---
title: Beats and Coupling
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 11
order: 711
summary: >
  Add two tones a few hertz apart and the sum swells and fades at their difference
  frequency — a beat — though neither source is changing. We work out that envelope,
  then ask the mechanical version of the same question: join two oscillators and a
  single resonance splits into normal modes, with energy sloshing between the
  coordinates at their frequency difference. The lesson identifies when a slow
  amplitude envelope signals genuine coupling rather than two independent sources,
  drift, or deliberate modulation, reading it from envelope timing, spectral
  sidebands, and the mode shapes.
topics: [Oscillations]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 14 — Oscillations; Ch. 16, §16-1 — Beats"
---

## 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:

$$
x(t)=A\cos(\omega_1t)+A\cos(\omega_2t).
$$

Define the average and separation

$$
\bar{\omega}=\frac{\omega_1+\omega_2}{2},
\qquad
\Delta\omega=\omega_2-\omega_1.
$$

The product-to-sum identity gives

$$
x(t)=2A\cos\!\left(\frac{\Delta\omega t}{2}\right)
\cos(\bar{\omega}t).
$$

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.

$$
% caption: Two equal-amplitude tones close in frequency add to a rapid carrier whose amplitude swells and fades. The outer curves are the slow envelope; it crosses zero at each beat node, where the tones momentarily cancel.
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$$

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

$$
f_{\mathrm{beat}}=|f_2-f_1|
=\frac{|\Delta\omega|}{2\pi}.
$$

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

$$
\delta(t)=(\omega_2-\omega_1)t+\delta_0.
$$

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

$$
R(t)^2=A_1^2+A_2^2+2A_1A_2\cos(\Delta\omega t+\delta_0).
$$

**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

$$
\begin{aligned}
x(t)&=\Re\!\left\{\mathcal A(t)e^{i\bar\omega t}\right\},\\
\mathcal A(t)&=A_1e^{-i\Delta\omega t/2}
+A_2e^{i(\Delta\omega t/2+\delta_0)}
=R(t)e^{i\psi(t)}.
\end{aligned}
$$

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:

$$
x(t)=R(t)\cos\!\left[\bar\omega t+\psi(t)\right],
\qquad
\psi(t)=\arg\mathcal A(t).
$$

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.

$$
% caption: At a chosen instant the complex envelope is the vector sum of the two slowly rotating component phasors. Its length is the envelope amplitude; its direction is the carrier-phase offset at that instant.
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$$

The envelope extrema are

$$
R_{\max}=A_1+A_2,
\qquad
R_{\min}=|A_1-A_2|.
$$

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

$$
C=\frac{R_{\max}-R_{\min}}{R_{\max}+R_{\min}}
=\frac{\min(A_1,A_2)}{\max(A_1,A_2)}.
$$

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.

$$
% caption: Unequal component amplitudes leave a nonzero envelope minimum. The dashed level marks the residual amplitude at each beat node; its depth measures the amplitude balance of the two tones.
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$$

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

$$
s(t)=A_{\mathrm c}\left[1+m\cos(\omega_{\mathrm m}t+\phi_{\mathrm m})\right]
\cos(\omega_{\mathrm c}t+\phi_{\mathrm c}),
$$

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

$$
\begin{aligned}
s(t)={}&A_{\mathrm c}\cos(\omega_{\mathrm c}t+\phi_{\mathrm c})\\
&+\frac{mA_{\mathrm c}}{2}
\cos[(\omega_{\mathrm c}+\omega_{\mathrm m})t+\phi_{\mathrm c}+\phi_{\mathrm m}]\\
&+\frac{mA_{\mathrm c}}{2}
\cos[(\omega_{\mathrm c}-\omega_{\mathrm m})t+\phi_{\mathrm c}-\phi_{\mathrm m}].
\end{aligned}
$$

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

$$
m=\frac{2A_{\mathrm{sb}}}{A_{\mathrm c}}.
$$

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
$m^2P_{\mathrm c}/4$, and total power is

$$
P_{\mathrm{total}}=P_{\mathrm c}\left(1+\frac{m^2}{2}\right).
$$

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

$$
% caption: A sinusoidal amplitude-modulated signal shows one carrier line and two equal first sidebands. Their equal spacing from the carrier is the modulator frequency; their height sets the modulation index.
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$$

For $0\leq m\leq1$, 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

$$
m=\frac{E_{\max}-E_{\min}}{E_{\max}+E_{\min}},
$$

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

$$
\begin{aligned}
m\ddot{x}_1&=-kx_1-k_{\mathrm c}(x_1-x_2),\\
m\ddot{x}_2&=-kx_2-k_{\mathrm c}(x_2-x_1).
\end{aligned}
$$

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.

