SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish beat recurrence from carrier oscillation

Two nearby frequencies produce a changing relative phase, even when both sources start in phase. The supplied signals have frequencies 200 Hz and 200+Δ Hz with peak amplitudes 1 Pa and b Pa. Their relative phase advances by 2πΔt, so the slowly varying resultant amplitude is √[1+b²+2b cos(2πΔt)]. This envelope is not the rapid pressure oscillation itself.

Learning outcome: Compare Δ=0,2,4 Hz and amplitude ratios .5 and 1. Count envelope maxima over one second before consulting the recurrence ledger. Distinguish the zero-frequency-difference case from a nonzero difference with incomplete cancellation.

Starting recap

Separate local oscillation from propagation, amplitude from its square, and a calibration fit from a fresh prediction. Use declared phase, medium frame and boundary conditions.

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Try

For 200 and 204 Hz at equal amplitudes, maxima recur every .25 s. The envelope ranges from 0 to 2 Pa, while the carrier oscillations remain near 202 Hz. If the second amplitude is .5 Pa, maxima still recur every .25 s but minima are .5 Pa and maxima 1.5 Pa.

Open the experiment

Predict

At 200 and 204 Hz with amplitudes 1 and .5 Pa, which statement is correct?

Your prediction and reason

Experiment

Compare Δ=0,2,4 Hz and amplitude ratios .5 and 1. Count envelope maxima over one second before consulting the recurrence ledger. Distinguish the zero-frequency-difference case from a nonzero difference with incomplete cancellation.

Distinguish beat recurrence from the carrier

Worked teaching model · synthetic values · Silent signals of 200 Hz and 200+Delta Hz, first amplitude 1 Pa, common initial phase. Envelope formula is sqrt(1+b²+2b cos(2pi Delta t)); its square gives normalized slow mean-square pressure. No audible playback, hearing test or safe-exposure claim.

Frequency difference: 4 Hz Second-to-first pressure amplitude: 1 ratio

Reset keeps your prediction, captures and practice history.

Slow pressure-amplitude envelope, not the carrier oscillation

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: positive envelope (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Slow pressure-amplitude envelope, not the carrier oscillation

positive envelope

time (s)	pressure envelope (Pa)
0	2
0.0125	1.97538
0.025	1.90211
0.0375	1.78201
0.05	1.61803
0.0625	1.41421
0.075	1.17557
0.0875	0.907981
0.1	0.618034
0.1125	0.312869
0.125	0
0.1375	0.312869
0.15	0.618034
0.1625	0.907981
0.175	1.17557
0.1875	1.41421
0.2	1.61803
0.2125	1.78201
0.225	1.90211
0.2375	1.97538
0.25	2
0.2625	1.97538
0.275	1.90211
0.2875	1.78201
0.3	1.61803
0.3125	1.41421
0.325	1.17557
0.3375	0.907981
0.35	0.618034
0.3625	0.312869
0.375	0
0.3875	0.312869
0.4	0.618034
0.4125	0.907981
0.425	1.17557
0.4375	1.41421
0.45	1.61803
0.4625	1.78201
0.475	1.90211
0.4875	1.97538
0.5	2
0.5125	1.97538
0.525	1.90211
0.5375	1.78201
0.55	1.61803
0.5625	1.41421
0.575	1.17557
0.5875	0.907981
0.6	0.618034
0.6125	0.312869
0.625	0
0.6375	0.312869
0.65	0.618034
0.6625	0.907981
0.675	1.17557
0.6875	1.41421
0.7	1.61803
0.7125	1.78201
0.725	1.90211
0.7375	1.97538
0.75	2
0.7625	1.97538
0.775	1.90211
0.7875	1.78201
0.8	1.61803
0.8125	1.41421
0.825	1.17557
0.8375	0.907981
0.85	0.618034
0.8625	0.312869
0.875	0
0.8875	0.312869
0.9	0.618034
0.9125	0.907981
0.925	1.17557
0.9375	1.41421
0.95	1.61803
0.9625	1.78201
0.975	1.90211
0.9875	1.97538
1	2

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
First carrier frequency	200	Hz
Second carrier frequency	204	Hz
Maximum envelope amplitude	2	Pa
Minimum envelope amplitude	0	Pa
Beat frequency	4	Hz
Envelope maximum recurrence time	0.25	s
Beating present	1	flag

Maxima recur at the frequency difference; unequal amplitudes leave nonzero minima.

Superposition of 200 Hz and 200+Delta Hz yields a slowly varying resultant amplitude. The envelope magnitude and mean-square maxima repeat every 1/Delta, even though a signed cosine factor in an equal-amplitude identity changes sign every half envelope-factor cycle. At Delta=0 the common phase is fixed and the amplitude is 1+b at all times. No frequency is inferred from a single amplitude sample, and no sound is emitted.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Maxima recur at the frequency difference; unequal amplitudes leave nonzero minima.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the observable

Build the bounded model

For nonzero Δ, envelope maxima recur every 1/Δ seconds and the beat frequency is Δ. The maximum amplitude is 1+b and the minimum is |1−b|. Unequal amplitudes therefore produce beats without complete silence in the ideal pressure signal. In the equal-amplitude trigonometric identity, a signed cosine factor changes sign; taking its magnitude is essential when counting successive amplitude maxima, or a factor-of-two error results.

Interpret the evidence

At Δ=0, relative phase never changes. With the stipulated common initial phase the amplitude stays exactly 1+b; there is no beat recurrence time to display. A single amplitude measurement does not identify either carrier frequency, and observing a frequency difference alone does not say which source is higher without another reference. The graph is a silent synthetic record and does not perform a hearing test or suggest exposing a listener to any sound level.

Worked original case

Predict and investigate

Changed-case practice

Try the changed case before feedback

For frequencies 200 and 205 Hz, find the time between successive envelope maxima in seconds.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original prose and cases checked 30 October 2026 against the primary source. These are silent, synthetic wave models: no sound playback, exposure prescription, hearing test or physical apparatus is requested. A correct public number is not independent mastery; retain first explanations and a changed-case review in the course evidence workspace.

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Which fresh observation separates a failed physical model from a fitted calibration parameter?

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Previous: Add pressure phase before squaring Next: Keep source and observer velocities in the medium frame

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