---
title: Wave Boundaries
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 15
order: 715
summary: >
  A pulse traveling along a string does something abrupt where the string's
  properties change: part reflects, part transmits, and which is which is set by the
  impedance mismatch alone. We impose continuity of displacement and transverse force
  at the join to get the reflection and transmission coefficients in terms of
  $Z=\sqrt{T\mu}$, fix their signs and the polarity flip, and balance the energy. The
  clean result assumes linear, nondispersive segments meeting at a localized join;
  pulse polarity, return timing, and energy ratios are the measurements that expose a
  real connector's mass, loss, or distributed transition.
topics: [Waves]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 15, §§15-1, 15-2, and 15-4; Ch. 16, §16-1 — Traveling pulses, barriers, and superposition"
---

## Incident pulses and boundary conditions

A transverse pulse on a taut string carries displacement from one part of the
string to another while the string material moves mainly up and down. At a join,
an end support, or a rapid change in string properties, the incoming disturbance
does not retain one travelling shape. A return pulse occupies the incident side,
and a forward pulse enters the other side when a second string is present. Their
amplitudes, orientations, widths, and energy shares follow from the mechanical
conditions at the boundary.

Place an ideal join at $x=0$. String 1 lies at $x<0$ and string 2 lies at $x>0$.
The positive $x$ direction points from string 1 into string 2. A right-moving
incident pulse, a left-moving return pulse, and a right-moving transmitted pulse
can be written

$$
y_i(x,t)=f\!\left(t-\frac{x}{v_1}\right),\qquad
y_r(x,t)=r\,f\!\left(t+\frac{x}{v_1}\right),\qquad
y_t(x,t)=\tau\,f\!\left(t-\frac{x}{v_2}\right).
$$

Here $f$ is the displacement history observed at the join before any return arrives,
$v_1$ and $v_2$ are the pulse speeds, $r$ is the displacement-amplitude reflection
coefficient, and $\tau$ is the displacement-amplitude transmission coefficient.
The factors $r$ and $\tau$ apply to every point of the pulse when the strings are
linear and nondispersive over the pulse bandwidth. A negative $r$ reverses the
upward and downward sense of the return pulse. A positive $\tau$ leaves the
transmitted displacement orientation unchanged.

$$
% caption: A pulse incident on the join at $x=0$ splits into a reflected pulse that returns along string 1 and a transmitted pulse that continues on string 2. Drawn for a higher-impedance second string, so the reflected pulse is inverted; the origin is the join used in every boundary equation below.
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$$

The formulas describe a local interaction. Before the incoming pulse reaches the
join, $y_r$ and $y_t$ are absent. During the encounter, the incident and return
portions overlap on string 1, so a camera sees their algebraic sum $y_1=y_i+y_r$.
After both portions clear a short observation window, their shapes can be measured
separately. A still image taken during overlap is therefore unsuitable for reading
$r$ from a single crest height without reconstructing the two contributions.

An ideal abrupt join preserves pulse duration. If a selected displacement landmark
lasts for $\Delta t$ at the join, it occupies $w_1=v_1\Delta t$ on string 1 and
$w_2=v_2\Delta t$ on string 2.
A slower second string therefore contains a shorter spatial copy of the same time
history. The coefficient $\tau$ scales its displacement, while the speed ratio
scales its horizontal width. Confusing these two changes leads to incorrect energy
comparisons: a shorter transmitted pulse can have less spatial extent even when its
displacement amplitude is substantial.

An isolated pulse requires its sign convention to be recorded with the raw data.
An upward crest may be assigned positive displacement, an image coordinate may
increase downward, and an accelerometer output may have its own polarity. The
physics of inversion is a sign reversal relative to a stated convention. Labelling a
trace only as “inverted” without that convention makes a correct measurement hard
to reproduce.

**Boundary conditions at a joined string.**

At an ideal massless join, the two string ends occupy the same transverse position.
The kinematic boundary condition is

$$
y_1(0,t)=y_2(0,t).
$$

The join also has no unbalanced transverse force. For small slopes, the transverse
component of tension is $T\,\partial y/\partial x$. Force balance gives

$$
T_1\left.\frac{\partial y_1}{\partial x}\right|_{0^-}
=T_2\left.\frac{\partial y_2}{\partial x}\right|_{0^+}.
$$

The signs arise from the tangents pointing away from the join on opposite sides;
writing both tension components in a common positive transverse direction yields
the equality above. These conditions are stronger than matching visible height at
one photograph. They must hold at every time during the incident, return, and
transmitted histories.

$$
% caption: For small slopes each tension resolves into a large horizontal component and a transverse component equal to tension times slope. Balancing the transverse components across the join keeps the massless junction from taking an unbounded transverse acceleration.
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$$

For harmonic components at one angular frequency, use complex displacement
amplitudes:

$$
\begin{aligned}
y_1&=A_i e^{i(\omega t-k_1x)}+A_r e^{i(\omega t+k_1x)},\\
y_2&=A_t e^{i(\omega t-k_2x)},
\qquad k_j=\frac{\omega}{v_j}.
\end{aligned}
$$

The real part gives the measurable string displacement. Substitution into the two
boundary conditions removes the common factor $e^{i\omega t}$ and gives

$$
A_i+A_r=A_t,
\qquad
T_1k_1(A_i-A_r)=T_2k_2A_t.
$$

The first equation joins displacement histories. The second compares transverse
force histories. A pulse can be built from harmonic components, so coefficients
that do not depend on $\omega$ apply unchanged to an arbitrary pulse shape. An
ideal string has that property because its wave speed is independent of frequency.
The condition fails for a dispersive string, cable, or connector, where different
spectral components can return with different amplitudes or phases.

