Wave Superposition

Superposition of travelling disturbances

In a linear medium, overlapping disturbances add point by point. If two independently produced displacements are $y_{1} (x, t)$ and $y_{2} (x, t)$ , the displacement measured when both are present is

y (x, t) = y_{1} (x, t) + y_{2} (x, t) .

The rule applies at the same position and the same time. It does not require either disturbance to stop while the other passes. Each pulse continues according to the wave equation appropriate to the medium; during their overlap, a probe reads their algebraic sum. Positive and negative displacements must use one common sign convention. A pulse above equilibrium and a pulse below equilibrium can therefore cancel locally, whereas two pulses on the same side of equilibrium reinforce.

The linear condition concerns the restoring physics rather than formal algebra. The restoring force must be approximately proportional to displacement and the material response must remain in the small-amplitude regime. A rope pulled into a sharp kink, a medium driven into plastic deformation, or a detector driven into saturation can produce a record that does not equal the sum of separate records. A superposition test should therefore begin with pulse amplitudes low enough that doubling one source approximately doubles its single-pulse signal without changing shape or speed.

Point-by-point superposition. Two same-sign pulses that overlap add to a resultant taller than either component; where the pulses had opposite signs the sum would be smaller than the larger component. Each pulse keeps propagating unchanged through a linear medium.

Constructive interference describes a region where contributions have the same sign and the sum has greater magnitude than either contribution alone. Destructive interference describes a region where opposite-signed contributions reduce the local displacement. Neither label is a statement that energy appears or vanishes. They name the shape of the summed displacement field at a particular place and time. For pulses of unequal amplitude, destructive overlap can reduce the signal without bringing it to zero; complete cancellation requires equal and opposite values at every point of the overlap region.

For periodic travelling waves with the same angular frequency, phase is a compact description of alignment. A sinusoidal component can be written

y = A cos (k x - ω t + ϕ) .

Two equal-amplitude components have maximum reinforcement when their phase difference is an integer multiple of $2 π$ and maximum cancellation when it is an odd multiple of $π$ . Their resultant amplitude is

A_{sum} = 2 A cos \frac{Δ ϕ}{2} .

Phase is not required to add isolated pulses. A one-time pulse has no repeating cycle from which a unique phase angle can be assigned. Its overlap is described directly by its displacement profile and arrival time. Phase language for a short pulse has a defined meaning only after a waveform convention has been specified, such as alignment of its peak or a fitted carrier cycle.

Equal-frequency components in and out of phase. In phase, the resultant reaches twice the component amplitude; a half-cycle out of phase, equal components cancel to a flat resultant. Intermediate phase differences give intermediate amplitude.

Energy interpretation needs the complete wave variables, not displacement alone. A small transverse disturbance on a stretched string has local energy density containing both the squared transverse speed and the squared slope. When two pulses overlap, squaring the summed variables produces cross terms. Constructive displacement overlap can place more energy temporarily in strain, while an instant of zero summed displacement can coincide with nonzero transverse speed and kinetic energy. The total energy of an ideal isolated system remains accounted for; superposition redistributes energy density in space and between kinetic and elastic forms.

Identical upward pulses travelling toward one another illustrate the point. At their full overlap, the slopes add while the transverse velocities can cancel, producing a large elastic-energy density. For equal pulses of opposite sign, the displacement and slope can cancel at full overlap while transverse velocities add, producing kinetic energy instead. A photograph at the instant of cancellation is therefore incomplete evidence about the energy. A time-resolved record or a measurement of velocity is needed before claiming that a region contains no wave energy.

Experimental pulse records test superposition directly. Record pulse 1 alone, pulse 2 alone, and both sources active using the same position calibration, time base, and trigger rule. Shift the single-source traces to a common launch reference and form the sample-by-sample sum. The residual

r (x, t) = y_{both} (x, t) - [y_{1} (x, t) + y_{2} (x, t)]

should be consistent with measurement noise and calibration uncertainty in a linear run. A residual concentrated near overlap can indicate nonlinear response, an arrival time error, or an amplitude calibration drift. A residual present before either pulse arrives indicates baseline or trigger mismatch rather than a failure of superposition.

