Beats and Coupling

Beats and amplitude modulation

Two harmonic signals add according to superposition. At one sensor, let their displacements have equal amplitude $A$, angular frequencies ${\omega }_1$ and ${\omega }_2$, and the same initial phase:

Define the average and separation

The product-to-sum identity gives

The rapid factor $\cos (\bar{\omega }t)$ is the carrier oscillation. The slow factor $2A\cos (\Delta \omega t/2)$ sets its signed amplitude. The product identity gives both factors directly for equal-amplitude sinusoids. A listener, microphone, accelerometer, or photodetector usually responds to the slow rise and fall when $|\Delta f|$ is much smaller than the average frequency.

The beat frequency is the repetition rate of the amplitude maxima:

The signed envelope changes sign after an interval $1/(2f_{\mathrm{beat}})$. A sign change shifts the carrier by half a cycle; the magnitude of the envelope is large on both sides of that sign change. Audible loudness, vibration energy, and mean-square detector output therefore repeat after $1/f_{\mathrm{beat}}$, not after the signed cosine completes only its positive half. Confusing signed envelope rate with observable beat rate creates a factor-of-two error.

The relative phase evolves as

At a constructive maximum, the components have the same phase. At the next destructive minimum, their phase difference has advanced by $\pi$. The initial phase ${\delta }_0$ shifts the timing of the beat pattern but leaves its repetition rate unchanged. A source trigger may establish a reproducible ${\delta }_0$; two independent oscillators generally begin with an unknown value.

Unequal amplitudes and beat visibility.

Equal amplitude gives complete cancellation at each beat minimum. Real sources often have amplitudes $A_1$ and $A_2$ that differ. The squared instantaneous amplitude of their sum can be arranged as

Complex envelope and carrier phase.

Set the phase of the first component to zero and collect the rapid average-rate factor. The sum can then be written as

The complex envelope $\mathcal{A}$ rotates slowly in the complex plane. Its modulus $R$ is the measured envelope amplitude, while $\psi$ is the carrier phase:

The complex-envelope representation separates two effects often hidden in a single waveform. As the component phasors approach opposition, $R$ decreases. Their vector sum also changes direction, shifting the apparent timing of carrier peaks. Equal amplitudes place the resultant at the origin at every complete cancellation; the signed envelope changes sign and the carrier phase advances by a half turn. Unequal amplitudes keep the resultant away from the origin, so its phase can be tracked continuously through the shallow minimum. A phase-sensitive measurement therefore contains information that a rectified envelope discards.

The envelope extrema are

The beat remains visible, but its minimum no longer reaches zero. A convenient contrast measure is

The last equality assumes nonnegative amplitude magnitudes. A small secondary source produces shallow fluctuations that may fall below sensor noise, even when the frequency separation lies inside the measurable range.

Beat frequency alone determines an absolute frequency separation, not its sign. Suppose a 440 Hz reference and an unknown tone produce 3 Hz beats. The unknown frequency can be 437 Hz or 443 Hz. A controlled change resolves the ambiguity. If the unknown source is raised slightly and the beat rate rises, it began above the reference. If the beat rate falls, it began below the reference. A tuning log records the adjustment direction and the resulting rate.

Deliberate amplitude modulation.

Beats arise from two nearby components already present in a signal. Amplitude modulation deliberately multiplies a carrier by a slower control waveform. A single-tone modulator has the form

where $A_{\mathrm{c}}$ is carrier amplitude and $m$ is modulation index. Expanding the product produces

The spectrum contains a carrier at $f_{\mathrm{c}}$ and equal sidebands at $f_{\mathrm{c}}\pm f_{\mathrm{m}}$. The slow envelope rate and the sideband spacing are both $f_{\mathrm{m}}$. Product expansion gives the same mathematical pattern as a two-frequency beat, but the source configuration differs: beats compare independent components; modulation specifies a carrier and a deliberately imposed control signal.

The spectral view gives a direct modulation measurement. With a calibrated amplitude spectrum, either sideband amplitude $A_{\mathrm{sb}}$ gives

With $A_{\mathrm{c}}$ interpreted as peak voltage across a frequency-independent resistive load, carrier power is $P_{\mathrm{c}}=A_{\mathrm{c}}^2/(2R)$. Each sideband has power $m^2{P}_{\mathrm{c}}/4$, and total power is

A pure sinusoidal modulator and a linear measurement chain give this power relation. Harmonic distortion produces extra spectral lines and changes the power balance.