The transverse wave impedance of a string segment is

$$
Z=\mu v=\sqrt{T\mu}=\frac{T}{v},
\qquad
v=\sqrt{\frac{T}{\mu}}.
$$

The units of $Z$ are $\mathrm{kg\,s^{-1}}$, also expressible as
$\mathrm{N\,s\,m^{-1}}$. The impedance converts transverse velocity into the
scale of transverse force carried by a travelling wave. It combines the string’s
inertia per unit length and its tension into the boundary quantity that controls
reflection. A large $Z$ can result from a heavy string, a large tension, or both;
wave speed alone does not identify impedance when tension differs between regions.

Replacing $T_jk_j$ by $\omega Z_j$ and solving the pair of amplitude equations
gives

$$
r=\frac{A_r}{A_i}=\frac{Z_1-Z_2}{Z_1+Z_2},
\qquad
\tau=\frac{A_t}{A_i}=\frac{2Z_1}{Z_1+Z_2}=1+r.
$$

The expression has a limiting-case check. Equal impedances give $r=0$ and
$\tau=1$, even if $T$ and $\mu$ differ individually. A large second impedance
gives $r\rightarrow-1$, the behaviour of a clamped end. A small second impedance
gives $r\rightarrow+1$, the behaviour of a loose end. Those limits concern
displacement; force and slope signals have different signs.

$$
% caption: Impedance sets the reflected pulse. A matched load ($Z_2=Z_1$) removes it; a larger $Z_2$ inverts it (return down) and a smaller $Z_2$ leaves it upright (return up). The incident crest is the same in all three.
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% smaller Z2
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\end{tikzpicture}
$$

## End conditions and abrupt interfaces

A clamped string end cannot move, so its total displacement must satisfy
$y(0,t)=0$. With an incident and a return pulse on the same string, that condition
requires $y_r(0,t)=-y_i(0,t)$ and hence $r=-1$. An upward incoming crest returns
as a downward crest. The midpoint of their temporary overlap lies at zero
displacement because the clamp supplies whatever transverse reaction force is
needed to enforce the constraint.

$$
% caption: A fixed (clamped) end reflects an incident crest as an inverted crest, $r=-1$. The endpoint stays at zero displacement while the two overlap; the clamp supplies whatever transverse force the constraint needs.
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$$

A loose end has no transverse force. In the small-slope approximation, the
boundary condition is $T\,\partial y/\partial x=0$, or $\partial y/\partial x=0$.
For the incident-plus-return description, that condition requires equal displacement
amplitudes, $r=+1$. The endpoint reaches twice the incident displacement during a
crest overlap. It can move because no rigid support supplies a transverse reaction.
The zero-slope condition is often easier to identify in a slow-motion recording than
the endpoint displacement: the string becomes horizontal at the end when the pulse
is maximally developed there.

$$
% caption: A free (loose) end reflects an incident crest as an upright crest, $r=+1$. The string tangent goes horizontal at the end, so the transverse tension component vanishes, and the endpoint reaches twice the incident height during overlap.
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$$

The distinction between a clamp and a loose end appears before the pulse reverses
direction. At a clamp, the leading edge begins building an opposite-signed return
as soon as it reaches the support. At a loose end, the return has the same sign. A
finite-width pulse can therefore produce a doubled hump near a loose endpoint or a
temporary flattened region near a clamp. The string is not locally “stopped” in
either case; each material element still follows a time-dependent transverse path.

An endpoint may be approximated through impedance limits. Joining a test string to
a segment with $Z_2\gg Z_1$ gives $r\simeq-1$, while $Z_2\ll Z_1$ gives
$r\simeq+1$. The approximation becomes poor when the added segment is short
enough for its far end to return a second pulse during the measurement gate. In that
case the apparent end response includes two boundaries and depends on pulse timing.

**Finite pulses at an abrupt interface.**

The harmonic calculation applies to a finite pulse without requiring a
sinusoidal crest. A nondispersive string transports every time sample of $f$ at one
speed. At the join, each sample has the same impedance ratio and therefore acquires
the same multipliers $r$ and $\tau$. A triangular pulse, a rounded Gaussian-like
pulse, and an irregular pluck all return as scaled copies when the join is short
compared with their spatial widths and the material response remains linear.

For $x<0$ and $x>0$, respectively, the total displacement is

$$
\begin{aligned}
y_1(x,t)&=f\!\left(t-\frac{x}{v_1}\right)
+r f\!\left(t+\frac{x}{v_1}\right),\\
y_2(x,t)&=\tau f\!\left(t-\frac{x}{v_2}\right).
\end{aligned}
$$

At $x=0$, the two equations agree because $1+r=\tau$. The equality holds point by
point in time, including the leading edge, peak, trailing edge, and any small
asymmetry created by the source. The same statement supports a strong laboratory
check. Time-shifted records acquired near the join should sum to the record on the
second string after the appropriate amplitude and travel-time corrections.

