---
title: Wave Superposition
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 5
order: 705
summary: >
  When two waves cross the same point, what does a probe read? In a linear medium
  the answer is arithmetic: the displacements add, $y=y_1+y_2$, and the pulses pass
  through each other unchanged. That one rule produces interference — reinforcement
  where the signs agree, cancellation where they oppose — and it guards against a
  common mistake, since displacement can vanish at an instant while the energy sits
  in transverse motion instead. We work out the signed sum, the phase bookkeeping
  for equal-frequency components, and why a null in the record is not a null in the
  wave.
topics: [Waves]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 16 — Superposition; §16-1"
---

## 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.

$$
% caption: Point-by-point superposition. Two same-sign pulses that overlap add to a
% resultant taller than either component; where the pulses had opposite signs the sum
% would be smaller than the larger component. Each pulse keeps propagating unchanged
% through a linear medium.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-0.30,-0.30) rectangle (6.65,3.20);
\draw[->,black] (.40,.55)--(6.15,.55);
\node[black,below] at (5.60,.52) {position};
\draw[black,thick] plot[domain=.70:5.90,samples=200]
  (\x,{.55+.85*exp(-((\x-2.85)/.55)^2)});
\draw[black,thick] plot[domain=.70:5.90,samples=200]
  (\x,{.55+.85*exp(-((\x-3.55)/.55)^2)});
\draw[acc,very thick] plot[domain=.70:5.90,samples=240]
  (\x,{.55+.85*exp(-((\x-2.85)/.55)^2)+.85*exp(-((\x-3.55)/.55)^2)});
\node[black] at (1.75,0.95) {$y_1$};
\node[black] at (4.65,0.95) {$y_2$};
\node[acc] at (3.20,2.15) {sum};
\end{tikzpicture}
$$

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

$$
y=A\cos(kx-\omega t+\phi).
$$

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

$$
A_{\rm sum}=2A\left|\cos\frac{\Delta\phi}{2}\right|.
$$

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.

$$
% caption: Equal-frequency components in and out of phase. In phase, the resultant reaches
% twice the component amplitude; a half-cycle out of phase, equal components cancel to a
% flat resultant. Intermediate phase differences give intermediate amplitude.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-0.30,-0.30) rectangle (6.65,3.25);
\draw[black] (.50,1.55)--(2.95,1.55);
\draw[black,thick] plot[domain=.55:2.90,samples=160]
  (\x,{1.55+.42*sin(240*(\x-.55))});
\draw[acc,very thick] plot[domain=.55:2.90,samples=160]
  (\x,{1.55+.84*sin(240*(\x-.55))});
\node[black,above] at (1.72,2.60) {in phase};
\node[acc,below] at (1.72,.60) {sum is $2A$};
\draw[black] (3.65,1.55)--(6.10,1.55);
\draw[black,thick] plot[domain=3.70:6.05,samples=160]
  (\x,{1.55+.42*sin(240*(\x-3.70))});
\draw[black,thick] plot[domain=3.70:6.05,samples=160]
  (\x,{1.55-.42*sin(240*(\x-3.70))});
\draw[acc,very thick] (3.70,1.55)--(6.05,1.55);
\node[black,above] at (4.87,2.60) {out of phase};
\node[acc,below] at (4.87,.60) {sum is zero};
\end{tikzpicture}
$$

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

$$
r(x,t)=y_{\rm both}(x,t)-\left[y_1(x,t)+y_2(x,t)\right]
$$

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 $+3\ \mathrm{mm}$ combined with
$-1\ \mathrm{mm}$ gives $+2\ \mathrm{mm}$, not $4\ \mathrm{mm}$; a value of
$+3\ \mathrm{mm}$ combined with $-3\ \mathrm{mm}$ 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.