Pulse experiments require timing resolution high enough to distinguish the overlap interval. Sensor bandwidth must preserve the pulse shape, and the amplitude scale must be checked separately for each channel. Repeating the three-record sequence at several amplitudes tests the linear range: normalized residuals should remain statistically similar as the drive is changed. Keep the source settings, medium tension or density, sensor locations, sampling rate, and trace-alignment rule with the data. These details turn a visual overlap into a reproducible measurement of superposition.

Phase measurements require a reference convention. For periodic records, choose one repeatable event such as an upward zero crossing and assign it a phase origin. The phase difference at a sensor is then the time difference between corresponding events multiplied by the common angular frequency. A time offset in the acquisition channels appears directly as a phase offset, so channel delay must be calibrated before using a measured phase difference to predict constructive or destructive sums. A phase value without a stated reference, frequency, and sign convention cannot be reproduced.

For unequal sinusoidal amplitudes, a phase shift changes the resultant but cannot force complete cancellation unless the amplitudes match. The general amplitude relation is best checked by comparing the vector sum of the two measured complex amplitudes with the measured combined record. In an ordinary time trace, an apparent reduction may also come from a small arrival-time shift that moves two narrow pulses past one another. Separating amplitude calibration from timing alignment avoids assigning every mismatch to a change in wave physics.

Energy checks should use quantities appropriate to the medium. On a string, estimate transverse slope and velocity from spatial and time-resolved displacement records; in another medium, use the corresponding elastic and kinetic variables. The local energy record need not resemble the displacement record. In particular, a cancellation point can have small displacement while its velocity estimate is large. Integrating the energy density over a region that contains both pulses before, during, and after overlap is a stronger check than inspecting one location at one instant. Losses or source work must be included if the chosen region is not isolated.

Residual analysis needs an uncertainty scale. Divide the record difference by the combined noise and calibration uncertainty at each sample to form a normalized residual. Random residuals with no systematic timing or overlap pattern support the linear model. A residual that grows faster than the input amplitude is evidence that the drive has left the linear range or that the measurement chain has compressed the signal. Record the baseline noise with both sources off, then verify that the simultaneous-source record remains within the detector’s linear output range before interpreting a failed sum as a property of the medium.

Signed pulse addition, phase control, and overlap energy

Algebraic addition is most reliable when it is performed sample by sample. At a chosen position and time, record the signed displacement due to each source using the same equilibrium reference. A value of $+ 3 mm$ combined with $- 1 mm$ gives $+ 2 mm$ , not $4 mm$ ; a value of $+ 3 mm$ combined with $- 3 mm$ gives zero displacement at that sample. The signs belong to physical direction, not to the order in which pulses were launched. Reversing a detector cable or redefining upward as negative changes the displayed signs of every trace but leaves the physical sum unchanged when the convention is applied consistently.

Pulse shape matters as much as peak amplitude. Two pulses may have equal peaks but different widths or asymmetry, so their pointwise sum can be constructive over one part of the overlap and destructive over another. A maximum-to-maximum comparison misses this structure. The practical calculation aligns the two single-source records on one calibrated time base, samples both at matching times, and adds their signed values. Interpolation should be stated if the records have different sample times; otherwise an apparent residual can come entirely from comparing neighbouring points on a steep pulse edge.

Signed sample addition during pulse overlap. At each common time sample the resultant is the algebraic sum of the two single-source displacements: a positive and a negative contribution partly cancel at sample 1, while two positive contributions add at sample 2.

For equal-frequency sinusoidal components, the time alignment is summarized by phase difference. With amplitudes $A_{1}$ and $A_{2}$ , the resultant amplitude obeys

A_{R}^{2} = A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} cos Δ ϕ .

