Standing Waves

Construction of a standing wave

A standing wave is produced when two waves of equal amplitude, equal frequency, and equal speed travel in opposite directions through the same medium. On a string, one convenient pair is

Adding them gives

The spatial factor fixes where the string can move; the time factor makes every nonzero point oscillate at the same angular frequency. Unlike a travelling pulse, the displacement pattern does not translate along the string. A point at a node remains at zero displacement for every time, while a point at an antinode has the largest allowed amplitude. The word “standing” denotes a fixed spatial pattern; material near an antinode still moves vigorously up and down.

The construction requires matched component waves. Equal amplitudes remove the travelling-wave remainder, equal frequencies keep nodes fixed, and equal wave speeds support the product form. Compare component amplitudes, frequency, and arrival speed before assigning fixed nodes to a measured pattern.

Node spacing follows directly from the spatial factor. Nodes occur where $\sin (kx)=0$, so adjacent nodes are separated by $\lambda /2$. Adjacent antinodes are also separated by $\lambda /2$, and a node-to-neighbouring-antinode distance is $\lambda /4$. These distances refer to the spatial envelope, not to the vertical distance travelled by a material point. Measuring node separation gives wavelength without timing a wave crest, provided the pattern is genuinely stationary over the measurement interval.

The phase of neighbouring segments differs by $\pi$. At one time, adjacent antinodes can be displaced in opposite directions; half a period later those directions reverse. The node between them remains fixed. This phase relation is a consequence of the sign change in the spatial sine factor and gives a direct record-level test: sensors placed on opposite sides of one node should have time traces with opposite sign but the same frequency and compatible amplitude when the geometry is symmetric.

Endpoint conditions select which wavelengths fit a finite system. A string fixed at an end must have zero displacement there, so the endpoint is a node. A string with both ends fixed has nodes at both ends and permits only patterns satisfying

where $L$ is the active string length. The corresponding normal-mode frequencies are

The integer $n$ counts the number of half-wavelength segments in the length. The lowest mode has one antinode and no interior nodes; the next has two antinodes and one interior node. Normal modes are not arbitrary sketches. They are the spatial patterns that satisfy the wave equation and the endpoint conditions simultaneously.

The wave speed in the normal-mode relation is still set by the medium. A uniform string has $v=\sqrt{T/\mu }$, so increasing tension raises every normal frequency while increasing linear density lowers every one. Changing length changes the allowed spatial patterns and therefore the frequencies even when tension and density remain fixed. An experiment can separate these effects by measuring or controlling $T$, $\mu$, and $L$, then comparing the measured mode frequencies with the common spacing $v/(2L)$.

Energy in a standing wave has a fixed spatial pattern only in a time-averaged sense. At an antinode, displacement and transverse velocity vary strongly through a cycle. At a node, displacement remains zero, but the local slope can be large and elastic energy need not vanish. At one instant, energy may be mainly kinetic near an antinode; a quarter cycle later, elastic energy can dominate elsewhere. Compare energy across the pattern with a time average rather than a single image of the string.

An experimental mode record should include the active length between constraints, the tension or other stiffness setting, linear density, drive frequency, sensor positions, and the method used to identify nodes. A spatial scan at fixed drive frequency can locate nodes from minimum displacement, but the sensor noise floor must be reported; an apparent nonzero node signal can be probe motion or baseline error. A time record at an antinode verifies the mode frequency, while paired records on opposite sides of a node verify the expected phase reversal. These measurements tie a drawn normal mode to quantitative endpoint and wave-speed conditions.

Endpoint modeling should match the actual support. A tightly clamped string is well approximated by a displacement node at the clamp, but a finite support can move slightly and shift the effective active length. The relevant length is the distance between the locations that enforce the observed node condition, not necessarily the distance between exterior hardware marks. Measure that length under operating tension. If a node scan places the endpoint minimum away from the assumed clamp position, use the measured minimum and include its position uncertainty in the normal-frequency comparison.

Mode-frequency checks use a sequence, not one isolated frequency. For fixed ends, the ratio $f_n/f_1$ should be close to the mode number when the string is uniform and the model applies. Plot measured frequency against mode number; the slope estimates $v/(2L)$ and any intercept indicates a systematic frequency or endpoint error. A single mode can agree by coincidence with an incorrect tension, density, or length. Several modes tested with the same calibrated length provide a much stronger check of the medium-speed relation and the endpoint assignment.