With $Z_2>Z_1$, the numerator of $r$ is negative. The return pulse has opposite
displacement sign, and $0<\tau<1$. For strings held at the same tension,
$Z\propto\sqrt{\mu}$ and $v\propto1/\sqrt{\mu}$, so the higher-impedance side is
also the slower side. The transmitted displacement peak is smaller than the
incident peak, while its spatial width is smaller by $v_2/v_1$. Both reductions are
visible in a photograph after the pulses separate.

With $Z_2<Z_1$, $r$ is positive and $\tau$ lies between one and two. A transmitted
crest can therefore be taller than the incident crest. That amplitude gain does not
violate energy conservation because the lower-impedance string stores less energy
per unit displacement velocity and the transmitted pulse may occupy a larger
distance. Energy accounting requires impedance and duration, developed in the next
section; comparing crest heights alone gives no power fraction.

The sign of the return is set by impedance, not by whether the incident pulse is a
crest or a trough. If an incident trough has $A_i<0$ and $Z_2>Z_1$, then $r<0$
makes $A_r>0$: the trough returns as a crest. Reversing the source polarity flips
all three pulse displacements but leaves $r$, $\tau$, and every energy fraction
unchanged.

The arrival sequence at a sensor near the join permits an alternative measurement
of the coefficients. Put one sensor at $x=-d$ on string 1 and another at $x=+d_2$
on string 2. The incident record reaches the first sensor before the return. The
return appears later by $2d/v_1$, while the second sensor receives the transmitted
history after $d_2/v_2$. A pulse duration shorter than $2d/v_1$ keeps the two traces
separate at the first sensor. If it is longer, deconvolution or a second spatial
measurement is required.

An abrupt join also preserves pulse time duration only when it introduces no
appreciable internal storage. A solder bead, knot, clip, or rigid ring has mass and
may oscillate while the pulse passes. Its response adds a frequency-dependent phase
and can release energy after the main crest. A record with a small trailing ripple
is evidence for such a nonideal junction, even if the largest peak agrees roughly
with the ideal coefficient.

## Power, pulse energy, and graded transitions

A string with small transverse displacement has mechanical energy per unit length
equal to the sum of kinetic and tension energy:

$$
u=\frac12\mu\left(\frac{\partial y}{\partial t}\right)^2
+\frac12T\left(\frac{\partial y}{\partial x}\right)^2.
$$

The instantaneous power crossing a point in the positive $x$ direction is

$$
P_x=-T\frac{\partial y}{\partial x}\frac{\partial y}{\partial t}.
$$

The derivatives of a right-moving pulse $y=f(t-x/v)$ satisfy
$\partial y/\partial x=-f'/v$ and $\partial y/\partial t=f'$. The local energy
and power reduce to

$$
u=\mu[f'(t-x/v)]^2,
\qquad
P_x=Z[f'(t-x/v)]^2.
$$

The two energy terms are equal for this one-way travelling wave. A left-moving
wave has the same positive energy density and a negative $P_x$ under the stated
coordinate convention. The sign of a return pulse displacement does not affect its
energy because the derivatives are squared in $u$ and $P_x$ magnitude.

The energy of a complete right-moving finite pulse follows by integrating over
space. Substituting $x=vt$ at a fixed observation time gives

$$
E_i=Z_1\int_{-\infty}^{\infty}[f'(t)]^2\d t,
\qquad
E_r=Z_1r^2\int_{-\infty}^{\infty}[f'(t)]^2\d t,
\qquad
E_t=Z_2\tau^2\int_{-\infty}^{\infty}[f'(t)]^2\d t.
$$

The integrals have the same time shape because the ideal boundary does not distort
the waveform. Dividing by $E_i$ gives the energy or power fractions

$$
\mathcal R=r^2=\left(\frac{Z_1-Z_2}{Z_1+Z_2}\right)^2,
\qquad
\mathcal T=\frac{Z_2}{Z_1}\tau^2
=\frac{4Z_1Z_2}{(Z_1+Z_2)^2},
\qquad
\mathcal R+\mathcal T=1.
$$

The quantities $\mathcal R$ and $\mathcal T$ are nonnegative. They should be
reported separately from $r$ and $\tau$, which are signed displacement ratios. A
return with $r=-0.60$ and one with $r=+0.60$ both carry $36\%$ of the ideal incident
energy, although their strings have opposite displacement orientation.

$$
% caption: At a lossless join, incident pulse energy divides into a return share and an output share. The displacement return can point up or down, but its energy share depends on the squared coefficient and is always positive.
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The time-average power of a harmonic travelling wave with displacement amplitude $A$ is

$$
\overline P=\frac12Z\omega^2A^2.
$$

The coefficient relations give the same energy fractions for a long sinusoidal
burst. The incident and return waves overlap on string 1, so instantaneous power
measured at a point can oscillate because of interference. Averaging over a full
period or integrating a separated finite pulse avoids mistaking that local exchange
for a loss of total energy.