The cosine factor retains the phase sign. When $Δ ϕ = 0$ , the cosine is one and amplitudes reinforce. When $Δ ϕ = π$ , the cosine is minus one and the resultant magnitude is $∣ A_{1} - A_{2} ∣$ . Equal amplitudes can then cancel, but unequal amplitudes leave a residual. Intermediate phase differences give intermediate resultant amplitudes; phase does not choose only two possible outcomes.

Phase control requires a stable frequency reference. Two electronic drives set to the same nominal frequency can drift relative to one another, changing the phase during a long record. A shared clock or a measured phase reference is needed when a prescribed phase difference is part of the experiment. The phase at a sensor also includes propagation delay from each source. Changing source timing by one amount does not necessarily produce the same phase change at every sensor position, so the location of the reported phase measurement must be given.

Phasor sum of two equal-frequency components. The angle between the component vectors is their phase difference; the resultant, from the common tail to the far corner of the parallelogram, gives the amplitude and phase of the combined wave.

Energy density during overlap shows why cancellation of displacement is not cancellation of the disturbance. A string with tension $T$ and linear density $μ$ has small-slope energy density

u = \frac{1}{2} μ (\frac{\partial y}{\partial t})^{2} + \frac{1}{2} T (\frac{\partial y}{\partial x})^{2} .

Both terms use the total displacement after superposition. Squaring that total introduces cross terms that move energy between the kinetic and elastic parts during overlap. Identical upward pulses approaching one another can have zero transverse velocity at their central instant while their slopes add, concentrating energy in the elastic term. Equal opposite pulses can have zero displacement and zero slope at that instant while transverse velocities add, concentrating energy in the kinetic term. Neither case permits a conclusion about energy from displacement alone.

Energy form at full overlap. Two same-sign pulses give a large summed slope and store energy elastically while the transverse velocities momentarily cancel; two opposite-sign pulses cancel the displacement and slope while the transverse velocities add, storing the energy as kinetic instead.

Waveform measurement needs checks that are specific to addition. First record the two single-source traces with identical sensor gain and offset settings. Then record the combined trace without changing the acquisition range, filter, or trigger criterion. Use a pre-pulse interval to remove a measured baseline rather than shifting traces by eye. Calibrate the time axis with a common clock, and record any fractional-sample interpolation used to align the two source runs. A small timing error on a steep edge can produce a residual larger than the sensor noise even when the medium is linear.

Amplitude checks are separate from time checks. Run each source at two or more drive levels and verify that the measured single-source waveform scales proportionally while its arrival marker remains fixed within uncertainty. Then form predicted sums at each drive level. If the residual pattern changes systematically with amplitude, inspect source saturation, sensor range, and material response before treating the effect as a new interaction. The uncertainty band on a predicted sum should include both input trace noise and gain calibration uncertainty; it is not enough to compare only the central curves.

A complete record includes source settings, sensor position, sign convention, sample rate, timing reference, baseline treatment, alignment method, predicted sum, residual, and uncertainty model. The experiment then distinguishes three outcomes cleanly: a residual consistent with noise supports the linear rule; a time-localized residual points to alignment or bandwidth; an amplitude-dependent residual points to a departure from the operating range in which pointwise superposition is valid.

Sinusoidal records permit phase measurement from a fitted sine function or from a complex Fourier component over a stated time interval. A fit should allow for an offset and amplitude uncertainty rather than forcing the trace through zero. The fitted phase is meaningful only when the frequency is resolved by the record duration and the signal-to-noise ratio supports it. A phase estimate from one cycle can be strongly shifted by noise or a small trigger error. Reporting the fit interval, reference clock, and uncertainty in phase difference prevents a nominally aligned pair from being treated as exact alignment.

The resultant phase is not generally the phase of either input. With unequal component amplitudes, the resultant vector lies closer to the larger component in the phase diagram. A measured amplitude reduction may therefore be accompanied by a substantial phase shift as well. A simultaneous fit to amplitude and phase at the sensor permits a stronger comparison with the predicted complex sum than separate peak and time measurements.

