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OscillationsIntroStarter tracks

Concept module

Beats

Superpose two nearby sound frequencies, watch the fast carrier sit inside a slower envelope, and connect beat rate to the frequency difference on one compact bench.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 8.00 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s8.00 s

Beats

Two nearby source frequencies stay on one compact bench so the source traces, resultant envelope, and heard loud-soft cue come from the same superposition.

Live setup

Beat state

A: 0.12 m
f1: 1 Hz
f2: 1.12 Hz
delta f: 0.12 Hz
f_beat: 0.12 Hz
T_beat: 8.33 s
envelope: 0.24 m
cue: 1

Carrier frequency f_avg = 1.06 Hz.
The loudness pulse repeats every 8.33 s because the frequency split is 0.12 Hz.
This bounded bench stays in the nearby-frequency regime where one swelling and fading sound is a useful perception model.

Graphs

Switch graph views without breaking the live stage and time link.

Source and resultant motion

Tracks the two source traces and their live superposition on the same time axis so the faster carrier stays visible.

time (s): 0 to 8displacement (m): -0.5 to 0.5

Source ASource BResultant

Hover or scrub to link the graph back to the stage.time (s) / displacement (m)

Controls

Adjust the physical parameters and watch the motion respond.

Source amplitude

A

0.12 m

Scales the source motion and the maximum possible envelope without changing the beat frequency.

Source A frequency

f_{1}

1 Hz

Sets the first source timing so you can tune the carrier and the beat difference.

Source B frequency

f_{2}

1.12 Hz

Sets the second source timing. Matching it to Source A removes beats, while widening the split speeds the envelope.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 1

Notice that the fast wiggle keeps running even while the envelope swells and fades. Beats are the slow envelope wrapped around that faster carrier.

Try this

Start from Near unison and watch the displacement graph before you read the envelope graph.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

A

Source amplitude

0.12 m

Scales the source traces and the largest possible envelope, but it does not set the beat rate.

Graph: Source and resultant motionGraph: Envelope and loudness cueOverlay: Envelope guideOverlay: Listener cue

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use one cue at a time. The prompts keep the fast carrier, the slow envelope, and the listening idea tied to the same superposition state.

ObservationPrompt 1 of 1

Graph: Source and resultant motion

Notice that the fast wiggle keeps running even while the envelope swells and fades. Beats are the slow envelope wrapped around that faster carrier.

Try this

Start from Near unison and watch the displacement graph before you read the envelope graph.

Why it matters

It prevents the loud-soft pulse from being mistaken for a separate low-frequency source.

Control: Source A frequencyControl: Source B frequencyGraph: Source and resultant motionGraph: Envelope and loudness cueOverlay: Envelope guideEquation

f_{1}

Equation

f_{2}

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

2 visible

Overlay focus

Envelope guide

Shows the slow upper and lower bounds that wrap the faster resultant trace.

What to notice

The fast oscillation keeps running inside the slow envelope instead of turning into a separate third wave.

Why it matters

It keeps the carrier and the beat envelope visible as two roles inside one honest superposition.

Control: Source amplitudeControl: Source A frequencyControl: Source B frequencyGraph: Source and resultant motionGraph: Envelope and loudness cueEquation

A

Equation

f_{1}

Equation

f_{2}

Challenge mode

Use the same compact bench to separate beat rate from carrier frequency and from source amplitude.

0/2 solved

MatchCore

1 of 5 checks

Tune slow pulses

Starting near unison, tune the source pair until the envelope pulses at about

0.2 Hz

while the source amplitude stays near the baseline.

Graph-linkedGuided start2 hints

Suggested start

Use the envelope graph and the frequency-difference callout together while you tune the two source frequencies.

Pending

Keep the envelope graph open.

Source and resultant motion

Pending

Keep the frequency-difference callout on.

Off

Matched

Keep the source amplitude near the baseline.

0.12 m

Pending

Tune the beat frequency to about

0.2 Hz

0.12 Hz

Pending

That should give a beat period close to

5 s

8.33 s

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the two nearby frequencies average 1.06 Hz while their difference is 0.12 Hz, so the beat frequency is 0.12 Hz and the loudness pulse repeats every 8.33 s. The instantaneous envelope is 0.24 m and the bounded loudness cue is 1.

Equation detailsDeeper interpretation, notes, and worked variable context.

Two nearby source waves

Each source keeps its own steady sinusoidal motion with the same amplitude but a slightly different frequency.

y_{1} = A sin (2 π f_{1} t), y_{2} = A sin (2 π f_{2} t)

A

Source amplitude 0.12 m

f_{1}

Source A frequency 1 Hz

f_{2}

Source B frequency 1.12 Hz

Resultant with envelope and carrier

The slow cosine factor sets the envelope, while the faster sine factor keeps the average-frequency carrier visible inside it.

y = y_{1} + y_{2} = 2 A cos (π (f_{2} - f_{1}) t) sin (2 π \frac{f _{1} + f _{2}}{2} t)

A

Source amplitude 0.12 m

f_{1}

Source A frequency 1 Hz

f_{2}

Source B frequency 1.12 Hz

Beat frequency

The envelope rate comes from the difference between the source frequencies.

f_{b e a t} = ∣ f_{2} - f_{1} ∣

f_{1}

Source A frequency 1 Hz

f_{2}

Source B frequency 1.12 Hz

Bounded loudness cue

This intro bench uses a normalized amplitude-squared cue so the heard loud-soft pulsing stays tied to the envelope instead of becoming a full acoustics model.

I_{c u e} \propto (\frac{A _{e n v}}{2 A})^{2}

The cue is bounded and conceptual. It is not a full psychoacoustic loudness law.

