The Decibel Scale: Compressing a Huge Range of Intensities

1 · Predict

Sound intensities that humans can hear span a factor of a trillion, from a whisper to a jet engine. Why do we measure loudness in decibels instead of directly in W/m²?

• Decibels use a logarithmic scale, so equal dB steps correspond to equal multiplicative jumps in intensity.
• Decibels are just a linearly rescaled version of W/m² for convenience.
• Decibels measure something physically different from intensity.

2 · Set Up

1. Open the sound-decibel preset and press Reset. The source radiates 0.12 W in all directions.
2. Enable the intensity and sound-level readouts at the observer.
3. Set the observer's distance from the source for each trial and record the intensity and sound level.

3 · Collect Data

Plot sound level β (y-axis) against log₁₀(intensity I in W/m²) (x-axis) for your three trials. Is the line straight?

4 · Analyze

1. For one trial, compute I = P/(4πr²) using P = 0.12 W (report in µW/m², 1 µW/m² = 10⁻⁶ W/m²), then β = 10·log₁₀(I/I₀) with reference intensity I₀ = 10⁻¹² W/m². Compare both to the table.
2. Even though your three intensities differ by almost a factor of 10, the sound-level column changes by only about 10 dB between the closest and farthest trial. Explain why, using the logarithm in the formula.

5 · Extend

1. The reference intensity I₀ = 10⁻¹² W/m² is defined as roughly the quietest sound a human can hear (0 dB). Using the formula, what sound level would a listener measure exactly at the threshold intensity?
2. Every +10 dB corresponds to a ×10 increase in intensity, not a doubling. Using the formula, roughly how many dB does an intensity that doubles (×2) actually add?

The Physics Behind This Experiment

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