Energy-density testing also requires spatial resolution. The slope term on a string is estimated from neighbouring displacement positions, so probe spacing must be small compared with the pulse-width scale. A coarse spatial grid can underestimate a sharp slope and falsely suggest that energy disappears at overlap. The velocity term requires time samples fine enough to resolve the motion near the selected instant. Finite differences amplify measurement noise, so a stated smoothing or derivative procedure is part of the energy estimate. Smoothing that removes a steep slope must be applied equally to the single-pulse and combined records.

An integrated energy comparison is often more stable than a point estimate. Select one region wide enough to contain the full pulse pair at several times, calculate the kinetic and elastic contributions from each sampled record, and integrate over position. In a low-loss setup with no active source work during the selected interval, the total should agree within uncertainty before, during, and after overlap. A local change from elastic to kinetic dominance is expected; a systematic loss of the integrated total calls for checks of damping, sensor calibration, and the limits of the small-slope energy model.

Timing alignment has a quantifiable effect on a predicted sum. If one trace is shifted by a small error $δ t$ , the induced displacement error is approximately $δ y ≃ (\partial y / \partial t) δ t$ . The error is largest on the steep time edges of a pulse and small near a broad peak. This pattern is diagnostic: residuals with opposite signs on the rising and falling sides of a pulse often indicate a timing shift rather than a changed amplitude. Fitting a sub-sample time offset before judging the residual can remove that artifact, provided the fitted offset is reported rather than hidden in a manual trace adjustment.

Detector linearity should be checked with known input scaling. If a sensor output is proportional to displacement over the required range, doubling a single-pulse input should double the output trace at every sample within uncertainty. A clipped crest or compressed large signal produces a combined trace that appears to violate superposition even when the mechanical medium remains linear. Independent calibration of the actuator, sensor, and acquisition chain narrows the diagnosis: a mechanical nonlinearity changes the physical pulse record, whereas an electronic nonlinearity changes only the recorded representation.

Uncertainty on the predicted sum can be formed sample by sample. Independent random noise from the two single-source traces adds in quadrature, while a shared gain calibration error is correlated and must not be counted twice as independent noise. Baseline uncertainty can be important when cancellation leaves a small resultant. Plotting the residual with its uncertainty band over the entire overlap interval makes the test auditable. A report that gives only a maximum residual loses the timing and sign pattern needed to identify its cause.

residual pattern in $y_{both} - (y_{1} + y_{2})$	first measurement check	interpretation after that check
Opposite-signed lobes on the rising and falling edges	Fit and report a relative time shift using the same arrival marker for all runs.	A small timing offset is sufficient when the residual follows $\partial y / \partial t$ .
Residual scales with the full pulse amplitude at every sample	Repeat each single-source trace at two drive levels with unchanged acquisition gain.	Gain or actuator calibration is more likely than a failure of linear propagation.
Flattened or clipped crests	Inspect raw sensor and amplifier ranges; reduce the drive without moving the sensors.	Saturation in the measurement chain can imitate a nonlinear medium.
Nearly constant offset before and after the pulses	Refit the baseline from a stated pre-pulse interval.	The trace alignment is acceptable, but offset subtraction has not been controlled.
Structured mismatch that persists after timing, gain, and baseline checks	Repeat at lower amplitude and at a second sensor position.	Amplitude dependence or a changing propagation pattern identifies the limit of the linear model.

Linear-system tests, timing offsets, and pulse-energy flow

Superposition is a condition on the complete input-output relation of a system. If $L$ denotes the operation that maps a drive history to a measured displacement, linearity requires both additivity and scaling:

L [a + b] = L [a] + L [b], L [α a] = α L [a] .

The first relation is the pulse-addition rule; the second is equally important. A medium may appear additive for one pair of weak inputs yet fail scaling as drive level increases. In a measurement chain, source, medium, sensor, amplifier, and digitizer must all remain in their linear ranges. A clipped sensor output can make an otherwise linear pulse experiment appear non-additive. The diagnostic sequence is therefore to test each input alone at several scales, then test pairs without changing any gain, filter, trigger, or timing setting.

