Wave Boundaries

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 (t - \frac{x}{v _{1}}), y_{r} (x, t) = r f (t + \frac{x}{v _{1}}), y_{t} (x, t) = τ f (t - \frac{x}{v _{2}}) .

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 $τ$ is the displacement-amplitude transmission coefficient. The factors $r$ and $τ$ 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 $τ$ leaves the transmitted displacement orientation unchanged.

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.

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 $Δ t$ at the join, it occupies $w_{1} = v_{1} Δ t$ on string 1 and $w_{2} = v_{2} Δ t$ on string 2. A slower second string therefore contains a shorter spatial copy of the same time history. The coefficient $τ$ 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} \frac{\partial y _{1}}{\partial x}_{0^{-}} = T_{2} \frac{\partial y _{2}}{\partial x}_{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.

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.

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

y_{1} y_{2} = A_{i} e^{i (ω t - k_{1} x)} + A_{r} e^{i (ω t + k_{1} x)}, = A_{t} e^{i (ω t - k_{2} x)}, k_{j} = \frac{ω}{v _{j}} .

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

A_{i} + A_{r} = A_{t}, T_{1} k_{1} (A_{i} - A_{r}) = T_{2} k_{2} A_{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 $ω$ 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 = μv = T μ = \frac{T}{v}, v = \frac{T}{μ} .

The units of $Z$ are $kg s^{- 1}$ , also expressible as $N s m^{- 1}$ . 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 $T_{j} k_{j}$ by $ω Z_{j}$ and solving the pair of amplitude equations gives

r = \frac{A _{r}}{A _{i}} = \frac{Z _{1} - Z _{2}}{Z _{1} + Z _{2}}, τ = \frac{A _{t}}{A _{i}} = \frac{2 Z _{1}}{Z _{1} + Z _{2}} = 1 + r .

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

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.

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.

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.

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.

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.

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 $Z_{2} ≫ Z_{1}$ gives $r ≃ - 1$ , while $Z_{2} ≪ Z_{1}$ gives $r ≃ + 1$ . 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 $τ$ . 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

y_{1} (x, t) y_{2} (x, t) = f (t - \frac{x}{v _{1}}) + r f (t + \frac{x}{v _{1}}), = τ f (t - \frac{x}{v _{2}}) .

At $x = 0$ , the two equations agree because $1 + r = τ$ . 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 $0 < τ < 1$ . For strings held at the same tension, $Z \propto μ$ and $v \propto 1/ μ$ , 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 $τ$ 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 $r$ , $τ$ , 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 $2 d / v_{1}$ , while the second sensor receives the transmitted history after $d_{2} / v_{2}$ . A pulse duration shorter than $2 d / v_{1}$ 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 = \frac{1}{2} μ (\frac{\partial y}{\partial t})^{2} + \frac{1}{2} T (\frac{\partial y}{\partial x})^{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 = μ [f^{'} (t - x / v)]^{2}, 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 $x = v t$ at a fixed observation time gives

E_{i} = Z_{1} - \infty \int \infty [f^{'} (t)]^{2} d t, E_{r} = Z_{1} r^{2} - \infty \int \infty [f^{'} (t)]^{2} d t, E_{t} = Z_{2} τ^{2} - \infty \int \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

R = r^{2} = (\frac{Z _{1} - Z _{2}}{Z _{1} + Z _{2}})^{2}, T = \frac{Z _{2}}{Z _{1}} τ^{2} = \frac{4 Z _{1} Z _{2}}{( Z _{1} + Z _{2} ) ^{2}}, R + T = 1.

The quantities $R$ and $T$ are nonnegative. They should be reported separately from $r$ and $τ$ , 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.

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.

The time-average power of a harmonic travelling wave with displacement amplitude $A$ is

\overline{P} = \frac{1}{2} Z ω^{2} A^{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.

Energy fractions versus impedance ratio

Z_{2} / Z_{1}

. At a match (ratio one) all incident power transmits (

T = 1

); a very large or very small ratio reflects nearly all of it, and

R + T = 1

at every ratio.

In a lossy measurement, use $R + T = 1$ 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 = \frac{T}{μ}, Z = T μ .

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

\frac{v _{2}}{v _{1}} = \frac{μ _{1}}{μ _{2}} = \frac{Z _{1}}{Z _{2}}, r = \frac{μ _{1} - μ _{2}}{μ _{1} + μ _{2}}, τ = \frac{2 μ _{1}}{μ _{1} + μ _{2}} .

A heavier second string has a smaller wave speed, a larger impedance, and a negative displacement return. Doubling $μ$ changes neither quantity by a factor of two: speed changes by $1/ 2$ and impedance changes by $2$ . 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 $Z_{⋆}$ can be matched by a string satisfying $T μ = Z_{⋆}^{2}$ . 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 $v = T / Z_{⋆}$ 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.

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.

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.

μ_{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 $0.10 m$ 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.

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.

