Add two sine waves in the sandbox and read off what their sum does — and predict, before the readouts confirm it, the beat frequency of two close tones, the carrier they ride, whether two equal tones reinforce or cancel, and how long one beat lasts.
Superposition is the whole game: two sines added sample by sample make one waveform. When their frequencies f₁ and f₂ are close, the sum swells and fades at the difference frequency — a beat far slower than either tone — while riding a carrier at their average. Two tones of the same frequency instead interfere: in step (φ = 0) they add to 2A; in opposition (φ = π) they cancel to 0. Reading these numbers off a live sum — instead of taking them on faith — is how beats, carriers and interference stop being words and become something you can compute.
Open the sandbox. It opens on two tones near each other (8 Hz and 9 Hz, equal amplitude, same phase), so a Reset gets you close. Each step names the one knob to move; leave everything else alone. Frequencies here are low so the beat and the carrier are both visible — the arithmetic is exact at any value.
Component 1 -> Frequency 8 Hz, Amplitude 1.0, Phase 0 (leave on)
Component 2 -> Frequency + Phase set per step Active components -> 2
Watch the Beat frequency, Carrier f̄, Combined peak and Beat period readout cards — every number you predict is printed there.
Leave Component 1 at 8 Hz and set Component 2 to 12 Hz, both at amplitude 1.0. The sum swells and fades at the difference of the two frequencies. Predict the beat frequency in hertz, then read the Beat frequency card.
The card reads 4 Hz — the difference |f₁ − f₂| = 12 − 8. The beat is far slower than either tone, and the closer the two frequencies, the slower it gets.
Keep the same two tones, 8 Hz and 12 Hz. Underneath the slow beat, the sum oscillates at the carrier — the average of the two frequencies. Predict the carrier in hertz, then read the Carrier f̄ card.
The card reads 10 Hz — the average (f₁ + f₂)/2 = (8 + 12)/2. The fast wiggle you see inside each beat is this carrier; the slow swelling is the beat from Step 1.
Set BOTH components to the same frequency, 8 Hz, both at amplitude 1.0, and Component 2's Phase at φ = 0. Now they are the same wave laid on itself. Predict the combined peak amplitude, then read the Combined peak card.
The card reads 2 — the two in-phase tones add to 2A. Now drag Component 2's phase to φ = π: the peak collapses to 0, total cancellation. Same amplitudes, opposite phase, nothing left.
Go back to the two tones of Step 1, 8 Hz and 12 Hz. The beat repeats once per cycle of the difference frequency, so its period is one over the beat frequency. Predict the beat period in seconds, then read the Beat period card.
The card reads 0.25 s — 1 / 4 Hz. A 4 Hz beat swells and fades four times a second, so one beat takes a quarter of a second — no matter what the carrier is doing underneath.
Everything above is waiting in the sandbox. Bring two frequencies together and watch the beat slow to a crawl, add a third component and watch the sum grow structure, and sweep Component 2's phase to see two equal tones swing between full reinforcement and total cancellation — all added sample by sample, never drawn.
- Read a 4 Hz beat from 8 Hz and 12 Hz — the difference of the two frequencies
- Found the carrier at 10 Hz, the average of the two tones
- Saw two equal tones add to 2A in phase and cancel to 0 at φ = π
- Measured one beat at 0.25 s, the inverse of the 4 Hz beat frequency
- Every value you predicted is the demo's own exact arithmetic, not a picture