Timing offsets are especially damaging for narrow pulses. If the correct trace is $y (t)$ but one record is delayed by $δ t$ , its displayed value is $y (t - δ t)$ . At small offset, the first-order error is $- δ t d y / d t$ , so the error follows the slope of the pulse: small on a flat crest and largest on a steep edge. A predicted sum can show an apparent positive residual on one side of a pulse and a negative residual on the other even when the physical response is perfectly linear. This paired residual shape is evidence for a timing mismatch, not immediate evidence for nonlinear propagation.

Pulse shape determines how tolerant the experiment is to such offsets. A broad smooth pulse changes slowly and can tolerate a larger timing uncertainty than a sharp pulse of the same amplitude. Asymmetric pulses require an unambiguous alignment rule because their peak, leading-edge crossing, and centroid occur at different times. Cross- correlation can estimate a relative delay from complete traces, but its peak must be checked against a direct feature-based alignment. A noisy baseline or a secondary pulse can shift a correlation maximum away from the physical arrival marker.

Timing-offset signature in a pulse sum. Two otherwise identical records separated by a small delay leave opposite-signed residual lobes on the rising and falling edges; aligning one common pulse marker removes the artifact.

Power and energy flux before, during, and after overlap must be interpreted from the combined wave variables. A small transverse wave on a string carries instantaneous power through position $x$ given by

P (x, t) = - T \frac{\partial y}{\partial x} \frac{\partial y}{\partial t} .

Substitution of $y = y_{1} + y_{2}$ creates terms from each pulse and cross terms that exist only while both contribute at the same place and time. Local power can change sign or magnitude during overlap as energy moves through a chosen section. That does not imply that the total energy of a low-loss region has changed. The integral of energy density over a region containing the full pair is the relevant quantity for a before-during- after comparison, together with any source work and distributed loss that crosses the chosen measurement region.

For identical pulses moving toward one another, the energy transport carried by each incoming pulse continues through the overlap. The displacement pattern may become large, small, or zero at selected times, but the energy variables retain the complete history through slope and transverse speed. A power probe at one location samples a local flux, not the energy stored in all of the overlapping disturbance. Comparing one probe value before overlap with another value during overlap therefore requires the same location, sign convention, and time reference.

Two unequal triangular pulses show why synchronized waveform values, not peak labels, decide the sum.

Worked example. Pulse A alone is a positive triangle of peak $4.0 mm$ and duration $6.0 ms$ ; pulse B alone is a negative triangle of peak $- 2.5 mm$ and duration $4.0 ms$ , its peak arriving $1.0 ms$ after the A peak. At the A-peak time, B has risen halfway up its $2.0 ms$ leading edge, so its value is $- 1.25 mm$ . The predicted combined displacement is

y = + 4.00 mm + (- 1.25 mm) = + 2.75 mm,

not zero, and not the difference of the two peak magnitudes read at unrelated times. At the B-peak time the A triangle is $1.0 ms$ down its $3.0 ms$ falling edge, at $+ 2.67 mm$ , so the predicted sum is

y = + 2.67 mm + (- 2.50 mm) = + 0.17 mm .

A measured $+ 0.20 \pm 0.10 mm$ is consistent with the prediction; a value of $+ 0.70 mm$ would call for a residual analysis. The uncertainty combines the two input-trace uncertainties and the timing-alignment uncertainty.

An experimental run should preserve the raw single-source and simultaneous-source records. State sensor calibration, source amplitude, timing offset correction, sample rate, and the method used to estimate derivatives for power or energy density. Include the complete time window around overlap so that baseline and post-overlap behaviour can be checked. With those records, superposition, flux, and energy claims can be repeated or challenged quantitatively rather than inferred from a single plotted crest.