Energy measurements require time resolution even when the spatial pattern is fixed. At each selected point, a displacement sensor and a time derivative estimate determine the kinetic contribution, while neighbouring spatial samples determine the slope contribution. Node measurements expose a nonzero slope that a small displacement can conceal. Antinode measurements give a time record with large amplitude. Comparing both locations over a full cycle checks the expected exchange between kinetic and elastic forms without relying on a single frame. Use the same coordinate origin for both scans. A displaced position reference changes the inferred slope and can falsely shift a node toward an antinode.

Drive level should remain within the regime where frequency and shape do not shift with amplitude. Increase the drive modestly and verify that node positions and measured frequencies remain stable within uncertainty. A changing node minimum, a frequency shift, or a distorted antinode trace can arise from a nonuniform tension, a support that moves, or a measurement chain leaving its linear range. The final mode report should preserve the spatial scan, time records, source frequency, tension record, and the uncertainty model used for node position and frequency fitting.

Frequency derivation and mode-selection checks

Normal-mode frequencies follow by combining a spatial fit condition with the wave relation $v=f\lambda$. The allowed patterns of a string fixed at both ends contain an integer number of half-wavelength segments in the active length:

Substitution of ${\lambda }_n=v/f_n$ gives $f_n=nv/(2L)$. Endpoint locations set $L$; tension and linear density set $v=\sqrt{T/\mu }$. The mode number does not alter wave speed; it selects a shorter wavelength that fits the same length. A frequency ratio alone is not a derivation unless the associated spatial pattern satisfies the endpoint condition.

An open air column has displacement antinodes at both open ends and gives the same integer sequence when its effective acoustic length is used. A column closed at one end has a displacement node at the closed end and an antinode at the open end. Its length contains an odd number of quarter-wavelength segments, so the allowed indices are $1,3,5,\dots$ and $f_n=nv/(4L)$. Pressure labels reverse the displacement labels: a closed end is a pressure antinode. The measured sensor variable therefore matters when comparing a spatial scan with an air-column mode diagram.

Mode-shape measurement checks the spatial half-wavelength count. Scan amplitude and phase along the active length with a sensor position referenced to the actual endpoint condition. A node is identified by a displacement minimum together with a phase sign change on its two sides. A single low-amplitude reading is insufficient because sensor noise, probe alignment, or a local gain error can imitate a node. Fit the full envelope and quote the fitted node positions with their uncertainty.

Frequency uncertainty has both timing and model components. A long time record gives finer frequency discrimination, but a calibrated clock and stable drive are still required. Uncertainty in a fitted sine trace depends on record duration, noise, and the number of observed cycles. Temperature changes can alter tension in a string or sound speed in air; record the temperature and interval of each scan. Frequency ratios from several modes suppress a common clock-scale error that affects all modes similarly, whereas an incorrect endpoint length changes the inferred spacing model.

Mode selection requires both shape and frequency. A measured frequency near twice the lowest value needs a scan showing one interior node before assignment to the second string mode. A mixed-end air column has its next allowed pattern near three times the lowest frequency; an even-index fit signals a model error. Compare measured node count, endpoint labels, and frequency index in one table. Disagreement often points to an effective-length error, a sensor-variable mismatch, or a support that fails to realize the assumed endpoint condition.

The final mode result should list active or effective length, wave-speed model, endpoint type, harmonic index, measured frequency, node positions, sensor calibration, and uncertainty budget. A fit across several modes is stronger than a single-frequency claim because it tests one common speed and length against multiple spatial patterns.

The fixed-end string relation can also be read geometrically. The fundamental has one half-wavelength across the active length, so its wavelength is twice the length. Each successive mode adds one more half-wavelength segment without changing the endpoint nodes. Harmonic wavelength decreases as $1/n$ while frequency rises as $n$: the medium transports every permitted pattern at the same wave speed, and the higher pattern has a shorter spatial period. Confusing mode number with wave speed reverses the derivation.

A string with one fixed and one free end has a different allowed set. The fixed end requires zero displacement, while the free end requires zero slope. The lowest pattern spans one quarter wavelength, then the next permitted pattern adds a half wavelength. Its harmonic sequence therefore has the same odd-index structure as the mixed-end air-column displacement pattern. Both systems demonstrate that endpoint conditions select the wavelength fit.