$$
% caption: Energy fractions versus impedance ratio $Z_2/Z_1$. At a match (ratio one) all incident power transmits ($\mathcal T=1$); a very large or very small ratio reflects nearly all of it, and $\mathcal R+\mathcal T=1$ at every ratio.
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$$

In a lossy measurement, use $\mathcal R+\mathcal T=1$ as a diagnostic. A measured
sum below one can arise from
distributed damping, a sliding support, energy stored temporarily in a heavy join,
or a calibration mismatch. A sum above one usually signals an amplitude, width,
or timing error unless an active driver continues to supply energy during the
measurement window. Record the observation interval and source status before
assigning a physical interpretation to the discrepancy.

**Speed, density, tension, and graded transitions.**

The speed and impedance of an ideal string depend on the same two mechanical
parameters in different combinations:

$$
v=\sqrt{\frac{T}{\mu}},
\qquad
Z=\sqrt{T\mu}.
$$

At a conventional knot or splice joining two strings held by the same applied
tension, the relevant ratios are

$$
\frac{v_2}{v_1}=\sqrt{\frac{\mu_1}{\mu_2}}
=\frac{Z_1}{Z_2},
\qquad
r=\frac{\sqrt{\mu_1}-\sqrt{\mu_2}}
{\sqrt{\mu_1}+\sqrt{\mu_2}},
\qquad
\tau=\frac{2\sqrt{\mu_1}}{\sqrt{\mu_1}+\sqrt{\mu_2}}.
$$

A heavier second string has a smaller wave speed, a larger impedance, and a
negative displacement return. Doubling $\mu$ changes neither quantity by a factor of
two: speed changes by $1/\sqrt2$ and impedance changes by $\sqrt2$. That square-root
dependence is a frequent source of incorrect visual estimates from wire thickness
alone. A larger diameter usually increases linear density, but braided geometry,
coating, and internal construction can alter the mass-per-length relation.

The common-tension assumption deserves an explicit check. A freely joined pair of
string segments in static equilibrium carries equal axial tension on both sides.
Different nominal tensions require an external support, a sliding contact, a driven
fixture, or a segment with longitudinal acceleration. The two-condition derivation
still applies when transverse force balance includes the actual tensions, but an
apparatus support can exchange energy and invalidate the simple two-port energy
balance. Measurements intended to test the ideal coefficients should use a single
hanging mass or force gauge loading the complete two-string assembly.

A prescribed target impedance $Z_\star$ can be matched by a string satisfying
$T\mu=Z_\star^2$. Matching does not require equal speed. A region with four times
the linear density can match a reference impedance if its tension is one quarter as
large, although such a tension step usually requires an engineered support. The
matched boundary has no ideal return pulse, but its transmitted spatial waveform
can stretch or compress because $v=T/Z_\star$ changes with tension.

An abrupt join is an approximation to a rapid material change. A tapered string
has a local impedance $Z(x)$ and sends a succession of small returns from many
locations. If $Z$ changes little over one local wavelength or one pulse rise
distance, those returns spread in time and can remain small. If the taper changes
over a distance much shorter than the pulse width, the aggregate response approaches
the abrupt-jump coefficient using the endpoint impedances. Compare the transition
length with the pulse’s shortest visible feature, especially a steep leading edge.

$$
% caption: A gradual taper spreads the impedance change over distance. Small reflections leave from many positions and return at different times, unlike the single prompt return from an abrupt splice.
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$$

A finite pulse can be distorted by a gradual transition even when its largest
return is small. The leading edge and trailing edge sample different local
speeds while the pulse is inside the taper. A time-domain comparison should align
the records by a measured group delay, then compare normalized shapes and peak
amplitudes. A broadened output, asymmetric return, or ringing tail shows
that one pair of frequency-independent coefficients is insufficient.

## Reflection measurement and error diagnosis

A measurement apparatus requires calibrated string geometry, axial tension, arrival time,
and transverse displacement. A practical arrangement
uses two strings joined at a marked splice, a common tension source, a short
transverse driver near the left end, and two cameras or position sensors on opposite
sides of the join. The driver must release before the return reaches it. Otherwise
the source boundary generates an additional pulse that overlaps the desired record.

Measure each linear density from a long, dry, untensioned sample.

$$
\mu_j=\frac{m_j}{L_j}.
$$

Use a balance resolution appropriate to the sample mass and a length long enough
that end fraying contributes little fractional error. Weighing a $0.10\ \mathrm{m}$
piece of light cord often gives a poor estimate because connector mass and edge
cuts dominate. A several-metre sample or multiple equal sections reduces those
effects. Record whether the measured material includes a coating, markers, or
adhesive that remains present in the pulse experiment.

$$
% caption: A calibrated two-string experiment. A driver launches the pulse, a marked splice is the join, cameras A and B sit on opposite sides, and one hanging mass sets a common tension. The left camera times the input and return; the right camera times the output.
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$$

The static tension may be estimated as $T=Mg$ only when the hanging mass is at
rest, the pulley friction is negligible, and the string’s angle at the pulley is
accounted for. A load cell inline with the string measures tension more directly.
Zero the load cell before attaching the sample, verify its response with at least two
known masses, and compare its reading with $Mg$ after the system settles. A sliding
pulley, swinging load, or driver that changes mean string length gives a time-varying
tension and changes the wave speed during a record.

