---
title: Standing Waves
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 6
order: 706
summary: >
  Clamp a string at both ends and only certain frequencies survive: the ends must be
  nodes, and that single geometric demand quantizes the wave into a discrete set of
  modes $f_n=nv/(2L)$. The travelling wave becomes a fixed pattern of nodes and
  antinodes — standing, not moving — because equal waves running in opposite
  directions superpose. We build the standing wave from its counter-propagating
  pieces, read the harmonic sequence off the boundary conditions (half-wavelengths
  for a fixed-fixed string, odd quarter-wavelengths for a closed pipe), and test the
  ideal model against node scans and resonance peaks.
topics: [Waves]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 16 — Standing Waves; §§16-2–16-3"
---

## 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

$$
y_1=A\sin(kx-\omega t),\qquad y_2=A\sin(kx+\omega t).
$$

Adding them gives

$$
y(x,t)=2A\sin(kx)\cos(\omega t).
$$

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.

$$
% caption: A standing wave as the superposition of two equal waves travelling in opposite directions. The right- and left-moving components (top two rows) share the same amplitude, frequency, and speed; their sum $2A\sin(kx)\cos(\omega t)$ is a fixed pattern whose zeros (nodes) stay put at intervals $\lambda/2$ while the antinodes between them oscillate.
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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

$$
L=n\frac{\lambda_n}{2},\qquad n=1,2,3,\ldots
$$

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

$$
f_n=\frac{nv}{2L}.
$$

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.

$$
% caption: The first three modes of a string clamped at both ends. Mode $n$ fits exactly $n$ half-wavelengths between the fixed ends and carries one more interior node (open circles) than the mode below it; the allowed lengths are $L=n\lambda_n/2$, giving frequencies $f_n=nv/2L$.
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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:

$$
L=n\frac{\lambda_n}{2}.
$$

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.

> **Worked example.** A string of active length $L=0.600\ \mathrm m$ carries tension
> $T=72.0\ \mathrm N$ and has linear density $\mu=5.00\ \mathrm{g\,m^{-1}}
> =5.00\times10^{-3}\ \mathrm{kg\,m^{-1}}$. The wave speed is
>
> $$
> v=\sqrt{\frac{T}{\mu}}
> =\sqrt{\frac{72.0\ \mathrm N}{5.00\times10^{-3}\ \mathrm{kg\,m^{-1}}}}
> =120\ \mathrm{m\,s^{-1}}.
> $$
>
> With both ends fixed, the first three normal-mode frequencies are
>
> $$
> f_n=\frac{nv}{2L}=n\,\frac{120\ \mathrm{m\,s^{-1}}}{2(0.600\ \mathrm m)}
> =n\times100\ \mathrm{Hz},
> $$
>
> so $f_1=100\ \mathrm{Hz}$, $f_2=200\ \mathrm{Hz}$, and $f_3=300\ \mathrm{Hz}$. The
> modes form an integer sequence because the fixed-fixed geometry admits only whole
> half-wavelengths; raising the tension to $T=108\ \mathrm N$ would raise $v$ to
> $147\ \mathrm{m\,s^{-1}}$ and every $f_n$ by the same factor $\sqrt{108/72}=1.22$.

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 $1,3,5,\ldots$ 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.

$$
% caption: Boundary conditions fix the harmonic set of an air column, drawn as the displacement envelope. An open-open column has displacement antinodes at both ends and admits every integer harmonic; a closed-open column has a node at the closed end and an antinode at the open end, so only odd harmonics fit and $f_n=nv/4L$.
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$$

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.

$$
% caption: A string clamped at one end and free at the other. The clamp forces a displacement node and the free end forces an antinode (zero slope), so the shortest fit is a quarter wavelength and only odd multiples follow, giving $f_n=nv/4L$ with $n=1,3,5,\dots$ Solid dots mark nodes, open circles the interior node of $n=3$.
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$$

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

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

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.

$$
% caption: Energy in a standing string mode. Antinodes carry the largest transverse motion, so kinetic energy peaks there through the cycle; nodes stay at zero displacement yet have the steepest slope, so elastic energy need not vanish at a node.
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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.

| measurement record | quantity extracted | role in the mode model | discrepancy it can expose |
|---|---|---|---|
| spatial displacement scan | node minima and antinode positions | active length and endpoint condition | A minimum displaced from the clamp or a mode shape inconsistent with the assigned index |
| support-motion channel | support amplitude and phase near an endpoint | finite termination impedance | A nominally fixed end that moves enough to shift the effective node |
| free-decay trace | amplitude envelope or energy-decay rate | damping and ring-down time | A decay that changes with amplitude or an endpoint condition that drifts during the record |
| synchronized force--velocity trace | cycle-averaged input power | energy supplied to string and support losses | A force-channel phase error or energy transfer into the support |
| upward and downward frequency scans | peak frequency, width, and scan repeatability | resonance sharpness and frequency calibration | Thermal tension drift, hysteresis, or a response sampled before settling |

## 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.

$$
% caption: The wave returned from a support builds the standing pattern, and the termination impedance sets the boundary. A near-rigid, high-impedance support (shown) returns the wave with a displacement node at the clamp; a low-impedance end approaches an antinode, and real supports lie between, shifting the effective node off the geometric endpoint.
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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

$$
X_n(x)=\sin\left(\frac{n\pi x}{L}\right),
\qquad n=1,2,3,\ldots.
$$

Different modes are orthogonal over the active length:

$$
\int_0^L X_m(x)X_n(x)\,\d x=0
\qquad (m\ne n).
$$

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.

$$
% caption: Two distinct fixed-end modes. Their product changes sign along the string,
% so the integral $\int_0^L X_m X_n\,\d x$ over the active length vanishes for $m\ne n$;
% this orthogonality lets each modal amplitude be projected out of a measured profile
% independently.
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Write the measured transverse displacement as

$$
y(x,t)=\sum_{n=1}^{N}q_n(t)X_n(x),
$$

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:

$$
q_n(t)\simeq
\frac{\sum_j w_j y(x_j,t)X_n(x_j)}
{\sum_j w_j X_n^2(x_j)}.
$$

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.

$$
% caption: Resonance scan at one selected sensor site. The peak locates the mode
% frequency $f_0$ and the half-power width $\Delta f$ sets the quality factor
% $Q\simeq f_0/\Delta f$; the sensor position must be stated because a node can hide
% the response even when the mode stores large energy.
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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

$$
Q\simeq\frac{f_0}{\Delta f},
$$

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.