Node and antinode geometry should be measured with a stated observable. A camera or displacement probe sees transverse string motion directly. A microphone in an air column responds mainly to pressure, whereas a small velocity probe responds to particle motion. Pressure nodes and displacement nodes occur at opposite positions in a simple air mode. A pressure scan can therefore look inverted relative to a displacement sketch without contradicting the same normal-mode solution. Report the sensor response before using a minimum or maximum to assign a spatial label.

Spatial resolution limits node placement. If sensors are separated by a substantial fraction of a half-wavelength, the true minimum may fall between them. Fit several nearby amplitudes to an envelope or scan the sensor through the minimum; do not promote the smallest sampled value to an exact node. Phase data make the fit stronger: the oscillation changes sign across a displacement node, so two nearby sensors on opposite sides should have an approximately half-cycle time shift. Noise can scramble phase where amplitude is very small, which is why node position uncertainty must be wider than the nominal sensor-coordinate uncertainty alone.

Frequency fitting also has an endpoint-length sensitivity. In $f_n=nv/(2L)$, a small fractional error in active length produces the same magnitude fractional error in the inferred frequency spacing with opposite sign. Measure the length under the operating tension and identify the physical points that enforce the node condition. A clamp, bridge, or tube opening can have an effective location that differs from a convenient external ruler mark. Comparing several measured modes with one fitted effective length is preferable to adjusting a separate length for each mode.

A reproducible mode-selection check proceeds in order. First survey the spatial shape at a drive frequency and locate endpoint minima or maxima appropriate to the sensor. Second count interior nodes and compare that count with a proposed harmonic index. Third measure the frequency with a calibrated time base. Finally, compare the complete set of measured frequencies with the index relation using one length and one speed model. This sequence prevents a frequency ratio from being assigned to a mode whose spatial pattern actually belongs to a different endpoint condition.

Uncertainty should be recorded separately for frequency, node position, active length, and wave-speed inputs. Repeating one frequency measurement estimates random timing and fit scatter. Changing tension, temperature, or sensor location tests systematic drift. The final table should contain each proposed mode index, its measured frequency, its node count, the active-length estimate, and the residual from the common frequency fit. An outlying row can identify a mistyped index, a disturbed mode shape, or an endpoint model requiring revision.

Energy, support loading, and damped mode sharpness

A standing string stores energy in a spatially patterned combination of kinetic and elastic forms. For small transverse motion, the local density is

At an antinode, transverse speed can be large, so kinetic energy changes strongly over a cycle. At a node, displacement remains zero but the spatial slope need not vanish, so elastic energy can remain important. Energy mapping requires both kinetic and elastic terms over time. The time average of the two terms supports location comparisons, while time-resolved records show their exchange during a cycle.

The ideal standing pattern has no net time-averaged power through an interior section. It can be regarded as two equal travelling components carrying equal average power in opposite directions. Their local instantaneous contributions do not vanish; they exchange energy with the string and with one another in the spatial pattern. A power probe at one point may alternate sign through a cycle even while its cycle average is zero. This differs from a travelling string wave, for which the time-averaged power has one preferred direction.

Real supports complicate the fixed-end idealization. A clamp can flex, a bridge can move, and a support can dissipate energy through friction or internal strain. Those effects shift the effective node location and allow some mode energy to enter the support. A support that moves slightly may leave a shallow displacement minimum rather than an exact node. The correct model is not chosen from the hardware name alone; measure the support motion or infer an effective endpoint from a spatial mode scan.

Support loading is especially visible when a string is driven strongly. The driver and support must supply energy lost to the medium and hardware each cycle. If their mechanical impedance changes with frequency, the measured mode amplitudes can differ even when the string mode shapes remain recognizable. Keep drive level low enough that support motion, tension, and endpoint location remain stable. Compare mode shapes at two nearby drive levels before treating an amplitude change as a property of the mode.

Damping removes energy from the mode and changes its observable sharpness. A lightly damped mode retains energy for many cycles after the drive changes, and its amplitude as a function of drive frequency has a narrow peak. Greater damping spreads the response over a wider frequency range and lowers the maximum amplitude. This statement does not require a detailed line-shape model: it follows from energy being lost each cycle, so the system has less time to accumulate a large mode amplitude at one selected frequency.