For $0\le m\le 1$, the prescribed envelope stays nonnegative. At $m=1$, the envelope reaches zero once per modulator cycle. Values above one force envelope reversal; the carrier changes phase by half a cycle around the reversal. An ideal linear multiplier still has the carrier and two first sidebands in the expansion. Envelope detectors and nonlinear transmitter stages can distort the zero crossing and add spectral lines. An oscilloscope measurement of the upper and lower envelope levels gives

after correcting any detector offset. A nonzero baseline, envelope detector droop, or insufficient carrier cycles per modulation period can bias this estimate.

Coupled oscillators and energy exchange

Two oscillators exchange force when a mechanical, electrical, acoustic, or optical link makes the state of one affect the other. The simplest mechanical model uses equal masses $m$, equal support springs of constant $k$, and a coupling spring of constant $k_{\mathrm{c}}$. Let $x_1$ and $x_2$ be displacements from the common equilibrium positions. The coupling spring extension is $x_1-x_2$, so Newton's second law gives

Each equation contains its own restoring force and a force proportional to relative displacement. Equal displacements leave the coupling spring unextended. Opposite displacements stretch or compress it strongly and add restoring force. Those two patterns become the normal modes.

The system has two degrees of freedom, so its undriven linear motion is a superposition of two independent normal coordinates. Define

Adding and subtracting the equations of motion gives

with

The plus coordinate has $x_1=x_2$ and is the in-phase mode. The coupling spring remains at its equilibrium length, so its stiffness does not enter ${\omega }_{+}$. The minus coordinate has $x_1=-x_2$ and is the opposite-phase mode. The link spring adds restoring force, so ${\omega }_{-}>{\omega }_{+}$ for positive coupling.

Normal coordinates are a change of basis, not an additional approximation. The vector of physical displacements can be reconstructed from them:

Generic initial conditions contain both mode vectors; a state aligned with one mode vector is the special case. The measured motion of one mass can therefore contain two nearby frequencies even when the system has no external modulation. Their superposition produces a beat-like envelope. The envelope describes an exchange of motion between physical coordinates, while the two normal coordinates evolve independently in the ideal model.

The total mechanical energy is

In normal coordinates it separates:

Each parenthesis is conserved in the ideal, undriven system. Coupling creates the force pathway through which energy assigned to one physical oscillator appears later in the other. The energy assigned to mass 1 alone includes part of the coupling-spring energy by convention, so an energy plot needs a stated partition rule. The total energy remains the unambiguous conserved quantity.

Frequency splitting at weak coupling

The uncoupled natural frequency is ${\omega }_0=\sqrt{k/m}$. When $k_{\mathrm{c}}\ll k$,

The splitting $\Delta \omega ={\omega }_{-}-{\omega }_{+}$ is small but measurable when a record is long enough. In ordinary frequency units,

Weak coupling therefore appears as a pair of nearby peaks around the isolated oscillator frequency. The mode split also sets the slow energy-transfer rate. Reducing coupling narrows the split and increases the time required to observe a complete transfer.

An experiment can estimate coupling in two complementary ways. Excite each normal mode separately and measure its frequency, then evaluate

Or excite one physical oscillator, measure the beat separation in its motion, and use the weak-coupling relation when $k_{\mathrm{c}}/k$ is small. Spectral measurement uses two clean peaks. Envelope timing uses the slow amplitude pattern and can be more sensitive to a small split, provided the exchange remains visible for several cycles.

Energy exchange is a two-mode interference pattern.

Release mass 1 from displacement $A$ with mass 2 at rest and both initial velocities zero. The initial state has equal amounts of the two normal coordinates. Solving the two normal-coordinate equations and transforming back gives

where $\bar{\omega }=({\omega }_{+}+{\omega }_{-})/2$ and $\Delta \omega ={\omega }_{-}-{\omega }_{+}>0$. Mass 1 begins with a large carrier oscillation whose envelope decreases. Mass 2 begins at zero and grows into a carrier oscillation with the complementary envelope. The sum and difference of normal modes determine both signals. A single normal mode keeps a fixed modal energy; their superposition changes the energy assigned to each physical coordinate.

