Mathematics 11–12 · Years 11–12
Sound level against distance: the inverse-square law on a decibel scale
Further graph transformations and modelling: Modelling with functions (Mathematics Advanced, Year 12); Exponential and logarithmic functions: Logarithmic functions (Mathematics Advanced, Year 11)
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
Sound intensity from a small source falls with the square of distance, and on the logarithmic decibel scale that appears as a fixed drop of about 6 dB for every doubling.
What you need
- A phone playing a steady 440 Hz tone at fixed volume on a stand
- A second phone running phyphox Audio Amplitude (relative readings of all the sound the microphone picks up, not only the 440 Hz tone; the app can be calibrated against a reference)
- A 10 m tape, a quiet outdoor space and a hall for comparison
How to do it
- With the source silent, read the background level at 1, 2, 4 and 8 m; keep only the tone readings that are at least 10 dB above the background at the same spot, so the background adds less than 0.5 dB.
- Measure the amplitude at 1, 2, 4 and 8 m from the source along a straight line outdoors, with the source and the phone at the same height; if one reading sits well off the trend, move 0.5 m and repeat, because sound reflected from the ground can cancel part of the direct sound at particular distances (the 440 Hz wavelength is 0.78 m).
- Convert each reading to a level relative to 1 m with L = 20 log10(amplitude / amplitude at 1 m).
- Plot the level against distance and against log10(distance) and find the gradient of the second plot.
- Compare with the inverse-square prediction of -20 dB per tenfold distance (-6.02 dB per doubling).
- Repeat in the hall and explain why the level falls less at the far points.
What you should see
An inverse-square source loses 6.02 dB per doubling of distance and 12.04 dB from 1 m to 4 m, so the level against log10(distance) is a line of gradient -20 dB per decade. In a hall reflected sound adds to the direct sound, so the far readings sit above the line. Outdoors the far readings also flatten when the tone comes close to the background level, because the app measures all sound, which is why only readings at least 10 dB above the background are kept. The learner knows it worked when the outdoor log plot, over the distances that stay at least 10 dB above the background, is straight with a gradient close to -20 dB per decade.
What changes
- What you change
- distance from the source
- What you measure
- measured amplitude and level in dB
- What you keep the same
- source volume
- microphone facing the source
- no obstacles on the line
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Twice as far is half as loud in decibels; it is 6 dB less, and half the intensity is 3 dB less.
- 0 dB means silence; it is the reference level, and negative decibel values are valid.
Safety card
Hazards
- a loud source at close range
Controls
- moderate volume with the source at bench height
Note
No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's grounds or excursion risk assessment (RiskAssess where the school uses it).
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-12-02
- Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Exponential and logarithmic functions; 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-11-07
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.