Scaling tests should include both positive and negative drive directions when the apparatus permits them. A source may be linear for a small upward displacement yet respond differently after a reversed preload or at a larger downward displacement. Plotting measured pulse amplitude against commanded drive amplitude tests whether the gain remains constant. The pulse width and arrival marker should be plotted as well; constant peak scaling with a changing width is not a complete linear-response result. If the source waveform changes with drive level, the single-source records still allow a valid predicted sum, but a simple amplitude-scaling shortcut does not.

Timing correction should be estimated from data that contain a common feature but do not depend on the desired overlap conclusion. A calibration pulse sent through matched channels, or a common source observed by both sensors, establishes a relative delay prior to the two-source measurement. The correction uncertainty is carried into each sample of the predicted sum. Re-fitting a timing shift only on the simultaneous-source record can artificially reduce its residual, so any such fit must be validated on independent single-source data or a held-out portion of the waveform.

Power estimates need sign discipline. On a string, the sign of instantaneous power is set by the product of local slope and transverse velocity under the chosen coordinate convention. Reversing the sensor displacement sign reverses both derivative signs and leaves a consistently computed power unchanged; reversing only one record does not. Numerical derivatives should be checked on a known pulse before being used in an overlap calculation. The integral of a noisy derivative can have a substantial bias, so derivative bandwidth and baseline handling belong beside any reported power trace.

In the triangular example, B rises over $2.0 ms$ with slope $1.25 mm m s^{- 1}$ , so a timing uncertainty of $0.10 ms$ shifts the B contribution by about $0.125 mm$ . That timing contribution may exceed a fine displacement readout uncertainty even when the sampled amplitudes look precise. At the second comparison time, the falling-side slope of A determines the analogous timing term. Propagating both pulse amplitude errors and timing error produces an uncertainty band for the predicted sum; only a residual outside that band supports a claim that linear addition has failed.

Before-during-after energy comparisons should use a fixed spatial region and identical derivative processing at all times. If the region is too short, energy can enter or leave it as a pulse moves and mimic a change caused by overlap. If it is wide enough to contain both pulses, the integrated kinetic-plus-elastic estimate is less sensitive to their internal redistribution. The measured energy may vary slightly in a real apparatus, but its change should be compared with independently estimated source work and dissipation rather than attributed automatically to superposition.

The final quantitative report benefits from retaining the full residual waveform. A single root-mean-square residual compresses overall agreement and can combine an opposite-signed pair caused by timing error with a broad offset caused by gain mismatch. Overlay the measured combined trace, predicted sum, residual, and uncertainty band on one shared time axis. That presentation exposes whether an apparent departure is tied to pulse overlap, to a baseline region, or to the largest amplitudes where an input or sensor may have left its linear range.

Nondispersive wavepackets, sampling, and two-source verification

A wavepacket is a localized disturbance assembled from several sinusoidal components. In the stated nondispersive model, every component relevant to the packet travels at the same medium-selected speed. The packet shape is therefore carried without a systematic spreading caused by unequal component speeds. Superposition still applies component by component and hence to the complete packet: two packets overlap according to their signed displacement profiles, then continue with the same shape they had before overlap if losses and nonlinear response are negligible.

This model is a control condition for a pulse experiment. A packet that broadens or changes asymmetry over the measurement path may no longer provide matching single-source templates at the overlap position. Before comparing a combined record with the sum of two source records, measure each packet at several locations and check whether a time shift alone aligns the profiles. If alignment also requires a change of width or amplitude beyond calibrated loss, the packet model and the residual analysis must include that evolution rather than treating it as a failure of superposition.

Two packets passing through each other in a nondispersive medium. Before overlap they are separated; during overlap the record is their pointwise sum; after overlap each packet re-emerges with its original shape and continues, having swapped positions.

Spatial sampling can make a correctly superposed packet look incorrect. A camera or sensor array records displacement only at discrete positions separated by $Δ x$ . To resolve a sinusoidal component of wavelength $λ$ , the sample spacing must be small enough to distinguish successive positive and negative parts of that component. At the minimum two-samples-per-wavelength limit, phase and amplitude estimates are fragile; a substantially finer grid is needed for reliable derivative and energy-flux estimates. The relevant design scale is the shortest wavelength or sharpest spatial feature present in either source packet, not the broad envelope alone.