At $t=0$, all displacement energy is associated with mass 1 and its support spring. At

the slow factors exchange roles: mass 1 has a small envelope and mass 2 has its largest envelope. At

the original envelope arrangement returns. These times apply to the ideal identical system and to the initial condition above. The carrier phase at a return may have advanced by many cycles, but the physical energy distribution repeats.

An envelope alone cannot label its mechanism. Two independent tuning forks make beats because their sensor signals add. Coupled oscillators produce a similar algebraic envelope because an initial physical displacement excites two normal frequencies. Independent tones require two sources at the sensor; a coupled pair can begin with one displaced mass and develops two frequencies through its equations of motion. In both cases, an FFT shows two frequency components and the time-domain envelope is governed by their separation.

The free-transfer model applies after the initial release. Weak damping slowly lowers both mode amplitudes and can make later transfers hard to resolve. Strong damping can remove the visible envelope before a transfer completes. Continuous driving injects energy throughout the record and requires the forced-response treatment developed in the damping and driven-resonance lessons.

Early maxima deserve extra care in laboratory data. A support spring may be nonlinear at large displacement, friction can depend on direction, and the coupling element can carry its own mass or internal modes. Each effect changes the mode frequencies with amplitude or adds extra spectral lines. Repeat a low-amplitude trial and compare the split before attributing an irregular envelope to a new coupling phenomenon.

Detuning competes with coupling.

Identical oscillators transfer energy completely because their two normal modes enter the initial state with equal weight. Fabricated oscillators are rarely identical. Keep equal masses $m$, let the support constants differ, and define

The normal-mode condition becomes

The two squared normal frequencies are

The square-root term combines detuning and coupling. Far from frequency matching, each mode is concentrated on one physical oscillator and transfer is weak. Near matching, the mode shapes mix strongly and the minimum separation remains set by coupling. Plotting the two measured mode frequencies while gradually changing one support stiffness gives a characteristic split near the crossing region.

One transfer indicator is the largest fraction of initial energy that can reach the other oscillator. In the weak-coupling, near-resonance limit, its scale is

It approaches one at matching and falls as detuning exceeds coupling. The formula assumes an initial displacement localized on one oscillator and neglects damping. It guides apparatus adjustment: tune the isolated frequencies close together before making a precision measurement of a weak link.

Frequency measurement and model checks

A beat measurement begins with a record long enough to contain several envelope cycles. If a trace contains $N_{\mathrm{b}}$ complete beat intervals during a measured duration $T_{\mathrm{b}}$, a direct estimate is

Counting intervals between the first and last selected maxima is better than dividing one interval by a stopwatch reading. A least-squares fit of peak index against peak time uses all selected maxima; residual structure can diagnose drift, missed peaks, or a changing envelope. The peak rule must be stated: raw-carrier maxima, rectified signal maxima, envelope maxima, or peaks of a correlation with a reference template lead to different timing jitter.

An envelope may be extracted from a narrow-band record by making a quadrature signal

Here $\mathcal{H}$ denotes the Hilbert transform. The procedure requires a band choice that retains both nearby components and rejects unrelated tones. A direct absolute-value trace creates additional carrier harmonics; it can still show beat timing, but it is less suitable for calibrated envelope amplitude. In a short record, transform end effects can distort $E(t)$ near the first and last few cycles. Discard edge regions or extend the record before interpreting envelope maxima.

The Fourier approach resolves the two nearby frequencies directly. A record of duration $T$ has nominal bin spacing

Resolving a split of 0.20 Hz requires a substantially longer record than 5 s. The exact record length depends on noise, window shape, and desired confidence. Windowing reduces spectral leakage from a finite gate, but broadens each displayed main lobe. Zero padding draws more points across that lobe and assists peak fitting; it does not provide the resolving power of a longer acquisition.

Frequency resolution and beat timing have complementary strengths. A 30 s record can resolve a small spectral split even when one complete energy transfer would take longer than the recording. Conversely, an envelope timing method can estimate a split smaller than one displayed FFT-bin spacing by fitting the complete time-domain model, provided its signal-to-noise ratio and phase stability are adequate. The reported method should identify whether the result came from spectral peak separation, envelope timing, or a fit of both traces.