Mode sharpness should be measured operationally. At a fixed drive level, scan frequency in small steps, wait a stated settling time, and record a calibrated antinode amplitude or integrated mode energy. Repeat the scan in both frequency directions to test for drift. A broad response can result from support loss, internal material loss, changing tension, or a drive whose frequency scale is inaccurate. The report should state the amplitude metric, frequency step, settling rule, and environmental conditions rather than assigning all broadening to one mechanism.

Energy and damping checks complement the normal-mode frequency audit. Use a time record after a short drive interruption to estimate how rapidly mode amplitude decays, and use spatial scans to verify that the shape remains associated with the selected mode while the amplitude falls. If node locations migrate during the decay, endpoint loading or tension drift may be changing the model itself. A sound report separates the measured decay, the support condition, and the frequency response instead of treating them as one unspecified loss parameter.

Average-power verification requires a defined observation interval. Record the force and transverse velocity at a selected string section, multiply them with a consistent sign convention, and average over an integer number of cycles. A nonzero average can indicate unequal counterpropagating components, distributed loss, or a probe location that includes driver work. The same instrument must be checked on a travelling-wave control if possible, because a sign error in one derivative can turn a physical power signal into an artificial cancellation.

The spatial energy estimate has its own resolution limit. Node regions require closely spaced displacement samples because the slope changes rapidly there. Antinode regions require fine time sampling because velocity changes rapidly through zero crossing. Apply the same derivative stencil and smoothing rule at every position. Comparing a high-resolution antinode record with a coarse node record can manufacture an apparent energy imbalance even when the underlying mode is well behaved.

Support uncertainty should be included in the active-length budget. A scanned node minimum can have an uncertainty from sensor position, amplitude noise, and the fit used to locate the minimum. Carry that uncertainty into the predicted mode frequencies. If a support motion sensor is available, compare its phase with the nearby string motion; a large support response signals that the ideal fixed-end assumption is weak. Repeating the scan after remounting the support is a practical check of whether the endpoint condition is reproducible.

Damping measurements also need a clear amplitude scale. A displacement sensor may report peak amplitude, root-mean-square amplitude, or a Fourier-component amplitude; these differ by fixed factors only for a clean sinusoidal record. State which quantity is used and keep it unchanged across the frequency scan and decay measurement. A change in sensor gain or automatic range selection can imitate a change in mode sharpness. Background vibration should be measured with the drive off and removed using a stated method before comparing low-amplitude points.

The final support-and-energy report should therefore contain the mode index, endpoint scan, support condition, drive level, time-averaged power result, spatial energy method, decay record, and frequency-response data. These independent checks show whether a measured normal mode is an ideal low-loss pattern, a support-loaded pattern, or a mode whose apparent sharpness is limited by the measurement chain.

Termination impedance, driven power, and frequency-systematics

A real termination has a mechanical impedance; the ideal words “fixed” and “free” give only limiting cases. A transverse string wave has characteristic impedance $Z_s=\sqrt{T\mu }$. It relates transverse force and transverse velocity for a travelling component. A termination with a very large mechanical impedance moves little under the string force and approaches a displacement node. A termination with a very small impedance develops little transverse force and approaches a displacement antinode. Intermediate loading produces a partial return of wave energy and shifts the node minimum away from the geometric endpoint.

The relevant comparison is the termination impedance relative to $Z_s$, not the absolute mass or stiffness of the support in isolation. A heavy support connected through a compliant clamp can have a low effective impedance at one frequency. A light support attached through a stiff fixture can have a larger impedance than expected. Measure or model the complete support path, including clamp compliance, added mass, and losses, before assigning an endpoint condition to a standing-wave calculation.

Termination loading affects mode frequencies through the effective length and phase condition at the end. A finite support motion can place the displacement minimum inside or outside the visible string span, changing the length that belongs in $f_n=nv/(2L)$. The effect is systematic: it can move every measured mode in the same direction while leaving approximate integer ratios intact. A frequency sequence that looks harmonic is therefore not sufficient evidence that ruler-to-ruler length is the correct active length.

Driven modes require a power balance. The driver exerts a transverse force and has a transverse velocity at its attachment point. Their signed product is the instantaneous power delivered to the string-support system. Over many cycles, positive average input power replaces losses in the string, air, clamp, and driver coupling. At a low-loss mode, the stored oscillation energy can be large even though the average driver power is set mainly by loss. Away from a mode frequency, the driver and local string motion have a different phase relation, so the same force amplitude can deliver less average power.