The static tension may be estimated as $T = M g$ 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 $M g$ 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 $t_{n} = t_{0} + n / f_{cam}$ . A least-squares line fitted to

x_{n} = x_{0} + v t_{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 $v = Δ x /Δ t$ 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_{meas} = \frac{A _{r}}{A _{i}}, τ_{meas} = \frac{A _{t}}{A _{i}}, R_{meas} = r_{meas}^{2} .

The ideal output-energy share may be obtained from $T_{meas} = (Z_{2} / Z_{1}) τ_{meas}^{2}$ 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 $v = L /Δ t$ with independent length and time uncertainties gives

(\frac{σ _{v}}{v})^{2} = (\frac{σ _{L}}{L})^{2} + (\frac{σ _{Δ t}}{Δ t})^{2} .

A linear-density measurement $μ = m / L_{s}$ and impedance estimated from a common tension have corresponding first-order expressions

(\frac{σ _{μ}}{μ})^{2} = (\frac{σ _{m}}{m})^{2} + (\frac{σ _{L_{s}}}{L _{s}})^{2}, (\frac{σ _{Z}}{Z})^{2} = \frac{1}{4} [(\frac{σ _{T}}{T})^{2} + (\frac{σ _{μ}}{μ})^{2}] .

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 $A_{i}$ ,

σ_{r}^{2} ≃ (\frac{σ _{A_{r}}}{A _{i}})^{2} + (\frac{A _{r} σ _{A_{i}}}{A _{i}^{2}})^{2} - \frac{2 A _{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}, \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 $\pm 1$ 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 $10.0 N$ common tension holds a light string with $μ_{1} = 0.0100 kg m^{- 1}$ against a heavy string with $μ_{2} = 0.0900 kg m^{- 1}$ . The measured incident crest is $A_{i} = 4.00 cm$ and its time duration at the join is $Δ t = 80.0 ms$ . The density ratio is $μ_{2} / μ_{1} = 9.00$ , so

v_{1} v_{2} = \frac{10.0}{0.0100} = 31.6 m s^{- 1}, = \frac{10.0}{0.0900} = 10.5 m s^{- 1}, Z_{1} Z_{2} = (10.0) (0.0100) = 0.316 kg s^{- 1}, = (10.0) (0.0900) = 0.949 kg s^{- 1} .

The impedance ratio is three. The amplitude and energy coefficients are therefore

r = \frac{1 - 3}{1 + 3} = - \frac{1}{2}, τ = \frac{2}{1 + 3} = \frac{1}{2}, R = \frac{1}{4}, T = \frac{3}{4} .

The return is a downward crest of amplitude $- 2.00 cm$ , and the output is an upward crest of amplitude $+ 2.00 cm$ . Its spatial widths follow from the shared time duration:

w_{1} = v_{1} Δ t = 2.53 m, w_{2} = v_{2} Δ t = 0.843 m .

If an independently gated input-energy measurement gives $E_{i} = 0.600 J$ , the ideal predictions are $E_{r} = 0.150 J$ and $E_{t} = 0.450 J$ . 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 $4.00 cm$ would signal a mismatch between the assumed and actual impedance ratio.

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.

Worked example. Place a sensor $d = 0.800 m$ from a loose ring on a uniform string whose measured pulse speed is $v = 20.0 m s^{- 1}$ . An incident crest of height $1.20 cm$ lasts $25.0 ms$ at the sensor. The return delay at that same sensor is

Δ t_{return} = \frac{2 d}{v} = \frac{2 ( 0.800 m )}{20.0 m s ^{- 1}} = 80.0 ms .

The input and return windows are separated because $25.0 ms$ is shorter than $80.0 ms$ . A loose ring gives $r = + 1$ , so the return crest at the sensor should have height $+ 1.20 cm$ after correcting for any damping between the two passages. At the ring, the incident and return crests superpose to give an endpoint height of $+ 2.40 cm$ . A clamp would produce a return height of $- 1.20 cm$ 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 $80.0 ms$ 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.

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

2 d / v = 80 ms

; a clamp would flip the second pulse.

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

\frac{\partial y}{\partial x} ≪ 1, ∣ y ∣ ≪ L_{tension},

where $L_{tension}$ 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 $ω = v k$ , 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, 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 $μ_{1}$ and $μ_{2}$ , 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 $r$ , $τ$ , $R$ , and $T$ .
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 $τ_{ab}$ as the corresponding forward displacement coefficient. The direct return at the first join has amplitude

A_{direct} = r_{12} A_{i} .

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

A_{echo} = τ_{12} r_{23} τ_{21} A_{i} .

Its extra delay relative to the direct return is $2 L / v_{2}$ . 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.

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

2 L / v_{2}

, where

v_{2}

is the middle-segment wave speed.

Pulse duration establishes the usable timing window. A pulse with meaningful time width $Δ t_{p}$ has distinct direct return and earliest echo at a sensor on string 1 when

Δ t_{p} ≪ \frac{2 L}{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{2 L}{t _{echo} - t _{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_{source} ≃ \frac{2 D}{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.