$$
% caption: Signed sample addition during pulse overlap. At each common time sample the
% resultant is the algebraic sum of the two single-source displacements: a positive and a
% negative contribution partly cancel at sample 1, while two positive contributions add at
% sample 2.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-0.55,-0.35) rectangle (6.65,3.25);
\draw[->,black] (.78,.55)--(5.95,.55) node[right] {time};
\draw[black] (.78,1.20)--(6.05,1.20);
\draw[black] (.78,1.90)--(6.05,1.90);
\draw[black] (.78,2.60)--(6.05,2.60);
\draw[black,very thick] plot[domain=.95:5.30,samples=180]
  (\x,{1.20+.52*exp(-((\x-2.05)/.27)^2)});
\draw[black,very thick] plot[domain=.95:5.30,samples=180]
  (\x,{1.90-.30*exp(-((\x-2.05)/.27)^2)+.46*exp(-((\x-3.95)/.27)^2)});
\draw[acc,very thick] plot[domain=.95:5.30,samples=200]
  (\x,{2.60+.22*exp(-((\x-2.05)/.27)^2)+.46*exp(-((\x-3.95)/.27)^2)});
\draw[dashed,black] (2.05,.78)--(2.05,3.00);
\draw[dashed,black] (3.95,.78)--(3.95,3.00);
\node[black,left] at (.74,1.20) {pulse 1};
\node[black,left] at (.74,1.90) {pulse 2};
\node[acc,left] at (.74,2.60) {sum};
\node[black,below] at (2.05,.52) {sample 1};
\node[black,below] at (3.95,.52) {sample 2};
\end{tikzpicture}
$$

For equal-frequency sinusoidal components, the time alignment is summarized by phase
difference. With amplitudes $A_1$ and $A_2$, the resultant amplitude obeys

$$
A_R^2=A_1^2+A_2^2+2A_1A_2\cos\Delta\phi.
$$

The cosine factor retains the phase sign. When $\Delta\phi=0$, the cosine is one
and amplitudes reinforce. When $\Delta\phi=\pi$, 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.

$$
% caption: Phasor sum of two equal-frequency components. The angle between the component
% vectors is their phase difference; the resultant, from the common tail to the far corner
% of the parallelogram, gives the amplitude and phase of the combined wave.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-0.30,-0.30) rectangle (6.65,3.20);
\coordinate (O) at (2.55,1.15);
\draw[->,black,very thick] (O)--(4.35,1.15);
\draw[->,black,very thick] (O)--(3.61,2.21);
\draw[black,thin,dashed] (4.35,1.15)--(5.41,2.21);
\draw[black,thin,dashed] (3.61,2.21)--(5.41,2.21);
\draw[->,acc,very thick] (O)--(5.41,2.21);
\draw[black] (3.15,1.15) arc (0:45:.60);
\node[black,below] at (3.95,1.10) {$A_1$};
\node[black,above left] at (3.55,2.18) {$A_2$};
\node[black] at (3.20,0.78) {phase};
\node[acc,above] at (4.95,2.18) {resultant};
\end{tikzpicture}
$$

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

$$
u=\frac{1}{2}\mu\left(\frac{\partial y}{\partial t}\right)^2+
\frac{1}{2}T\left(\frac{\partial y}{\partial x}\right)^2.
$$

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.

$$
% caption: Energy form at full overlap. Two same-sign pulses give a large summed slope and
% store energy elastically while the transverse velocities momentarily cancel; two
% opposite-sign pulses cancel the displacement and slope while the transverse velocities
% add, storing the energy as kinetic instead.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-0.30,-0.30) rectangle (6.65,3.20);
\draw[black,thick] (.55,.60) rectangle (3.05,2.55);
\draw[black] (.70,1.25)--(2.90,1.25);
\draw[acc,very thick] plot[domain=.80:2.80,samples=140]
  (\x,{1.25+.92*exp(-((\x-1.80)/.30)^2)});
\node[black,above] at (1.80,2.60) {same sign};
\node[black,below] at (1.80,.55) {elastic energy};
\draw[black,thick] (3.60,.60) rectangle (6.10,2.55);
\draw[black] (3.75,1.55)--(5.95,1.55);
\draw[black,thin] plot[domain=3.85:5.85,samples=140]
  (\x,{1.55+.62*exp(-((\x-4.85)/.30)^2)});
\draw[black,thin] plot[domain=3.85:5.85,samples=140]
  (\x,{1.55-.62*exp(-((\x-4.85)/.30)^2)});
\draw[acc,very thick] (3.85,1.55)--(5.85,1.55);
\node[black,above] at (4.85,2.60) {opposite sign};
\node[black,below] at (4.85,.55) {kinetic energy};
\end{tikzpicture}
$$