When the spacing is too coarse, distinct spatial patterns can produce the same sampled values. This aliasing error can shift an apparent wavelength, reverse an apparent phase trend, or turn a narrow cancellation feature into a false broad maximum. More signal averaging does not repair inadequate spatial spacing because the missing information was never recorded. A sampling plan should set sensor spacing before the experiment, then verify it against a measured single-source spectrum or the narrowest observed packet feature.

Spatial sampling of an interference record. A dense sensor grid resolves each crest and cancellation; a sparse grid spaced one wavelength apart returns the same reading at every sensor and misses the oscillation entirely.

Sampling in time and space must be coordinated. A moving packet observed by a scanned single sensor can be confused with a spatial pattern if the scan timing is not locked to repeated launches. A fixed array avoids that ambiguity but requires channel-to- channel timing and gain calibration. In either arrangement, record a common trigger, the launch-to-sensor timing relation, and the actual sensor coordinates. A location error translates directly into phase and spatial-derivative error, particularly for short features.

Energy-flux verification uses the same combined variables as the displacement test. On a string, calculate local slope and transverse velocity from the combined record, then evaluate the signed power through a selected section. Compare this record with the sum of power contributions only after retaining the cross terms generated by overlap. Power itself is not generally additive when fields overlap because it is bilinear in slope and velocity. The total energy of a region is checked by integrating the kinetic and elastic density over that region before, during, and after the packet encounter.

The flux measurement needs a region wide enough to contain the complete packets at the comparison times. A fixed local section answers a different question: how energy is passing that location at one instant. Mixing these two measurements creates a false energy discrepancy. Use a local power probe to verify directional transport and a spatial integral to verify the energy budget. Both calculations require the same calibrated displacement scale, derivative method, and baseline treatment used for the superposition residual.

A complete two-source procedure begins with survey and calibration. Measure every sensor coordinate along the medium, synchronize all channels to one trigger, and record baseline noise before launching a packet. Run source A alone and source B alone at the intended drive level; repeat each run enough times to estimate launch-time jitter and amplitude scatter. Confirm that translating each single-source packet by the measured travel time aligns its shape at the analysis site. Then run both sources with the same drive settings and retain the unedited records.

Process the three data sets with one locked pipeline. Subtract the baseline, apply the documented gain calibration, align records by a predeclared marker, and form the pointwise predicted sum. Compute residuals at every sensor and time sample. In parallel, use the combined record to calculate the spatial energy density and local flux with a stated derivative stencil. Repeat the analysis after changing source amplitude within the linear range. Agreement of normalized residuals, packet shape, and integrated energy across these checks supports the nondispersive linear model; a patterned departure identifies which calibration or model condition requires revision.

Sampling uncertainty has a direct quantitative signature. If a sensor position has an uncertainty $σ_{x}$ , a local spatial phase estimate for wavelength $λ$ carries an uncertainty of order $2 π σ_{x} / λ$ . The same position error enters a finite-difference slope estimate more strongly when sensors are close together, because the displacement difference is divided by their spacing. An array design uses survey precision much smaller than the shortest spacing and repeats a static position check after the apparatus is mounted. Temperature expansion, sag, or a moved sensor can otherwise create a spatial residual that resembles a phase shift.

Aliasing checks should use a deliberately shifted grid when possible. Repeat a single-source measurement after moving the array by half of its nominal spacing, or move the packet path relative to a fixed array by a known amount. A well-resolved spatial record changes smoothly under that shift. A sparse aliased record can change its apparent wavelength or cancellation site substantially. This test addresses the sampling plan itself, whereas ordinary repeated launches mainly estimate random noise on the same inadequate grid.