### Timing and position calibration

Track a reproducible pulse landmark across several camera frames. The half-height
point on the leading edge is often more stable than the peak because a broad crest
can cover many pixels and move imperceptibly between frames. Let $x_n$ be the
calibrated longitudinal coordinate of that landmark in frame $n$, and let
$t_n=t_0+n/f_{\rm cam}$. A least-squares line fitted to

$$
x_n=x_0+vt_n
$$

uses every frame in the selected travel interval. Its slope gives $v$ and its
residuals quantify any systematic departure from constant-speed motion. A two-frame
estimate $v=\Delta x/\Delta t$ is appropriate only when the pulse moves across a
large, accurately known distance and the frame timestamps are reliable.

Camera timing must be verified independently when possible. Nominal frame rate is
often rounded in the camera interface, and variable-frame-rate video can assign
unequal frame intervals. A flashing LED driven by a known clock, a frame-timestamp
file, or a camera operating in a hardware-timed mode checks the timing.
When pulse duration approaches one frame interval, the camera cannot resolve its
shape; increase the frame rate, decrease the tension to slow the pulse, or use a spatial
array of sensors instead.

Spatial calibration requires a ruler or fiducial marks in the plane of string
motion. A ruler placed far behind the string creates parallax: a crest moving toward
or away from the camera shifts relative to the scale even when its longitudinal
position is unchanged. Put the scale beside the equilibrium string line, align the
camera optical axis approximately perpendicular to the measurement plane, and
retain the pixel-to-length conversion with the data. Lens distortion matters near
wide-angle image edges; use the central image region or correct the image with a
calibration grid.

The camera measures a projected coordinate. If the string departs appreciably from
the calibration plane, the conversion to physical displacement needs a geometric
correction. A side-view camera measures transverse displacement well but may be
poor for longitudinal travel; an overhead camera reverses that tradeoff. Two
synchronized views or a mirror arrangement can recover both coordinates when a
large-amplitude experiment requires them. For the small-slope model used in this
lesson, keep transverse amplitudes much smaller than the camera distance and use
one view aligned to the relevant coordinate.

### Extracting coefficients from records

Determine a baseline before measuring a crest or trough. Let $y_b(t)$ be a local
baseline obtained from nearby undisturbed string pixels or from a low-order fit to
the equilibrium line. The signed pulse height is $A(t)=y(t)-y_b(t)$. A clean,
separated crest uses the maximum of that signed height; a noisy or asymmetric
pulse, use a matched shape fit or an integrated squared-velocity estimate. The same
amplitude definition must be used for input, return, and output.

For separated pulse records, compute

$$
r_{\rm meas}=\frac{A_r}{A_i},
\qquad
\tau_{\rm meas}=\frac{A_t}{A_i},
\qquad
\mathcal R_{\rm meas}=r_{\rm meas}^2.
$$

The ideal output-energy share may be obtained from
$\mathcal T_{\rm meas}=(Z_2/Z_1)\tau_{\rm meas}^2$ when the three shape histories
agree after scaling. A direct energy measurement is preferable when they do not.
Differentiate a smoothed displacement trace to estimate transverse velocity, then
integrate $Z[y'(t)]^2$ over a time gate containing the full pulse. Differentiation
amplifies camera noise, so the smoothing bandwidth and gate endpoints belong in the
lab record.

An interface measurement should include a zero-contrast control. Join two pieces
cut from the same spool, keep their diameter and tension alike, and run the same
driver and camera procedure. The expected return is near the apparatus noise floor.
A large apparent return in this control points to a knot, clamp, camera threshold,
or source-reflection problem. The control result then bounds any material-mismatch
claim and sets an empirical lower bound on the coefficient magnitude the setup
can resolve.

**Uncertainty, repeatability, and error diagnosis.**

Begin uncertainty analysis with measured observables. A time-of-flight estimate
$v=L/\Delta t$ with independent length and time uncertainties gives

$$
\left(\frac{\sigma_v}{v}\right)^2
=\left(\frac{\sigma_L}{L}\right)^2
+\left(\frac{\sigma_{\Delta t}}{\Delta t}\right)^2.
$$

A linear-density measurement $\mu=m/L_s$ and impedance estimated from a common
tension have corresponding first-order expressions

$$
\left(\frac{\sigma_\mu}{\mu}\right)^2
=\left(\frac{\sigma_m}{m}\right)^2
+\left(\frac{\sigma_{L_s}}{L_s}\right)^2,
\qquad
\left(\frac{\sigma_Z}{Z}\right)^2
=\frac14\left[\left(\frac{\sigma_T}{T}\right)^2
+\left(\frac{\sigma_\mu}{\mu}\right)^2\right].
$$

The formulas assume independent, small random errors. A common ruler scale used
for two lengths creates correlation; the scale error largely cancels in a ratio of
lengths but does not cancel in an absolute speed. A hanging load used to set both
segments’ tension likewise introduces a common systematic uncertainty. Keep shared
calibration terms together until the final propagation; adding them twice as
independent contributions overstates the uncertainty.