Measure input power with synchronized force and velocity records. A force sensor alone does not determine energy transfer, and a displacement amplitude alone does not determine average power. Calibrate sensor signs at the driver attachment, subtract any static preload, and average over an integer number of drive cycles. Repeat with the string disconnected or with the driver held stationary to estimate electrical and mechanical backgrounds not transferred into the mode.

Systematic frequency errors often dominate a mode table after random timing scatter has been reduced. A clock-scale error multiplies all measured frequencies by one factor. An active-length error changes the expected spacing. Tension drift changes wave speed; temperature can alter both tension and linear density. Sensor placement affects mode-shape assignment, while a frequency-dependent support load can shift modes by different amounts. Treat these as separate model terms. One generic error bar cannot identify their distinct residual patterns.

A practical audit holds one variable fixed at a time. Check the time base against a reference frequency, measure tension before and after the mode scan, survey the active length from fitted node minima, and repeat selected modes after remounting the support. Compare residuals from the frequency fit with residuals from the spatial shape scan. A length error changes the full frequency sequence coherently, whereas a support-load effect may be larger in modes with substantial motion at that termination.

Report a mode result with the measured frequency, harmonic index, active-length model, support configuration, drive level, and uncertainty budget. Include the endpoint model and evidence for it from the spatial scan. The resulting frequency table tests the physical string-and-support model against the measured peaks.

The returned wave establishes the standing pattern. Its amplitude and phase must match the termination model. A nearly rigid support produces a return with a displacement sign change at the endpoint, while a nearly free support produces a different displacement condition. An impedance-loaded end lies between these limits. Measure the endpoint neighbourhood directly instead of assigning a return phase from an ideal support description.

Power balance can be checked over one selected mode shape. Estimate the string's stored energy from the calibrated displacement record, then compare the average driver input with the observed decay when the drive is reduced or interrupted. The comparison need not assume zero loss; it tests whether the measured loss rate and input power are compatible with the stored energy scale. If the driver input is much larger than the observed loss estimate, check force-sensor phase, attachment slip, and energy carried into the support.

Frequency scans need a settling criterion. After each drive-frequency change, wait a fixed number of cycles or a fixed time relative to the observed decay before recording amplitude and phase. A reading taken during transient growth can shift the apparent mode maximum and make repeated scans disagree. Store both upward and downward scans; a difference between them can indicate slow tension drift, thermal change, or a support whose response changes with loading history.

Use residual patterns to separate systematic causes. A nearly constant fractional offset over all modes suggests clock scale or tension calibration. A residual that grows with mode index can arise from active-length error or a frequency-dependent support load. A mismatch in node positions with otherwise accurate frequencies points to sensor-coordinate or endpoint-shape error. These diagnostic patterns separate the causes that a root-mean-square frequency residual conceals.

The final uncertainty budget should list time-base calibration, tension measurement, linear-density measurement, active-length fit, support repeatability, sensor position, and amplitude-dependent shift separately. Some terms are correlated across every mode; others vary from run to run. Preserve the raw force, velocity, displacement, and mode scan records so that an updated support model can be tested without repeating the entire experiment.

This documentation keeps termination loading distinct from measurement error.

Orthogonality, modal coordinates, and spatial projection

The normal-mode shapes of an ideal string with fixed endpoints are

Different modes are orthogonal over the active length:

The result is a spatial cancellation. The product of two different mode shapes has positive regions and negative regions whose signed areas balance. Orthogonality decomposes a measured displacement profile into modal coordinates.

Write the measured transverse displacement as

where $q_n(t)$ is the generalized coordinate of mode $n$. In the ideal linear model, each coordinate has its own angular frequency and damping. A drive or initial shape contributes to mode $n$ in proportion to its overlap with $X_n$. A point driver located at a node of one mode has zero ideal coupling to that mode because its local displacement coordinate vanishes there. A driver near an antinode has stronger coupling, subject to its force orientation and mechanical impedance.

Modal projection requires a spatial measurement rule. At sampled positions $x_j$, fit the profile to the known shapes or approximate the continuous projection with a weighted sum:

The weights $w_j$ represent the physical spacing of the samples. Uniformly spaced measurements use equal interior weights in a basic rectangular rule; endpoint or nonuniform samples need weights appropriate to the actual spatial grid. A sensor located only at one antinode cannot distinguish a pure mode from a mixture that happens to have the same local displacement at that point.