Amplitude-modulation data require a different spectral check. A pure sinusoidal modulator gives a carrier and two equal first sidebands. Unequal sidebands can arise from detector response, frequency-dependent path loss, a non-sinusoidal modulator, or phase and gain mismatch in an instrument. Inspect higher sidebands before interpreting unequal first sidebands as a physical asymmetry. A square or clipped modulator contains harmonics, each of which produces an additional sideband pair.

The time-domain envelope estimate and the sideband estimate should agree after calibration. Measure $E_{\max }$ and $E_{\min }$ from an envelope with its DC offset removed, then calculate

Measure carrier and either sideband from a linear-amplitude spectrum, then calculate

A disagreement points to a finite envelope-detector response, analogue attenuation of one spectral region, or an FFT-window bias when a line lies between bins. Repeat the test with a known modulation index before assigning an uncertainty to an unknown signal.

Identifying the normal modes

Measure both coupled coordinates during a low-amplitude free trial. The low mode has the same displacement sign at the two sensors; the high mode has opposite sign. A spectral peak alone cannot label a mode if sensor polarity is unknown. Verify the channel polarity with a common test displacement, then inspect the cross-spectrum

Near a clean low-mode peak, ${\theta }_{12}$ is near zero. Near a clean high-mode peak, it is near a half turn modulo full cycles. Coherence should also be reported. Low coherence means the cross-phase estimate has little physical value, even when the plotted phase happens to lie near an expected number.

Prepare the initial condition close to one mode to isolate it. Move both masses together and release them with matched velocities to excite the low pattern. Move them oppositely to excite the high pattern. Small differences in spring constants, release timing, or support friction add the other mode. A fitted two-mode model reports the contaminating contribution.

For each sensor trace, fit

The fitted amplitudes and phases test the mode shapes. In the low mode, $B_{1+}$ and $B_{2+}$ are equal within calibration uncertainty; in the high mode, the corresponding fitted phases differ by a half turn. Residual plots expose frequency drift, nonlinear stiffness, or an unmodeled third mode before those effects are hidden inside a broad spectral peak.

Resolution, uncertainty, and model checks.

Close-frequency measurements are often limited by record length before they are limited by sample rate. A 20 kHz digitizer can represent a 500 Hz carrier easily, yet a 0.05 Hz beat split still needs a long, stable record. The carrier must be sampled adequately, the record must retain several slow envelope cycles, and the instrument clock must remain stable across the full duration. These are separate requirements. Increasing sample rate without increasing duration improves waveform shape but leaves the nominal frequency-bin spacing unchanged.

Time-domain beat fitting uses the covariance from the fitted peak-time slope or from a direct two-frequency nonlinear fit. A simple estimate from repeated complete beat intervals is

The counting term is usually negligible after a peak index has been fitted rather than rounded by eye. Timing uncertainty includes sample clock error, interpolation error, trigger jitter, and uncertainty in defining an envelope maximum. A long record reduces random timing scatter, but a drifting source can turn a long average into a biased result. Plot peak-time residuals against time; a curved trend suggests that the instantaneous split is changing.

Spectral peak uncertainty is wider than one line of code suggests. The transform bin spacing $1/T$ is a sampling grid, whereas a line-center uncertainty depends on signal-to-noise ratio, leakage, window shape, and the fitted line model. A well-calibrated sinusoid can be estimated between bins by fitting a complex sinusoidal model. Two closely spaced lines need a joint fit; fitting one peak at a time biases both centers when their spectral lobes overlap. Preserve the complex data or the raw time series, because a plotted magnitude spectrum alone discards phase information used by the stronger estimators.

The uncertainty of a normal-mode split follows from both peak frequencies. If $\Delta f=f_{-}-f_{+}$ and the estimates are independent,

Common clock error is correlated and can cancel partly in a frequency difference. Separate channel timing errors affect phase measurements but do not necessarily shift a frequency measured independently from each channel. State the acquisition architecture before deciding which errors are independent.