Amplitude-ratio uncertainty has a different structure. For independently estimated
signed heights $A_r$ and $A_i$,

$$
\sigma_r^2\simeq
\left(\frac{\sigma_{A_r}}{A_i}\right)^2
+\left(\frac{A_r\sigma_{A_i}}{A_i^2}\right)^2
-\frac{2A_r}{A_i^3}\cov(A_r,A_i).
$$

The covariance term matters when both heights use one image scale, one baseline
procedure, or one source calibration. A common multiplicative scale largely cancels
from $r$, while a baseline offset can affect a small return disproportionately. The
formula becomes awkward near $r=0$ because relative uncertainty in the return
height diverges. Report an absolute confidence interval for $r$ near an impedance
match, along with the control-run noise floor.

An uncertainty budget identifies the measurement that limits the result. If the
pulse crosses only two pixels per frame, timing dominates and a more accurate ruler
does little. If a large pulse spans many frames but its peak lies on a curved or
tilted baseline, amplitude extraction dominates. Estimate each contribution by
holding other measurements at their best values and varying one component within
its calibration interval. A table or bar chart of those contributions gives a more
diagnostic experimental account than a single final standard deviation.

The theoretical coefficient is sensitive to impedance ratio $q=Z_2/Z_1$ through

$$
r(q)=\frac{1-q}{1+q},
\qquad
\frac{\d r}{\d q}=-\frac{2}{(1+q)^2}.
$$

Near $q=1$, a small ratio error produces a comparable absolute coefficient error.
Far from match, $r$ approaches $\pm1$ and changes more slowly with $q$. The
laboratory difficulty reverses in one respect: a near-match return may be too small
to distinguish from control noise, while a large-mismatch return has a clear sign
but can overlap the source or an end return if the string segment is short.

Repeated trials should change one physical variable at a time. For example, retain
the same string 1, tension, cameras, pulse duration, and source geometry while
replacing string 2 with several measured linear densities. Plot the observed signed
$r$ against the predicted impedance ratio. Random scatter about the curve estimates
repeatability. A consistent displacement of all points signals a calibration bias or
an unmodeled join property; a trend that becomes worse for narrow pulses signals
dispersion or finite join dynamics.

> **Worked example.** A $10.0\ \mathrm{N}$ common tension holds a light string with
> $\mu_1=0.0100\ \mathrm{kg\,m^{-1}}$ against a heavy string with
> $\mu_2=0.0900\ \mathrm{kg\,m^{-1}}$. The measured incident crest is
> $A_i=4.00\ \mathrm{cm}$ and its time duration at the join is
> $\Delta t=80.0\ \mathrm{ms}$. The density ratio is $\mu_2/\mu_1=9.00$, so
>
> $$
> \begin{aligned}
> v_1&=\sqrt{\frac{10.0}{0.0100}}=31.6\ \mathrm{m\,s^{-1}},
> &Z_1&=\sqrt{(10.0)(0.0100)}=0.316\ \mathrm{kg\,s^{-1}},\\
> v_2&=\sqrt{\frac{10.0}{0.0900}}=10.5\ \mathrm{m\,s^{-1}},
> &Z_2&=\sqrt{(10.0)(0.0900)}=0.949\ \mathrm{kg\,s^{-1}}.
> \end{aligned}
> $$
>
> The impedance ratio is three. The amplitude and energy coefficients are therefore
>
> $$
> r=\frac{1-3}{1+3}=-\frac12,
> \qquad
> \tau=\frac{2}{1+3}=\frac12,
> \qquad
> \mathcal R=\frac14,
> \qquad
> \mathcal T=\frac34.
> $$
>
> The return is a downward crest of amplitude $-2.00\ \mathrm{cm}$, and the output
> is an upward crest of amplitude $+2.00\ \mathrm{cm}$. Its spatial widths follow
> from the shared time duration:
>
> $$
> w_1=v_1\Delta t=2.53\ \mathrm{m},
> \qquad
> w_2=v_2\Delta t=0.843\ \mathrm{m}.
> $$
>
> If an independently gated input-energy measurement gives $E_i=0.600\ \mathrm J$,
> the ideal predictions are $E_r=0.150\ \mathrm J$ and $E_t=0.450\ \mathrm J$.
> The three numbers sum to the input energy. An output pulse with twice the predicted
> width would signal a timing or speed error; an output crest of $4.00\ \mathrm{cm}$
> would signal a mismatch between the assumed and actual impedance ratio.

$$
% caption: Worked heavy-string result with impedance ratio three. The reflected pulse returns inverted at half height (down 2 cm), the transmitted pulse stays upright at half height (up 2 cm), and the slower second string compresses its width from 2.53 m to 0.843 m.
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$$

