How fast does a speaker get quieter? Decibels, logarithms and distance

AA SLAI SLAA HLAI HL Accessible Also in: Modelling, Kinematics

Research question

How does the sound level from a speaker (in decibels) fall with distance outdoors and indoors, does it follow the inverse-square law's prediction of about 6 dB per doubling, and why do rooms behave differently?

Adapt it: change the place, the data or the comparison until the question is yours.

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Why it makes a good exploration

Logarithms and an inverse-square model meet a measurement you can make with a phone. The indoor-outdoor difference gives something real to explain.

The mathematics you'll need

Logarithmic scales: decibels
Inverse-square model turned into a log model
Linear regression of dB against log distance
Comparing gradients with the predicted value
Residuals and measurement error

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Measure sound level with phyphox at measured distances from a speaker playing a steady tone, outdoors and in a room.

phyphox (RWTH Aachen) — Free app that turns your phone's accelerometer, microphone, barometer and gyroscope into data loggers with CSV export.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Derive the 6 dB per doubling prediction.
Measure outdoors and indoors.
Fit models and compare gradients.
Explain the difference.
Reflect on phone microphones and background noise.

Pitfalls that cost marks

Uncalibrated readings treated as absolute; compare differences instead.
Background noise not recorded.
Too few distances.

Showing personal engagement

Use a speaker you own.
Predict the level at the back of your school hall.
Compare frequencies.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Logarithmic scales: decibels; Inverse-square model turned into a log model
AA HL	Fits, but add an HL technique	Logarithmic scales: decibels; Inverse-square model turned into a log model
AI SL	Good fit	Logarithmic scales: decibels; Inverse-square model turned into a log model
AI HL	Fits, but add an HL technique	Logarithmic scales: decibels; Inverse-square model turned into a log model

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Use an artwork, instrument or design you know or make yourself, and measure it yourself. Explain the aesthetic question as well as the mathematical one.

Reflection (D)

Test popular claims honestly (the golden ratio is often not where people say it is) and reflect on how measurement choices affect the result. For this idea, start with: uncalibrated readings treated as absolute; compare differences instead — say how it affects your answer.

Use of mathematics (E)

SL: Geometry, trigonometry, sequences or functions used to model or test a real piece of art or music, with measurements and error discussed.

HL: Complex numbers, parametric curves, transformations as matrices, or Fourier-style sums of sinusoids, used to explain the structure rigorously.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Model a room by adding a constant reverberant level to the inverse-square term.

Extending it for HL

Describe the construction with parametric or complex-number methods, or prove a property of the pattern or scale you studied.

See a complete IA, marked

Our annotated exemplar Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) asks a different question, but shows how a complete art & music exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).