The coupling estimate from normal frequencies is

For independent standard uncertainties, first-order propagation gives

The three propagation terms identify the limiting measurement. In a small split, frequency uncertainty often dominates because the two squared frequencies nearly cancel. Heavy masses with poorly known added hardware can instead be dominated by mass uncertainty. A sensitivity calculation before the experiment indicates whether to lengthen the record, improve mass calibration, or increase coupling temporarily for a coarse measurement.

Amplitude-modulation uncertainty has its own calibration problem. If

then independent envelope-level uncertainties give

At high modulation depth, $E_{\min }$ becomes small and baseline offset becomes important. At low modulation depth, the difference $E_{\max }-E_{\min }$ is small and noise dominates. Sideband measurements provide a second estimate, but their calibration requires a spectrum in linear amplitude units, not logarithmic display units.

Several systematic effects can imitate a split or obscure it in a measured record. A nonlinear spring changes frequency with amplitude; a record that decays in amplitude then shows a moving frequency even with no change in coupling. A support can introduce a third mode near one intended mode, producing an extra envelope. Reflections in a mechanical guide can add a delayed copy to a sensor trace. A microphone, accelerometer, or optical sensor can saturate and create harmonics that resemble modulation sidebands. Each effect has a test: vary amplitude, change sensor position, inspect the residual spectrum, and repeat with a calibrated single-tone source.

Worked measurements and laboratory workflow

Tuning with a frequency reference

A 440.0 Hz reference tone and an unknown instrument string give a measured beat rate of 3.0 Hz. The frequency candidates are

so the data support 437.0 Hz and 443.0 Hz before a directional test. Raise the string tension slightly. If the beat rate rises from 3.0 Hz to 3.6 Hz, the string was already above the reference and the initial value was 443.0 Hz. If the rate falls to 2.4 Hz, the string was below the reference and the initial value was 437.0 Hz. Tightening further until the beat rate vanishes gives the tuning target, provided the reference remains stable.

The adjustment must be small enough that the system stays on the same frequency branch. A large tension change can cross the reference, making the beat rate fall to zero and rise again. Record the adjustment direction and several beat rates rather than relying on a single before-and-after observation. A plot of beat rate against the control setting has a V shape whose minimum marks the reference frequency.

The precision of an acoustic tuning result is usually limited by pitch drift and ambient sound before it is limited by the arithmetic. Isolate a short section with stable amplitude, repeat the measurement after the source has warmed up, and use a microphone position that avoids a deep room cancellation. An FFT can confirm the two carrier lines, while the time-domain envelope confirms that the counted slow fluctuation arises from their separation. Together, the two checks distinguish a genuine beat from instrument harmonics near the reference frequency.

A sideband method can give a reliable modulation index even when the carrier is too rapid for a simple envelope detector. It loses reliability when the analyser displays logarithmic power without a documented reference, when its resolution bandwidth merges carrier and sideband, or when a non-sinusoidal modulator creates neighboring harmonics. Preserve the amplitude unit, detector mode, resolution bandwidth, window, and averaging count with the spectrum.

Coupled-oscillator split and transfer time

Let two equal carts have $m=0.500\mathrm{kg}$, support stiffness $k=80.0\mathrm{N}{\mathrm{m}}^{-1}$, and coupling stiffness $k_{\mathrm{c}}=4.00\mathrm{N}{\mathrm{m}}^{-1}$. Their normal frequencies are

The split is about $0.099\mathrm{Hz}$. A one-cart release therefore transfers its energy envelope to the second cart after

The full envelope return takes about 10.1 s. A record shorter than several seconds can still contain both spectral peaks if the frequency fit is strong, but it cannot show the full transfer directly. Plan the recording length from the slow five-second envelope time scale; the approximately 2 Hz carrier period is too short for this purpose.

The stiffness can also be inferred from measured frequencies. With a calibrated mass, substitute the two measured angular frequencies into $k_{\mathrm{c}}=m({\omega }_{-}^2-{\omega }_{+}^2)/2$. Compare that result with a static force-versus-extension measurement of the link spring. Agreement tests both the lumped-mass approximation and the assumption that the link has negligible mass. Disagreement can arise from support-spring mismatch, rotational motion of the carts, or a coupling element whose stiffness depends on extension.