> **Worked example.** Place a sensor $d=0.800\ \mathrm m$ from a loose ring on a uniform string whose
> measured pulse speed is $v=20.0\ \mathrm{m\,s^{-1}}$. An incident crest of height
> $1.20\ \mathrm{cm}$ lasts $25.0\ \mathrm{ms}$ at the sensor. The return delay at
> that same sensor is
>
> $$
> \Delta t_{\rm return}=\frac{2d}{v}
> =\frac{2(0.800\ \mathrm m)}{20.0\ \mathrm{m\,s^{-1}}}
> =80.0\ \mathrm{ms}.
> $$
>
> The input and return windows are separated because $25.0\ \mathrm{ms}$ is shorter
> than $80.0\ \mathrm{ms}$. A loose ring gives $r=+1$, so the return crest at the
> sensor should have height $+1.20\ \mathrm{cm}$ after correcting for any damping
> between the two passages. At the ring, the incident and return crests superpose to
> give an endpoint height of $+2.40\ \mathrm{cm}$. A clamp would produce a return
> height of $-1.20\ \mathrm{cm}$ at the sensor and an endpoint height of zero.
>
> The endpoint height is a geometry check, while the sensor delay is a speed check.
> They test different parts of the model. A measured delay of $80.0\ \mathrm{ms}$ with
> a return of the wrong sign indicates an end-condition or camera-polarity problem.
> A correct sign with a delayed return points instead to an error in distance, speed,
> or an unrecognized extra segment between sensor and ring.

$$
% caption: Worked loose-end result. An incident crest and its upright return appear as two separated positive pulses at the sensor, delayed by the round-trip time $2d/v=80\ \mathrm{ms}$; a clamp would flip the second pulse.
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$$

## Ideal-coefficient limits and multiple interfaces

The boundary formulas require a one-dimensional, linearly elastic string with small
transverse slope. The tension-energy expression assumes

$$
\left|\frac{\partial y}{\partial x}\right|\ll1,
\qquad
|y|\ll L_{\rm tension},
$$

where $L_{\rm tension}$ is the apparatus length scale over which mean tension
changes appreciably. Large slopes change the string length during motion, modulate
tension, and couple transverse and longitudinal motion. Return amplitude can then
depend on the incident pulse height. Repeat the measurement at several driver
amplitudes while retaining the same pulse width. Agreement of the signed $r$ values
within uncertainty supports the linear range used for the coefficient calculation.

String bending rigidity introduces dispersion at sufficiently short wavelengths. An
ideal flexible string has $\omega=vk$, whereas a stiff wire has a higher-frequency
correction. A narrow sharp pulse contains more high-$k$ content than a broad rounded
pulse, making it more susceptible to dispersion. Compare time-normalized records
at two pulse widths. Shape changes that grow for the narrow pulse show that the
single nondispersive waveform $f$ is only an approximation over that bandwidth.

Distributed damping removes energy during travel. Air drag, internal friction,
sliding through a guide, and contact with a support can attenuate input, return, and
output by unequal path lengths. Measure a control pulse on a uniform string over
the same outbound and return distances. Its attenuation factor can be applied as a
separate path correction only when it is independent of pulse amplitude and shape.
Otherwise report the raw coefficients and describe the propagation loss.

The reflection coefficient also assumes that the interface is small compared with
the distance over which the pulse varies. A thick knot, a loop of extra string, or a
massive sensor at the join is better treated as an intervening element with its own
inertia and compliance. Two-dimensional strings, membranes, and real ropes can
scatter transverse motion into additional polarizations or directions. Their
energy balance remains valid, but the two one-dimensional amplitude equations do
not account for every outgoing channel.

Repeated end returns can build the conditions used in
[standing-wave normal modes](/mechanics/oscillations-waves/standing-waves),
but an isolated-pulse measurement should stop before those multiple returns overlap.
The present coefficients describe one encounter at one boundary. A periodic drive
and repeated returns require phase bookkeeping over many encounters.

**Minimum reporting record.**

A complete pulse-reflection report should state the following items.

- **String properties.** Sample masses, measured lengths, calculated $\mu_1$ and
  $\mu_2$, the applied tension, and the method used to calibrate that tension.
- **Geometry.** Join position, sensor distances, end conditions, camera orientation,
  and every support or guide that contacts the string.
- **Pulse definition.** Source motion, signed displacement convention, amplitude
  definition, duration definition, and the particular leading-edge landmark used
  for timing.
- **Data reduction.** Pixel scale, frame-time calibration, baseline method,
  smoothing used before differentiation, time-gate endpoints, and the formulas for
  $r$, $\tau$, $\mathcal R$, and $\mathcal T$.
- **Checks.** Zero-contrast control result, comparison of measured and predicted
  speed, energy-share sum, repeated-trial scatter, and any visible trailing ripple
  or pulse-shape change.

These records make the claimed coefficient traceable to physical quantities and
make a discrepancy diagnosable. They also permit a later reader to distinguish a
material mismatch from a source, imaging, or end-boundary artifact without relying
on a single illustrative frame.

**Two interfaces, separated echoes, and pulse windows.**

A finite segment between two joins produces more than one return. This occurs in a
spliced repair, a string section carrying a sensor, a coated cable, or a laboratory
sample placed between two reference strings. The earliest return comes from the
near join. A later echo has crossed the middle segment, returned from the far join,
and crossed the near join on its way back. The echo time distinguishes the two
boundaries when the pulse is short enough.