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 $\delta t$, the induced displacement error is approximately
$\delta y\simeq (\partial y/\partial t)\delta t$. 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 $y_{\rm both}-(y_1+y_2)$ | 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
$\mathcal L$ denotes the operation that maps a drive history to a measured displacement,
linearity requires both additivity and scaling:

$$
\mathcal L[a+b]=\mathcal L[a]+\mathcal L[b],\qquad
\mathcal L[\alpha a]=\alpha\mathcal L[a].
$$

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 $\delta t$, its displayed value is
$y(t-\delta t)$. At small offset, the first-order error is
$-\delta t\,\d y/\d t$, 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.

$$
% caption: Timing-offset signature in a pulse sum. Two otherwise identical records separated
% by a small delay leave opposite-signed residual lobes on the rising and falling edges;
% aligning one common pulse marker removes the artifact.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-0.30,-0.35) rectangle (6.65,3.25);
\draw[->,black] (.60,.35)--(5.95,.35) node[right] {time};
\draw[black] (.60,1.60)--(6.05,1.60);
\draw[acc,very thick] plot[domain=.85:5.30,samples=180]
  (\x,{1.60+.85*exp(-((\x-2.15)/.32)^2)});
\draw[black,very thick] plot[domain=.85:5.30,samples=180]
  (\x,{1.60+.85*exp(-((\x-2.55)/.32)^2)});
\draw[black,very thick] plot[domain=.85:5.30,samples=220]
  (\x,{.90+.55*exp(-((\x-2.15)/.32)^2)-.55*exp(-((\x-2.55)/.32)^2)});
\draw[black,dashed] (2.15,2.50)--(2.15,3.00);
\draw[black,dashed] (2.55,2.50)--(2.55,3.00);
\draw[<->,black] (2.15,2.90)--(2.55,2.90);
\node[black,above] at (2.35,2.88) {delay};
\draw[black,thin] (1.98,2.30)--(1.45,2.55);
\node[black,left] at (1.47,2.57) {record A};
\draw[black,thin] (2.78,2.28)--(3.30,2.12);
\node[black,right] at (3.28,2.10) {record B};
\node[black] at (1.25,1.18) {residual};
\end{tikzpicture}
$$

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 $4.0\ \mathrm{mm}$
> and duration $6.0\ \mathrm{ms}$; pulse B alone is a negative triangle of peak
> $-2.5\ \mathrm{mm}$ and duration $4.0\ \mathrm{ms}$, its peak arriving
> $1.0\ \mathrm{ms}$ after the A peak. At the A-peak time, B has risen halfway up its
> $2.0\ \mathrm{ms}$ leading edge, so its value is $-1.25\ \mathrm{mm}$. The predicted
> combined displacement is
>
> $$
> y=+4.00\ \mathrm{mm}+(-1.25\ \mathrm{mm})=+2.75\ \mathrm{mm},
> $$
>
> not zero, and not the difference of the two peak magnitudes read at unrelated times.
> At the B-peak time the A triangle is $1.0\ \mathrm{ms}$ down its $3.0\ \mathrm{ms}$
> falling edge, at $+2.67\ \mathrm{mm}$, so the predicted sum is
>
> $$
> y=+2.67\ \mathrm{mm}+(-2.50\ \mathrm{mm})=+0.17\ \mathrm{mm}.
> $$
>
> A measured $+0.20\pm0.10\ \mathrm{mm}$ is consistent with the prediction; a value of
> $+0.70\ \mathrm{mm}$ 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 $2.0\ \mathrm{ms}$ with slope
$1.25\ \mathrm{mm\,ms^{-1}}$, so a timing uncertainty of $0.10\ \mathrm{ms}$ shifts
the B contribution by about $0.125\ \mathrm{mm}$. 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.