A defensible laboratory workflow.

Start with geometry and sensor checks. Measure the effective mass of each moving body, including fixtures that move with it. Locate the motion-sensing points and verify that both channels use a common clock. Test sensor polarity by moving both targets in the same direction. The low mode should then produce matched signs. A polarity error swaps the visual meaning of low and high modes and can survive a frequency-only analysis.

Characterize the isolated oscillators before connecting the link. Measure each uncoupled frequency at the amplitude intended for the coupled trial. If their frequencies differ substantially, adjust support stiffness or mass before expecting full transfer. Then measure the link spring statically over the small extension range used dynamically. These preliminary measurements separate a later coupling split from support-spring mismatch.

Use three complementary trials:

1. Release both oscillators together to emphasize the low mode.
2. Release them in opposite directions to emphasize the high mode.
3. Release one oscillator while the other starts near rest to observe a two-mode envelope and energy exchange.

Low- and high-mode trials establish normal-mode frequencies and phase patterns. The one-mass-release trial tests whether their superposition predicts the observed transfer. Use one sensor gain, sample clock, and analysis gate across all trials. Changing instrument settings between normal-mode and transfer recordings confounds mechanical changes with measurement changes.

Store raw traces before extracting a peak or applying an FFT. Each processed result should retain a link to its input gate, baseline treatment, calibration factors, sampling rate, and analysis code settings. Save the full complex spectrum when phase is used. A screenshot of a magnitude plot cannot be reanalysed for channel phase, timing offset, or mode-shape sign.

The written result needs more than a final frequency. Report the two mode frequencies with uncertainties, relative sensor phase near each peak, calculated split, predicted transfer time, observed transfer time, and the method used for each estimate. Report modulation index from envelope and sideband methods when amplitude modulation is under study. State whether amplitudes are peak, rms, or calibrated displacement. These details let another reader reproduce the reduction and judge the limits of the model.

The linear two-mode model applies over a defined operating range. Keep amplitudes small enough that spring stiffness and sensor response are stable. Keep damping weak enough that several envelope periods remain visible. Avoid nearby support resonances, loose mounts, and contact friction that add frequencies outside the two-mode model. If a third spectral line persists as amplitude is reduced and the sensor position is changed, expand the model to include a third mode and report that structure separately from the beat rate.

Intentionally amplitude-modulated signals require source calibration at the measurement point. A cable, transducer, or acoustic path can change sideband balance between source and sensor. Compare a known unmodulated carrier, a known sinusoidal modulator, and the unknown signal with one acquisition configuration. The sequence separates source modulation from path response and keeps a spectrum of independent beats from being mislabeled as deliberate amplitude modulation.

Compare time-domain and frequency-domain predictions by using the two fitted normal frequencies to synthesize both sensor traces with their measured amplitudes and phases. Compare the synthetic envelopes with the observed one-mass-release envelope. Agreement across more than one initial amplitude and sensor gap supports the split and coupling interpretation beyond a single spectral plot. Structured residuals direct the next refinement: amplitude dependence points to nonlinear stiffness, a late delayed copy points to reflection, and a stable extra line points to an additional mode.

Envelope diagnosis and model limits

An envelope alone does not identify its cause. A recorded signal can show slow amplitude variation because two independent tones superpose, because a source was amplitude modulated, because a coupled system contains two normal modes, or because a sensor chain has frequency-dependent gain. The diagnosis starts with the source configuration and then checks phase, spectrum, and complementary coordinates.

Independent-tone beats require two components that reach the same measurement point. Turning either source off removes one spectral line and removes the beat. Changing one source frequency moves one line while the other remains fixed. The envelope rate follows the new line separation. Source amplitudes set beat contrast: unequal amplitudes leave a nonzero envelope minimum. Independent tones do not produce a complementary-coordinate energy-transfer signature.

A controlled amplitude-modulation source has a different hierarchy. The applied carrier establishes the central line. The modulating waveform establishes symmetric sideband spacing, and a sinusoidal modulator predicts sideband amplitude from one index $m$. Altering modulator frequency moves both sidebands by equal amounts in opposite directions around the carrier. Altering modulation index changes both sideband heights together. These tests distinguish a controlled AM signal from two unrelated sources whose amplitudes and phases drift independently.

