Mathematics 11–12 · Years 11–12

A 440 Hz tone on a phone oscilloscope: amplitude, period and transformations of sin x

Further graph transformations and modelling: Transformations of trigonometric functions (Mathematics Advanced, Year 12); Trigonometry and measure of angles: Radians (Mathematics Advanced, Year 11)

Practical, model not builtLow risk

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The idea

A pure tone is a sine wave whose period is the reciprocal of its frequency, and changing volume, pitch or start time performs the dilations and translations the syllabus describes.

What you need

Three phones: two running the phyphox Tone Generator, one set to 440 Hz (the note A4) and one to 880 Hz (the generator plays a single tone at a fixed amplitude, so loudness is set with the phone's volume buttons), the third running phyphox Audio Scope and Audio Spectrum
A ruler for reading the scope trace, a spreadsheet

How to do it

Play 440 Hz and view the trace on the scope; count the cycles across a 10 ms span and compute the period and frequency.
Lower the phone's volume and describe what changes on the trace; then play the 880 Hz phone instead and describe what changes.
Write y = A sin(2 pi f t) for each setting and sketch the graphs on the same axes, naming each change as a vertical or horizontal dilation.
Use the spectrum view to confirm the peak frequency for each tone.
Play the 440 Hz and 880 Hz phones together and sketch the sum of the two sine functions.

What you should see

At 440 Hz the period is 2.273 ms, so 4.4 cycles fit in 10 ms; at 880 Hz the period is 1.136 ms and 8.8 cycles fit. Lowering the volume shrinks the amplitude without changing the period (a vertical dilation); doubling the frequency halves the period without changing the amplitude (a horizontal dilation by factor 1/2). In air at 20 degrees Celsius (343 m/s) the 440 Hz wavelength is 0.780 m. The learner knows it worked when the period counted from the trace agrees with 1/f within one scope division and the spectrum peak sits at the set frequency.

What changes

What you change: frequency and volume settings
What you measure: period and amplitude of the trace
What you keep the same: phone separation
background noise
scope time span

Common misconceptions

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

A louder sound has a shorter period; volume changes the amplitude only.
The 2 pi in sin(2 pi f t) stretches the graph upward; it converts cycles to radians and sets the period.

Safety card

Low riskLearners carry it out

Hazards

loud tones close to the ears

Controls

keep the volume moderate and the phones on the bench

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 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
Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Trigonometry and measure of angles; 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-11-04
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities