Library / Foundations: Physics / FPH-56
Foundations: Physics
Sound intensity level
Loudness on the decibel scale: ten times the base-ten logarithm of the intensity compared with the quietest sound a young ear can detect.
- Ref
- FPH-56
- Rev
- v1
- Field
- Foundations: Physics
- Result
- LI [dB]
The decibel scale compresses the huge range of sound intensities into a convenient one: LI = 10 log10(I/I₀), with I₀ = 10−12 W/m² the faintest sound a young ear can detect. Every factor of ten in intensity adds 10 dB, and doubling the intensity adds about 3 dB. This is the intensity level, which is not the same as the A-weighted level used in noise regulations.
Loudness on the decibel scale: ten times the base-ten logarithm of the intensity compared with the quietest sound a young ear can detect. Every factor of ten in intensity adds 10 dB.
Calculator
- Intensity relative to the threshold, I/I₀1,000,000
- Level if the intensity doubles63.01 dB
- Level if the intensity falls by a factor of 1050 dB
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- An intensity of 1×10−6 W/m² is 1,000,000 times the threshold, a level of 60 dB, typical of ordinary conversation.
- Doubling the intensity adds only 3.01 dB: two identical machines are barely 3 dB louder than one, and ten identical machines 10 dB.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (dB) | Local step | Change in result |
|---|---|---|---|---|---|---|
| I Sound intensity | 1e-12 to 10 W/m² | 0 | 130 | 130 | +10% | +0.414 dB |
Sweep of Sound intensity
| Sound intensity (W/m²) | Sound intensity level (dB) |
|---|---|
| 1e-12 | 0 |
| 4.217e-11 | 16.25 |
| 1.778e-9 | 32.5 |
| 7.499e-8 | 48.75 |
| 3.162e-6 | 65 |
| 1.334e-4 | 81.25 |
| 0.005623 | 97.5 |
| 0.2371 | 113.8 |
| 10 | 130 |
Sweep one variable
Variables and how each one contributes
Sound intensity level
What the equation gives you. Everything else feeds it.
Sound intensity
The sound intensity, in W/m². Normal speech is around 10−6 W/m². The slider is logarithmic. Raises the level by 10 dB for every factor of ten in intensity, so the dependence is logarithmic. Raising it by 1×10−6 (to 2×10−6) moves the result by +5.02 %, from 60 to 63.01 dB.
Worked example with the values above
- Ratio to the threshold: I/I₀ = 1×10−6 / 10−12 = 1,000,000.
- Take the base-ten logarithm: log10(1,000,000) = 6.
- LI = 10 × 6 = 60 dB.
Assumptions and limits
- Intensity measured in W/m² at the point of interest, with the standard reference I₀ = 10−12 W/m².
- Intensity level without frequency weighting.
Common mistakes
- Adding decibel values directly instead of adding the intensities and then converting.
- Using 20 log₁₀ for intensity: that factor is used for pressure or amplitude.
Statistics and uncertainty
Because the scale is logarithmic, a 10 % error in the intensity changes the level by only 0.4 dB. Measuring the level with a meter is usually good to about ±1 dB, which is a ±26 % uncertainty in the intensity.
Where it is used
- Noise assessment of vehicles, engines and workplaces.
- Hearing-protection decisions and combining several noise sources.
Maths you need
- Logarithm: the power to which 10 is raised to give a number
- Intensity: power per unit area