Coupled oscillators add mode-shape evidence. At one normal frequency, the two physical coordinates have matched signs; at the other, they have opposite signs. After a one-oscillator release, the second coordinate grows as the first coordinate loses envelope energy. Varying link stiffness changes the separation between normal frequencies and changes transfer time. These coupled signatures remain meaningful even when a single sensor trace resembles an ordinary two-tone beat.

Sampling can imitate a close frequency pair. Temporal aliasing folds a component above half the sample rate into a lower apparent frequency. Leakage from a strong nearby line can resemble a weak sideband when a short rectangular gate is used. Repeated mains interference can create fixed frequency offsets unrelated to the apparatus. Check the sample rate, analogue bandwidth, window, and background spectrum before interpreting a small line as a beat component or a normal mode.

Frequency drift requires a time-resolved view. Divide a long record into overlapping segments, fit the frequency pair or sidebands in each segment, and plot their centers. A stationary beat has a stable separation. A drifting source can retain a visually regular envelope while its peak intervals change gradually. In a coupled apparatus, a drifting support stiffness shifts the normal frequencies; a change in link stiffness changes the separation more directly. Track both centers and their separation to distinguish those effects.

Amplitude variation also needs a source check. A slowly changing source amplitude multiplies an entire record and can mimic a broad envelope without two close frequencies. The spectrum then carries one carrier line with low-frequency broadening or sidebands set by the amplitude drift. Compare the trace against an independent source monitor, or record a reference channel that bypasses the propagation path. A genuine two-component beat keeps two carrier frequencies visible even when the envelope becomes shallow.

Spatial measurements add another test. Place sensors at several positions along a mechanical guide or sound path. Two propagating tones can acquire different path phases while retaining their source frequencies. A normal mode of a compact two-oscillator apparatus gives a consistent sign pattern tied to the two masses. Sensor movement that changes only amplitude may indicate spatial cancellation; sensor movement that changes relative phase at one spectral line can indicate propagation geometry. Record physical sensor positions with every trace.

An analysis report should name the observed quantity without overreaching. Use beat rate for a measured envelope repetition, sideband spacing for a spectrum around an identified carrier, and normal-mode split for a pair with measured mode shapes. Use transfer time for the interval between specified energy-envelope landmarks. Those labels distinguish the similar algebraic forms of independent tones, modulation, and coupling by their physical mechanism.

A controlled intervention pairs a model prediction with a measurement. For beats, adjust one source frequency. For AM, adjust the modulator. For coupled oscillators, change link stiffness or detuning. Predict the line movement, sideband movement, split movement, or transfer-time change before taking the next record. Agreement after a controlled change outweighs a visually persuasive envelope in one unperturbed trace.

Conditions behind the formulas.

The beat identity requires linear superposition over the amplitudes in use. A nonlinear oscillator can generate harmonics and combination frequencies even when only one source is present. A nonlinear coupling spring can make normal-mode frequencies depend on amplitude. Check this by repeating the same frequency and transfer measurement at a lower release amplitude. Stable line centers and a stable split support the linear approximation across that interval.

The three-line amplitude-modulation spectrum assumes a sinusoidal control waveform and a linear multiplier. A real source can add carrier phase noise, DC offset, harmonic content, clipping, and frequency-dependent output response. Treat the three-line result as a testable model: measure the residual spectral lines and compare their size with the sideband uncertainty before calling the signal single-tone AM.

The two-mode coupled model assumes two dominant degrees of freedom. It remains valid when weak loss lowers amplitudes slowly, provided the normal frequencies and relative phase stay well defined over the analysis gate. Strong loss, repeated forcing, nearby apparatus modes, and time-varying stiffness require an expanded model. A clean normal-mode split, complementary transfer envelope, and agreement between static and dynamic coupling estimates define the validity range of the two-oscillator approximation.

Retain both the raw data and the intermediate products used to reach a result. Those products include the selected time gate, envelope trace, fitted peak times, complex spectrum, frequency-fit residuals, channel-polarity check, and calibration record. They expose whether a quoted split came from a stable pair of modes or from a convenient peak selection. They also permit a later comparison against a refined model after source state or apparatus alignment has changed.