$$
% caption: Two equal mass-and-spring oscillators joined by a coupling spring. Equal displacements leave the link relaxed; relative displacement stretches it and transfers force between the masses.
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$$

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

$$
q_+=\frac{x_1+x_2}{\sqrt2},
\qquad
q_-=\frac{x_1-x_2}{\sqrt2}.
$$

Adding and subtracting the equations of motion gives

$$
\ddot q_++\omega_+^2q_+=0,
\qquad
\ddot q_-+\omega_-^2q_-=0,
$$

with

$$
\omega_+=\sqrt{\frac{k}{m}},
\qquad
\omega_-=\sqrt{\frac{k+2k_{\mathrm c}}{m}}.
$$

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.

$$
% caption: The two normal modes of the equal coupled pair. In the low mode both masses move the same way and the link stays relaxed; in the high mode they move oppositely, stretching the link and raising the natural frequency.
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$$

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

$$
\begin{bmatrix}x_1\\x_2\end{bmatrix}
=\frac{q_+}{\sqrt2}\begin{bmatrix}1\\1\end{bmatrix}
+\frac{q_-}{\sqrt2}\begin{bmatrix}1\\-1\end{bmatrix}.
$$

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

$$
E=\frac12m\dot x_1^2+\frac12m\dot x_2^2
+\frac12k(x_1^2+x_2^2)+\frac12k_{\mathrm c}(x_1-x_2)^2.
$$

In normal coordinates it separates:

$$
E=\left(\frac12\dot q_+^2+\frac12\omega_+^2q_+^2\right)m
+\left(\frac12\dot q_-^2+\frac12\omega_-^2q_-^2\right)m.
$$

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.

$$
% caption: In normal coordinates the total energy splits into two conserved modal parts, while the energy carried by each physical mass oscillates in antiphase and sums to a constant.
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$$

### Frequency splitting at weak coupling

The uncoupled natural frequency is $\omega_0=\sqrt{k/m}$. When
$k_{\mathrm c}\ll k$,

$$
\omega_+\approx\omega_0,
\qquad
\omega_-\approx\omega_0\left(1+\frac{k_{\mathrm c}}{k}\right),
\qquad
\Delta\omega\approx\omega_0\frac{k_{\mathrm c}}{k}.
$$

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

$$
\Delta f\approx f_0\frac{k_{\mathrm c}}{k}.
$$

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

$$
k_{\mathrm c}=\frac{m}{2}\left(\omega_-^2-\omega_+^2\right).
$$

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

$$
\begin{aligned}
x_1(t)&=\frac{A}{2}\left[\cos(\omega_+t)+\cos(\omega_-t)\right]\\
&=A\cos(\bar{\omega}t)\cos\!\left(\frac{\Delta\omega t}{2}\right),\\
x_2(t)&=\frac{A}{2}\left[\cos(\omega_+t)-\cos(\omega_-t)\right]\\
&=A\sin(\bar{\omega}t)\sin\!\left(\frac{\Delta\omega t}{2}\right),
\end{aligned}
$$

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.

$$
% caption: A one-mass release excites both normal modes. The first mass starts at full amplitude and its envelope decays; the second starts at rest and its envelope grows, exchanging the motion between them.
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$$

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

$$
t_{\mathrm{transfer}}=\frac{\pi}{\Delta\omega}
=\frac{1}{2\Delta f},
$$

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

$$
t_{\mathrm{return}}=\frac{2\pi}{\Delta\omega}
=\frac{1}{\Delta f},
$$

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.

$$
% caption: Independent tones and a coupled pair can show the same two-line spectrum and envelope rate. Only the coupled system adds a complementary second-coordinate motion and an energy-transfer reading.
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\node[above] at (1.45,2.20) {tone 1};
\node[above] at (2.45,1.90) {tone 2};
\node[black,below] at (1.88,-.22) {two sources};
\draw[->,black] (4.05,.45)--(6.75,.45);
\draw[->,black] (4.05,.45)--(4.05,3.00);
\node[below left] at (6.55,.45) {frequency};
\draw[acc,very thick] (4.95,.45)--(4.95,2.20);
\draw[acc,very thick] (5.95,.45)--(5.95,1.90);
\node[above] at (4.95,2.20) {low mode};
\node[above] at (5.95,1.90) {high mode};
\node[black,below] at (5.45,-.22) {one system};
\end{tikzpicture}
$$

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

$$
\omega_1^2=\frac{k_1}{m},
\qquad
\omega_2^2=\frac{k_2}{m},
\qquad
\kappa=\frac{k_{\mathrm c}}{m}.
$$

The normal-mode condition becomes

$$
\left(\omega_1^2+\kappa-\omega^2\right)
\left(\omega_2^2+\kappa-\omega^2\right)-\kappa^2=0.
$$