$$
% caption: Two packets passing through each other in a nondispersive medium. Before overlap
% they are separated; during overlap the record is their pointwise sum; after overlap each
% packet re-emerges with its original shape and continues, having swapped positions.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-0.95,-0.25) rectangle (6.65,3.25);
\draw[black] (.55,2.55)--(6.05,2.55);
\draw[acc,very thick] plot[domain=.70:6.00,samples=180]
  (\x,{2.55+.46*exp(-((\x-1.55)/.30)^2)});
\draw[black,very thick] plot[domain=.70:6.00,samples=180]
  (\x,{2.55+.46*exp(-((\x-4.55)/.30)^2)});
\node[black,left] at (.48,2.55) {before};
\node[acc,left] at (1.32,2.90) {A};
\node[black,right] at (4.78,2.90) {B};
\draw[black] (.55,1.50)--(6.05,1.50);
\draw[acc,very thick] plot[domain=.70:6.00,samples=180]
  (\x,{1.50+.46*exp(-((\x-2.80)/.30)^2)});
\draw[black,very thick] plot[domain=.70:6.00,samples=180]
  (\x,{1.50+.46*exp(-((\x-3.30)/.30)^2)});
\node[black,left] at (.48,1.50) {during};
\draw[black] (.55,.45)--(6.05,.45);
\draw[acc,very thick] plot[domain=.70:6.00,samples=180]
  (\x,{.45+.46*exp(-((\x-4.55)/.30)^2)});
\draw[black,very thick] plot[domain=.70:6.00,samples=180]
  (\x,{.45+.46*exp(-((\x-1.55)/.30)^2)});
\node[black,left] at (.48,.45) {after};
\end{tikzpicture}
$$

Spatial sampling can make a correctly superposed packet look incorrect. A camera or
sensor array records displacement only at discrete positions separated by $\Delta x$.
To resolve a sinusoidal component of wavelength $\lambda$, 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.

$$
% caption: Spatial sampling of an interference record. A dense sensor grid resolves each
% crest and cancellation; a sparse grid spaced one wavelength apart returns the same
% reading at every sensor and misses the oscillation entirely.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-0.75,-0.35) rectangle (6.65,3.20);
\draw[black] (.65,1.10)--(6.05,1.10);
\draw[acc,very thick] plot[domain=.75:5.85,samples=220]
  (\x,{1.10+.62*sin(240*(\x-.75))});
\foreach \x in {.75,1.05,1.35,1.65,1.95,2.25,2.55,2.85,3.15,3.45,3.75,4.05,4.35,4.65,4.95,5.25,5.55,5.85}
  {\draw[fill=white,draw=black,line width=.35pt] (\x,1.10) circle (.045);}
\node[black,left] at (.60,1.10) {dense};
\draw[black] (.65,2.35)--(6.05,2.35);
\draw[black,thick] plot[domain=.75:5.85,samples=220]
  (\x,{2.35+.62*sin(240*(\x-.75))});
\foreach \x in {.75,2.25,3.75,5.25}
  {\draw[fill=white,draw=black,line width=.35pt] (\x,2.35) circle (.06);}
\node[black,left] at (.60,2.35) {sparse};
\node[black,below] at (3.35,.35) {sensor spacing};
\end{tikzpicture}
$$

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 $\sigma_x$, a local spatial phase estimate for wavelength $\lambda$ carries
an uncertainty of order $2\pi\sigma_x/\lambda$. 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.