The nondispersive condition can be tested from packet records without adding a broader wave model. At each sensor, align the isolated packet to a reference profile by one time translation and one calibrated amplitude factor. Plot the residual versus sensor position. If the residual remains within noise and does not develop a systematic leading-or-trailing pattern, the shape-preserving approximation is supported over the measured path. If a packet becomes wider, narrower, or skewed, retain that measured shape in the two-source prediction rather than using the source waveform as an unchanged template.

Energy-flux verification benefits from a control run with only one source active. The integrated energy for the control packet establishes the displacement-to-energy scale and shows whether the derivative calculation has a baseline bias. Then repeat the calculation for the second source and for the simultaneous record using identical spatial limits. The simultaneous energy estimate is not expected to equal a pointwise sum of two energy-density plots because overlap cross terms are real. It should instead remain compatible with the energy budget set by the two calibrated packet inputs and the stated losses over the selected region.

Finally, freeze the analysis choices before inspecting the simultaneous residual. The sensor subset, temporal window, baseline interval, interpolation rule, derivative stencil, and uncertainty model should be selected from the separate-source data or a calibration run. Changing these choices until the combined trace agrees is a fitting operation, not an independent test of superposition. A reproducible laboratory result keeps the raw traces, survey coordinates, clock metadata, calibration records, and analysis code together with the reported packet sum.

Reporting an interference result

A final interference result should report a prediction and a measurement on the same coordinate system. Give the two calibrated single-source records, their signed pointwise sum, the simultaneous-source record, and the residual with its uncertainty band. State the sensor position, time reference, phase convention for periodic data, source settings, and the interval over which the comparison was made. A sentence such as “the waves cancelled” is incomplete without the measured displacement range, the time window, and the criterion used to call a residual consistent with noise.

Phase and energy checks give independent limits on the interpretation. For periodic records, the measured phase gap should predict the resultant amplitude within the amplitude and timing uncertainty. For pulse records, the signed waveform sum should predict the overlap shape. In either case, a small displacement resultant does not establish small energy. Use the combined slope and transverse-speed records to check whether the local kinetic and elastic terms have exchanged as expected, and compare integrated energy over one fixed region before, during, and after overlap.

Sampling resolution sets the smallest trustworthy feature in a report. State the sensor spacing, sample interval, effective bandwidth, and spatial or temporal interpolation rule. A cancellation notch narrower than two adjacent sensor positions cannot be assigned a reliable width or depth. A time trace sampled too slowly can miss the steep sides that determine timing alignment and energy derivatives. Increasing the number of repeated launches reduces random noise but does not restore a feature absent from the sampled grid.

The completed result should separate three conclusions: whether the displacement sum matches within uncertainty, whether phase alignment is consistent with the stated source timing, and whether the energy estimate closes over the selected region. Each uses related records but a different calculation. Keeping the sampling limit beside all three prevents an overconfident claim based on a visually persuasive but under-resolved overlap pattern.

Uncertainty should be propagated into the final decision rather than attached after a visual comparison. Timing calibration contributes most strongly where the waveform slope is steep; gain calibration contributes where the single-source amplitudes are large; sensor-coordinate uncertainty contributes to spatial phase and derivative estimates. Report correlated terms, such as a shared gain scale, separately from random sample noise. A residual near zero is informative only when its uncertainty band is also small enough to distinguish the proposed sum from plausible alternatives.

Use a held-out check when possible. Set the timing offset and gain factors from a calibration record or from part of the separate-source data, then apply those fixed choices to a later simultaneous record. This prevents the analysis from tuning the same trace used to claim agreement. If the held-out residual grows, identify whether the change follows source amplitude, sensor location, or sampling interval before changing the model.

The archived report should include raw traces, calibrated traces, sensor coordinates, clock metadata, source settings, baseline interval, and the code or equations used for the sum, derivatives, and energy integral. A reader can then repeat the signed-addition test, inspect the phase convention, and determine whether the sampling grid supports the stated interference result. That is the appropriate endpoint for a laboratory superposition claim: a result tied to its resolution, uncertainty, and measurement conditions.