Let string 1 join string 2 at $x=0$, and let string 2 join string 3 at $x=L$.
Define $r_{ab}$ as the displacement return coefficient for incidence from region
$a$ toward region $b$, and define $\tau_{ab}$ as the corresponding forward
displacement coefficient. The direct return at the first join has amplitude

$$
A_{\rm direct}=r_{12}A_i.
$$

The earliest echo that returns to string 1 after one visit to the far join has
amplitude

$$
A_{\rm echo}=\tau_{12}r_{23}\tau_{21}A_i.
$$

Its extra delay relative to the direct return is $2L/v_2$. The expression carries
the sign of all three factors. A positive direct return can be followed by a
negative echo, or vice versa, depending on the two impedance steps. The amplitudes
of later echoes acquire additional factors from repeated internal returns and
usually decay when $|r_{12}r_{23}|<1$ or when damping is present.

$$
% caption: A middle segment between two joins gives a prompt return at the near join A and a later echo after a round trip to the far join B. Their time separation is $2L/v_2$, where $v_2$ is the middle-segment wave speed.
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\node[above] at (4.94,.62) {join B};
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$$

Pulse duration establishes the usable timing window. A pulse with meaningful time
width $\Delta t_p$ has distinct direct return and earliest echo at
a sensor on string 1 when

$$
\Delta t_p\ll\frac{2L}{v_2}.
$$

The stronger condition gives room for a gate around each waveform and allows a
background interval between them. When the widths overlap, a single peak can be
misread as an anomalous reflection coefficient. A model-based fit to the whole
trace can still estimate the joins, but it needs the input waveform, both travel
times, and an assumed attenuation law. A short-pulse measurement avoids that
parameter coupling.

The two-interface arrangement supports a speed measurement inside an otherwise
inaccessible segment. If the two join positions are known and the direct and echo
arrival times are read at one upstream sensor, then

$$
v_2=\frac{2L}{t_{\rm echo}-t_{\rm direct}}.
$$

This difference cancels the travel time from the sensor to the near join and is
therefore less sensitive to the upstream distance than a single time-of-flight
measurement. It still depends on correct identification of the earliest far-boundary
echo. A control arrangement with the far join removed, or with a known matched
termination, helps identify which feature comes from the far boundary.

A periodically driven string can contain many echoes and eventually form the
interference pattern treated in the standing-wave lesson. The pulse method keeps the
arrivals temporally separate and treats each path as an event with a travel time and
amplitude factor. In a laboratory record, a few unwanted echoes usually indicate
extra propagation paths outside a normal-mode experiment.

**Source-end isolation and acquisition windows.**

The pulse source is another boundary. A hand, a moving paddle, a servo arm, or a
clamp can return part of the pulse after the desired interface interaction. Source
behaviour depends on whether the driver remains constrained, is released, or is
actively position-controlled. A rigid servo holding zero displacement resembles a
clamp for a returning pulse; a freely released lightweight loop can resemble a loose
end. A motor controller may impose an intermediate, frequency-dependent mechanical
impedance. Treating the source as transparent without a check produces false late
returns in the interface record.

Put the source a distance $D$ from the near sensor or join. A return from the source
cannot reappear at that point before a round-trip interval of approximately

$$
t_{\rm source}\simeq\frac{2D}{v_1}.
$$

Select a measurement gate ending before that interval, after allowing for the
initial pulse duration and camera timing uncertainty. The condition is especially
important for a low-impedance source that sends a same-sign pulse back toward the
join. A source return can resemble a low-impedance material transition unless its
time path is calculated from the apparatus geometry.

The driver should generate the incident pulse reproducibly and then cease
transverse motion. A displacement-controlled driver can use a programmed short
ramp followed by a hold or release. The ramp duration sets the pulse bandwidth. A
shorter ramp has sharper edges and increases sensitivity to bending stiffness and
camera sampling. Use several ramp durations during apparatus characterization. If
the measured coefficient changes as the ramp becomes shorter, the string, join, or
driver response has frequency dependence outside the ideal model.

Record one input trace before adding the second string. The trace establishes source
polarity, pulse duration, and the amplitude range that remains linear. Then attach
the test string without changing the driver setting. Subtracting a source-only
control trace from an interface trace is unsafe when their end conditions differ,
because a return alters the driver motion itself. A sensor located between the
driver and join gives a direct incident record for each trial and avoids that
assumption.

For hand-generated pulses, place the hand far enough from the measurement region
that the hand-boundary return arrives outside the gate. Trigger recording before the
pluck and retain pre-pulse baseline frames. Repeat only trials whose input amplitude
and duration fall within stated acceptance bounds. Selecting a return trace after
viewing its apparent coefficient biases the result; select trials from the incident
trace before inspecting the return or output record.

The same timing analysis applies to the far end of string 2. If the output reaches
that end before the output sensor has completed its record, the far-end return can
overlap the transmitted pulse. Increase the length beyond the sensor, shorten the
record, add an absorbing termination, or place the output sensor closer to the join.
The physical path lengths and the selected time gate determine whether a displayed
waveform represents one boundary encounter or several superposed encounters.