Spatial aliasing can corrupt the projection. A high mode has shorter wavelength and more nodes; a sampling grid that resolves the fundamental can miss sign changes of a higher mode. Increase spatial sample density until fitted modal amplitudes remain stable under grid refinement. The scan should extend over the full active length. Truncating one end changes orthogonality and can leak amplitude from one mode into another in the fitted result.

Real strings can couple modes through geometric nonlinearity, time-varying tension, or a support whose motion has its own resonance. In that case, modal coordinates remain a valid basis, but their time records can exchange energy and exhibit sidebands or amplitude modulation. Report whether the experiment is in the low-amplitude linear regime before treating each projected coordinate as an independently damped oscillator.

A modal analysis report should include the assumed endpoint shapes, active length, spatial grid, projection weights, number of retained modes, residual profile, and grid-refinement result. A low residual concentrated near a support can indicate an endpoint model error. The fitted modal spectrum remains tied to the physical string and support configuration.

Resonance scans, linewidth, and mode identification

A frequency sweep measures a driven response, not the undamped natural frequency in isolation. At each source setting, record the drive frequency, the amplitude at a stated sensor position, and the phase relative to the drive. A narrow response peak often marks a lightly damped normal mode, but the peak location can shift when the driver has appreciable stiffness or mass, when the support is compliant, or when the amplitude is large enough to alter tension. The spatial pattern measured near the peak therefore belongs in the mode assignment alongside the frequency value.

A response curve expressed as squared amplitude or power has a full width at half maximum that defines a damping scale. With weak damping and a well isolated mode, the quality factor is estimated by

where $f_0$ is the resonance frequency and $\Delta f$ is the separation between the two half-power frequencies. The approximation requires a response curve with a single identifiable maximum and a bandwidth small compared with the frequency. It should not be applied to two unresolved modes, to a sweep that changes drive force with frequency, or to a sensor chain with an uncorrected resonance of its own.

Phase data expose several such failures. A simple driven mode changes its response phase rapidly through the resonance interval. The exact reference depends on whether the instrument reports displacement, velocity, or acceleration and on the polarity of the drive transducer. Calibrate that convention at a frequency where the mechanical response is understood, preserve it in the data file, and compare phase curves from several positions. Sites on opposite sides of a node should differ by approximately one half-cycle after the common instrument delay is removed.

Sweep direction must be logged. A frequency scan upward can disagree with a downward scan if the tension warms, the support creeps, or a nonlinear response develops. A linear, time-independent system returns the same curve after a sufficiently long settling interval. A separated pair of peaks or a direction-dependent jump calls for shorter steps, longer settling, and a spatial scan at each peak. Labeling the larger amplitude maximum as a single harmonic without those checks can join two physical modes or mask a support resonance.

The response amplitude also depends on the observation point. At an ideal node, the mode displacement signal tends to zero even though the mode can store substantial energy. A fixed sensor position can thus make a strong resonance appear absent. A mode map avoids that ambiguity: use the same drive setting at a series of positions, record complex amplitude, and compare the measured sign changes and node minima with the proposed shape. The map tests the mode identity; the frequency scan supplies its dynamic scale.

Report the frequency-grid spacing, dwell time, source-amplitude control method, sensor locations, response calibration, and the rule used to define the bandwidth. A result such as $Q=f_0/\Delta f$ has no reproducible meaning if the half-power convention, amplitude variable, or background subtraction is omitted. A response table with frequency, complex amplitude, repeat number, and drive monitor also supports later checks for drift and nonlinearity. The final mode identification rests on agreement between endpoint conditions, spatial shape, phase relation, frequency sequence, and the dynamic response curve.

Modal closure is a final check on a multi-sensor record. Reconstruct the measured spatial trace from the retained modal coordinates at each sampled time, then examine the residual by position and frequency. A residual concentrated at one clamp can identify an endpoint-load error. A residual concentrated at one sensor can identify gain, timing, or alignment error. A residual growing with drive amplitude can identify nonlinear coupling. The reconstruction should use the same coordinate origin, spatial weights, and sensor calibration as the mode projection. This closes the connection between a frequency peak, a spatial node pattern, and the stated mechanical model.