A

Source amplitude 0.12 m

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 5 of 90 / 9 complete

Waves

Earlier steps still set up Beats.

1. Simple Harmonic Motion2. Wave Speed and Wavelength3. Sound Waves and Longitudinal Motion4. Pitch, Frequency, and Loudness / Intensity+5 more steps

Previous step: Pitch, Frequency, and Loudness / Intensity.

Start from track beginning

Starter track

Step 3 of 50 / 5 complete

Sound and Acoustics

Earlier steps still set up Beats.

1. Sound Waves and Longitudinal Motion2. Pitch, Frequency, and Loudness / Intensity3. Beats4. Doppler Effect+1 more steps

Previous step: Pitch, Frequency, and Loudness / Intensity.

Start from track beginning

Short explanation

What the system is doing

Beats appear when two nearby frequencies reach the same listener or probe and superpose. The fast oscillation does not disappear. Instead, it sits inside a slower amplitude envelope, so the combined motion swells and fades even though each source keeps oscillating steadily at its own frequency.

This bench stays bounded on purpose. It shows two equal-amplitude source traces, one live resultant, and one normalized loudness cue from the same superposition state. That keeps the physics honest: beat frequency comes from the frequency difference, while the faster carrier still follows the average source frequency.

Key ideas

01Beat frequency depends on the difference between the two source frequencies, not on their average amplitude or on the average frequency by itself.

02The loud-soft pulse is an envelope created by superposition, so neither source needs to get louder or quieter on its own.

03Keeping the same frequency difference keeps the same beat rate even if both source frequencies shift upward or downward together.

Live beat checks

Solve the exact state on screen.

These checks read the same source frequencies and live envelope state that the stage and graphs already show, so the math stays tied to one superposition bench.

Live valuesFollowing current parameters

For the current sources with $f_{1} = 1 Hz$ and $f_{2} = 1.12 Hz$ , what beat frequency and average carrier frequency follow from that pair?

f_{1}

Source A frequency

1 Hz

f_{2}

Source B frequency

1.12 Hz

f_{b e a t}

Beat frequency

0.12 Hz

f_{a v g}

Average carrier frequency

1.06 Hz

1. Start from the nearby-frequency relations

Use

f_{b e a t} = ∣ f_{2} - f_{1} ∣

for the envelope rate and

f_{a v g} = \frac{f _{1} + f _{2}}{2}

for the faster carrier underneath it.

2. Substitute the live source pair

f_{b e a t} = ∣1.12 - 1∣ = 0.12 Hz

and

f_{a v g} = \frac{1 + 1.12}{2} = 1.06 Hz

3. Interpret the pulse rate

So the envelope repeats at

0.12 Hz

and each loud-soft cycle takes

8.33 s

while the faster oscillation still centers on

1.06 Hz

Current beat pair

f_{b e a t} = 0.12 Hz, f_{a v g} = 1.06 Hz

The frequency difference is small, so the carrier oscillates many times before the loudness envelope completes one slow beat cycle.

Envelope checkpoint

You want the loud-soft pulsing to disappear without changing the source amplitudes. What should you do to the two source frequencies?

Prediction prompt

Decide whether you should move them farther apart, bring them together, or shift both upward together.

Check your reasoning

Bring the two source frequencies together until they match.

The beat envelope comes from the frequency difference. When

f_{1}

and

f_{2}

become equal,

Δ f

goes to zero and the separate beat cycle disappears.

Common misconception

If you hear beats, each source must be turning its own volume up and down.

Each source keeps a steady amplitude in this model. The pulsing comes from how the two waves add together.

The superposed resultant grows when the waves reinforce and shrinks when they nearly cancel, which is why the loudness cue pulses.

Quick test

Variable effect

Question 1 of 4

Use the live bench, not memory alone. These checks separate beat rate, carrier frequency, and source amplitude.

Which quantity sets the beat frequency in this page's model?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows two source drivers on the left, a shared time-trace region in the middle, and one listener cue on the right. The upper trace compares the two source motions, while the lower trace shows their combined resultant with an optional envelope guide around it.

Optional overlays label the envelope, the loudness cue, and the current frequency difference. In compare mode, the same compact bench appears in two rows so the learner can contrast beat rate and carrier frequency without leaving the shared layout.

Graph summary

The first graph plots Source A, Source B, and the resultant displacement against time so the faster carrier stays visible inside the superposition.

The second graph plots the normalized envelope ratio and a bounded loudness cue against time so the slow beat cycle can be read separately from the fast oscillation.

Carry beats into superposition and resonance

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Motion and pitchOscillations

Beats

See each variable before you move it.

Envelope guide

Tune slow pulses

Waves

Sound and Acoustics

What the system is doing

Solve the exact state on screen.

For the current sources with $f_{1} = 1 Hz$ and $f_{2} = 1.12 Hz$ , what beat frequency and average carrier frequency follow from that pair?

Which quantity sets the beat frequency in this page's model?

Keep moving through the concept map.

Doppler Effect

Wave Interference

Standing Waves

Beats

See each variable before you move it.

Envelope guide

Tune slow pulses

Waves

Sound and Acoustics

What the system is doing

Solve the exact state on screen.

For the current sources with f1​=1Hz and f2​=1.12Hz, what beat frequency and average carrier frequency follow from that pair?

Which quantity sets the beat frequency in this page's model?

Keep moving through the concept map.

Doppler Effect

Wave Interference

Standing Waves

For the current sources with $f_{1} = 1 Hz$ and $f_{2} = 1.12 Hz$ , what beat frequency and average carrier frequency follow from that pair?