The two squared normal frequencies are

$$
\omega_{\pm}^2=
\frac{\omega_1^2+\omega_2^2+2\kappa}{2}
\pm\sqrt{
\left(\frac{\omega_1^2-\omega_2^2}{2}\right)^2+\kappa^2}.
$$

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.

$$
% caption: As one support stiffness is tuned through resonance the uncoupled frequencies (dashed) would cross, but coupling repels the true normal modes (solid) into an avoided crossing whose gap is set by the link stiffness.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-.95,-.60) rectangle (7.05,3.45);
\draw[->,black] (.60,.45)--(6.75,.45);
\draw[->,black] (.60,.45)--(.60,3.10);
\node[below left] at (6.62,.45) {detuning};
\node[left] at (.60,3.02) {frequency};
\draw[black,dashed] (.90,.951)--(5.60,2.549);
\draw[black,dashed] (.90,2.549)--(5.60,.951);
\draw[acc,very thick] plot[domain=.90:5.60,samples=160]
  (\x,{1.75+sqrt(.10+.1156*(\x-3.25)*(\x-3.25))});
\draw[acc,very thick] plot[domain=.90:5.60,samples=160]
  (\x,{1.75-sqrt(.10+.1156*(\x-3.25)*(\x-3.25))});
\draw[<->,black] (3.25,1.434)--(3.25,2.066);
\node[black,right] at (3.34,1.75) {gap};
\draw[black,dashed] (3.25,.45)--(3.25,1.30);
\node[below] at (3.25,.45) {matching};
\node[acc,left] at (5.55,2.98) {high mode};
\node[acc,left] at (5.55,.62) {low mode};
\end{tikzpicture}
$$

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

$$
\eta_{\max}\approx
\frac{\kappa^2}{\kappa^2+
\left[(\omega_1^2-\omega_2^2)/2\right]^2}.
$$

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

$$
\widehat f_{\mathrm{beat}}=\frac{N_{\mathrm b}}{T_{\mathrm b}}.
$$

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

$$
z(t)=s(t)+i\mathcal H\{s(t)\},
\qquad
E(t)=|z(t)|.
$$

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

$$
\Delta f_{\mathrm{bin}}=\frac{1}{T}.
$$

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

$$
m_{\mathrm{time}}=\frac{E_{\max}-E_{\min}}{E_{\max}+E_{\min}}.
$$

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

$$
m_{\mathrm{spec}}=\frac{2A_{\mathrm{sb}}}{A_{\mathrm c}}.
$$

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

$$
G_{12}(f)=X_1(f)X_2(f)^\ast,
\qquad
\theta_{12}(f)=\arg G_{12}(f).
$$

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

$$
x_j(t)=B_{j+}\cos(\omega_+t+\phi_{j+})
+B_{j-}\cos(\omega_-t+\phi_{j-}),
\qquad j\in\{1,2\}.
$$

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

$$
u(f_{\mathrm{beat}})\approx
f_{\mathrm{beat}}\left[
\left(\frac{u(N_{\mathrm b})}{N_{\mathrm b}}\right)^2+
\left(\frac{u(T_{\mathrm b})}{T_{\mathrm b}}\right)^2
\right]^{1/2}.
$$

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,

$$
u(\Delta f)=\sqrt{u(f_-)^2+u(f_+)^2}.
$$

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

$$
k_{\mathrm c}=\frac{m}{2}\left(\omega_-^2-\omega_+^2\right).
$$

For independent standard uncertainties, first-order propagation gives

$$
\begin{aligned}
u(k_{\mathrm c})^2\approx{}&
\left[\frac{\omega_-^2-\omega_+^2}{2}u(m)\right]^2\\
&+\left[m\omega_-u(\omega_-)\right]^2
+\left[m\omega_+u(\omega_+)\right]^2.
\end{aligned}
$$

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

$$
m=\frac{E_{\max}-E_{\min}}{E_{\max}+E_{\min}},
$$

then independent envelope-level uncertainties give

$$
u(m)^2\approx
\left[\frac{2E_{\min}}{(E_{\max}+E_{\min})^2}u(E_{\max})\right]^2
+\left[\frac{2E_{\max}}{(E_{\max}+E_{\min})^2}u(E_{\min})\right]^2.
$$

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

$$
f_{\mathrm u}=440.0\ \mathrm{Hz}\pm3.0\ \mathrm{Hz},
$$

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.

$$
% caption: Tuning against a fixed reference gives a V-shaped beat-rate curve. The rate falls to zero at the match; which arm the data sit on tells whether the unknown began above or below the reference.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-.95,-.60) rectangle (7.05,3.15);
\draw[->,black] (.60,.45)--(6.75,.45);
\draw[->,black] (.60,.45)--(.60,3.00);
\node[below left] at (6.62,.45) {tuning setting};
\node[left] at (.60,2.92) {beat rate};
\draw[acc,thick] (.90,2.55)--(3.35,.58);
\draw[acc,thick] (3.35,.58)--(5.80,2.55);
\draw[fill=white,draw=black,line width=.4pt] (3.35,.58) circle (2.4pt);
\node[below] at (3.35,.45) {match};
\node[acc] at (2.05,1.15) {below};
\node[acc] at (4.65,1.15) {above};
\end{tikzpicture}
$$

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.

