Mathematics 11–12 · Years 11–12

Beats from two close tones: a sum of sines rewritten as a product

Further trigonometry: Further trigonometric identities (Mathematics Extension 1, Year 11); Further graph transformations and modelling: Transformations of trigonometric functions (Mathematics Advanced, Year 12)

Practical, model not builtLow risk

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The idea

Two tones a few hertz apart add to a tone whose loudness swells and fades, and the sum and difference expansions rewrite the sum as a product that predicts the beat rate exactly.

What you need

Two phones running the phyphox Tone Generator, set to 440 Hz and 444 Hz
A laptop running Audacity (free), recording through its microphone, to show the waveform over several seconds (phyphox Audio Scope shows only 10 ms segments and phyphox Audio Amplitude gives one reading about every 0.1 s, both too coarse for a 0.25 s beat)
A stopwatch

How to do it

Play 440 Hz and 444 Hz together from phones side by side and count the loudness swells in 10 s.
Record the sound in Audacity for about 5 s, zoom to about 1 s of the waveform and measure the time between quiet points.
Starting from sin(A + B) and sin(A - B), show that sin p + sin q = 2 sin((p + q)/2) cos((p - q)/2).
Apply the identity with p = 2 pi (444) t and q = 2 pi (440) t and identify the fast carrier and the slow envelope.
Change the second tone to 442 Hz and to 446 Hz and predict, then count, the new beat rate.

What you should see

sin(2 pi 444 t) + sin(2 pi 440 t) = 2 sin(2 pi 442 t) cos(2 pi 2 t): a 442 Hz tone inside an envelope whose loudness peaks twice per cycle of cos(2 pi 2 t), so the beat rate is 444 - 440 = 4 per second and the quiet points are 0.25 s apart. At 442 Hz the rate falls to 2 per second and at 446 Hz rises to 6. The learner knows it worked when 10 s gives about 40 swells and the recorded waveform shows quiet points 0.25 s apart.

What changes

What you change: difference between the two frequencies
What you measure: beat rate
What you keep the same: equal volumes
phones side by side
fixed listening position

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

The beat is a third tone at 4 Hz; it is a change in loudness of a 442 Hz tone.
Two tones always add to a louder steady tone; tones that differ slightly drift in and out of step.

Safety card

Low riskLearners carry it out

Hazards

loud tones close to the ears

Controls

keep the volume moderate

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Extension 1 11–12 Syllabus (2024), Year 11 focus area Further trigonometry; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22ME1-11-03
Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Further graph transformations and modelling; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-01
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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