation, but it can also increase dispersion and attenuation. A model with two paths should be fitted to the full waveform when a direct arrival and an echo overlap; selecting the tallest sample discards the structure that identifies the two paths.

Several modes can share the same source and sensor pair. A plate may carry bending and extensional components; a fluid guide may carry more than one acoustic branch; a structured material can split a packet into polarization or path families. Each branch has its own $\omega (k)$ and its own group speed. A broad spatial-temporal transform can separate two ridges, while a single time trace may show a leading packet and a slower tail. Treating the combined record as one broadened mode gives an effective speed whose value changes with gate placement.

Spatially varying media require a local interpretation. If tension changes along a string, water depth changes along a path, or a material has a temperature gradient, the packet can change its central wave number and bandwidth as it travels. A single global dispersion curve is then an approximation. Short sensor separations can estimate a local group delay, while a longer experiment measures the integrated travel time through the changing medium. The geometry and medium-state profile belong in the model before a speed difference is attributed to dispersion alone.

A compact analysis sequence keeps the stages visible.

1. Calibrate the sample clock, sensor locations, and channel delays.
2. Record a source reference and several repeated propagation trials.
3. Plot raw time traces before baseline removal, gating, or smoothing.
4. Select a gate from arrival geometry and state the gate edges in physical time.
5. Check temporal and spatial spectra against the sampling limits.
6. Extract a named landmark, correlation lag, or unwrapped phase slope.
7. Compare the result across at least two sensor gaps or propagation distances.
8. Test a refined numerical grid or an independent measurement method when a model claims dispersive broadening.
9. Preserve raw data, code settings, transform convention, and uncertainty inputs.

The sequence applies across instruments and exposes ambiguous packets, aliased spectra, and reflection-selected delays before they become fitted parameters.

Limits of group-speed statements

The derivative $\mathbf{d}\omega /\mathbf{d}k$ has a direct packet interpretation when the occupied spectrum is narrow around one smooth dispersion branch. A practical bandwidth indicator is

A small value of ${ϵ}_k$ supports a carrier-envelope description when $k_0$ is safely away from zero and from a cutoff. The accumulated quadratic phase, roughly $|{\omega }_2|(\Delta k{)}^{2}t$, must also remain small enough across the propagation interval. It determines how much the initially narrow spectrum spreads in phase.

Use a feature-specific name when the packet conditions fail. A leading-edge speed comes from a stated threshold. A peak speed comes from the maximum of a stated envelope. A centroid speed comes from a stated weighting and gate. A correlation speed comes from a stated lag convention and search range. In a lossy or multi-mode system, report those feature speeds by their own definitions and reserve $\mathbf{d}\omega /\mathbf{d}k$ for the smooth-branch derivative.

An apparently superluminal or negative envelope peak in an absorbing, sharply filtered, or strongly reshaped signal is also a feature-tracking result. The peak can move because the waveform has changed shape. A causal signal front and a newly introduced modulation require a separate analysis of source turn-on and high-frequency response. The packet formulas here describe linear spectral evolution inside the measured band and apply to the defined packet landmarks.

A reported group velocity needs three views of the same experiment: a measured $\omega (k)$ slope over the occupied band, a direct two-sensor packet delay, and a propagation record showing limited distortion over the chosen path. Agreement among those views supports the stated model range. Disagreement identifies a source chirp, an unmeasured mode, an alias, a reflection, a lossy band, or numerical grid dispersion before those effects are folded into a single misleading speed.

Bandwidth-sweep check.

A bandwidth sweep directly tests the packet model. Hold the central frequency, source position, sensor positions, and medium state fixed. Launch at least three envelopes with different durations, and measure each packet's input spectrum, delay, width, and chirp over the same path. A nearly linear dispersion branch gives equal centroid delays within uncertainty and little width change across the sweep. Curvature produces a systematic dependence: the shorter, broader-band packet accumulates more relative phase and broadens sooner.

Use the same width definition at the source and receiver. A threshold width can change because attenuation lowers the peak even when the pulse shape is unchanged. Root-mean-square width or a fitted envelope parameter separates that gain effect from genuine temporal spreading more reliably. Plot width squared against propagation time for a Gaussian-like packet. Under the quadratic model,

Comparison with the predicted functional form tests the width model, while the sign of ${\omega }_2$ appears in the chirp rather than in width alone. Repeating the sweep after reducing source amplitude checks the linear-superposition assumption. Repeating it on a refined numerical grid checks the propagation algorithm. A model that matches one packet width but fails the bandwidth sweep has identified a local fit, not a validated dispersion law.