Mathematics 11–12 · Year 12

Two measured tones added as phasors: the Argand plane on a phone oscilloscope

Introduction to complex numbers: Geometric representation of complex numbers; Powers and roots of complex numbers, including complex numbers as vectors (Mathematics Extension 2, Year 12)

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

Two tones of the same frequency add like two complex numbers: the modulus of the sum is the amplitude a microphone reads and its argument is the phase of the combined wave, so head-to-tail addition on the Argand plane predicts a measurement.

Safety card

Low riskAn adult supervises

Hazards

sustained tones close to the ears
trailing speaker and power leads across a walkway
speakers toppling from a bench edge

Controls

keep the level at conversational loudness, take readings in short bursts and stand back from the speakers between readings
tape the leads to the floor and keep them out of walkways
set the speakers back from the bench edge on stable stands

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; the hearing exposure and the mains leads belong in the school's risk assessment before the lesson.

What you need

A laptop running Audacity with the project sample rate set to 44,100 Hz
A 3.5 mm stereo splitter lead and two identical small powered speakers on stable stands
A phone running the phyphox Audio Scope and a second phone running phyphox Audio Amplitude
A tape measure and masking tape for marking positions on the floor
A third phone for the phyphox Speed of sound measurement, and a thermometer reading the room temperature
A spreadsheet or GeoGebra for the Argand plots

How to do it

Measure the speed of sound in the room with the phyphox Speed of sound experiment and two phones a measured distance apart, and record the room temperature.
In Audacity at 44,100 Hz generate a 441 Hz sine tone of amplitude 0.5 lasting 5 s, so that one cycle is exactly 100 samples, and duplicate the track.
Set the selection format to samples and insert silence at the start of the second track: 25 samples for a quarter cycle, 50 samples for a half cycle, and 0 samples for the in-step case. Pan one track hard left and the other hard right, then export each case as a stereo file.
Repeat the generation at 420 Hz, where one cycle is 105 samples, and insert 35 samples for a one-third cycle shift.
Repeat at 490 Hz, where one cycle is 90 samples, with the second track at amplitude 0.3 and a 15 sample shift for a one-sixth cycle.
Stand the two speakers side by side 0.20 m apart facing the Audio Scope phone 1.00 m away on the centre line. Play each exported file in turn and record the peak amplitude of the trace, and the level from Audio Amplitude, for each case and for each channel alone.
For each case write the two arriving waves as complex numbers in modulus-argument form, add them, and compare the modulus of the sum and its principal argument with the measured amplitude ratio and the measured shift of the trace. Convert each amplitude ratio to decibels with 20 log10.
Plot each pair on the Argand plane as vectors added head to tail, with the resultant drawn, and mark the triangle inequality on the diagram.
Return to the in-step file, separate the speakers to 1.00 m, and walk the Audio Amplitude phone along a line 2.00 m in front of them, recording the level every 0.10 m out to 1.50 m from the centre line.
Compute the path difference at each position from the measured geometry, convert it to a phase angle, and compare the predicted modulus with the measured level. Locate the first minimum and compare it with the far-field estimate.

What you should see

At 441 Hz the period is 2.268 ms, and with a measured speed of sound near 343 m/s the wavelength is 0.778 m. Two equal tones in step give a modulus of 2.000, an amplitude twice one channel alone and a level 6.02 dB higher; a quarter-cycle shift gives 1.414 and 3.01 dB with a principal argument of 45 degrees; a one-third cycle shift gives a modulus of exactly 1.000, so two speakers sound no louder than one; a half-cycle shift gives 0, and the measured level falls well below that of either channel alone rather than to silence, because the two speakers are not identical and the room reflects. With amplitudes 1.00 and 0.60 at a one-sixth cycle shift the sum is 1.30 + 0.520 i, of modulus exactly 1.400 and principal argument 21.79 degrees, which is 2.92 dB above the stronger channel alone. With the speakers 1.00 m apart and the line 2.00 m in front, the exact geometry puts the first minimum 0.866 m from the centre line, where the far-field estimate lambda D / (2 d) gives 0.778 m, 10 per cent smaller. The learner knows it worked when the measured amplitude ratios match the four computed moduli within 10 per cent, when the one-third cycle case shows no rise in level as the second speaker is switched in, and when the measured minimum sits nearer the exact path-difference position than the far-field estimate.

What changes

What you change: phase difference set by the number of samples inserted, and the position along the measuring line
What you measure: amplitude of the combined trace and the level in decibels
What you keep the same: frequency within each case
the same two speakers at the same volume setting
microphone distance and orientation
room temperature and the same speed of sound
background noise checked before each reading

Common misconceptions

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

Two sounds together are always louder; equal tones half a cycle apart cancel on the centre line.
Two equal amplitudes always give twice the amplitude; the resultant is 2 cos(phi / 2) times one amplitude, so at a one-third cycle shift two speakers sound like one.
The modulus of a sum is the sum of the moduli; the triangle inequality allows that only when the two arguments agree.
The number i cannot describe a measurement because it is not real; here the real and imaginary parts carry the in-step and quarter-cycle parts of one measured wave, and the modulus is the amplitude the microphone reads.
Cancellation destroys the sound energy; the energy turns up at the maxima elsewhere along the line.
Multiplying by i makes a number larger; it turns the phasor through a quarter turn without changing the modulus.
A cancellation on the centre line means the speakers are faulty; it is the phase difference written into the file.

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 2 11–12 Syllabus (2024), Year 12 focus area Introduction to complex numbers; Extension 2 is a Year 12 course taught from Term 4 2026 with the first HSC examination in 2027, and it replaces the Mathematics Extension 2 Stage 6 Syllabus (2017) taught until then; page read 2026-09-23ME2-12-03
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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