> **Worked example.** Take a carrier of peak amplitude $A_{\mathrm c}=2.00\ \mathrm V$ across a
> frequency-independent 50.0 ohm load. Suppose a calibrated spectrum gives each
> first sideband amplitude $A_{\mathrm{sb}}=0.300\ \mathrm V$. The modulation index
> is
>
> $$
> m=\frac{2(0.300)}{2.00}=0.300.
> $$
>
> The carrier power is
>
> $$
> P_{\mathrm c}=\frac{(2.00\ \mathrm V)^2}{2(50.0\ \mathrm\Omega)}
> =0.0400\ \mathrm W.
> $$
>
> Each sideband power is
>
> $$
> P_{\mathrm{sb}}=\frac{m^2}{4}P_{\mathrm c}
> =0.000900\ \mathrm W,
> $$
>
> and the predicted total is
>
> $$
> P_{\mathrm{total}}=0.0400\left(1+\frac{0.300^2}{2}\right)
> =0.0418\ \mathrm W.
> $$
>
> The waveform then has envelope ratio
> $E_{\min}/E_{\max}=(1-m)/(1+m)=0.538$. Checking that ratio can identify a baseline
> offset or compression in the detector chain. The calculation uses peak amplitudes;
> rms amplitudes require a consistent conversion before power is evaluated.

$$
% caption: The worked amplitude-modulated signal: a 2.00 V carrier with two equal 0.300 V first sidebands. Sideband height fixes the modulation index; the three lines together carry the total power.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\useasboundingbox (-.95,-.60) rectangle (7.05,3.15);
\draw[->,black] (.60,.45)--(6.80,.45);
\draw[->,black] (.60,.45)--(.60,3.00);
\node[below left] at (6.72,.45) {frequency};
\node[left] at (.60,2.92) {amplitude};
\draw[acc,very thick] (1.80,.45)--(1.80,1.18);
\draw[acc,very thick] (3.30,.45)--(3.30,2.55);
\draw[acc,very thick] (4.80,.45)--(4.80,1.18);
\node[above] at (1.80,1.18) {0.300 V};
\node[above] at (3.30,2.55) {2.00 V};
\node[above] at (4.80,1.18) {0.300 V};
\node[black,below] at (1.80,.42) {lower};
\node[black,below] at (3.30,.42) {carrier};
\node[black,below] at (4.80,.42) {upper};
\end{tikzpicture}
$$

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
$k=80.0\ \mathrm{N\,m^{-1}}$, and coupling stiffness
$k_{\mathrm c}=4.00\ \mathrm{N\,m^{-1}}$. Their normal frequencies are

$$
\begin{aligned}
f_+&=\frac{1}{2\pi}\sqrt{\frac{80.0}{0.500}}
=2.01\ \mathrm{Hz},\\
f_-&=\frac{1}{2\pi}\sqrt{\frac{80.0+2(4.00)}{0.500}}
=2.11\ \mathrm{Hz}.
\end{aligned}
$$

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

$$
t_{\mathrm{transfer}}=\frac{1}{2(0.099\ \mathrm{Hz})}
\approx5.1\ \mathrm{s}.
$$

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.

| candidate cause of the envelope | required spectral or coordinate evidence | controlled intervention | predicted change |
|---|---|---|---|
| two independent tones | Two resolved lines at $f_1$ and $f_2$ at the same sensor | Shift one source frequency while leaving the other source unchanged. | The beat rate follows $\lvert f_1-f_2\rvert$; only one spectral line moves. |
| deliberate amplitude modulation | Carrier plus symmetric sidebands at $f_c\pm f_m$ | Change the modulation frequency or index. | Both sidebands move together with $f_m$, or their heights change together with the modulation index. |
| two coupled normal modes | Two line frequencies together with in-phase and out-of-phase coordinate patterns | Alter link stiffness or detuning and record both coordinates. | The normal-mode split and transfer time change with the coupling; the mode-shape signs remain paired with the lines. |
| source or sensor artifact | A reference channel, background spectrum, and acquisition settings | Change sample rate, window, gain, or analogue bandwidth while preserving the mechanical state. | Alias lines, leakage features, or clipping signatures change with the acquisition chain rather than the apparatus. |

**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.