Sound Intensity Converter
Convert between decibels (dB), watts per square meter, and pascals for sound intensity. Calculate combined sources, distance attenuation, and hearing risk
Sound Intensity Converter
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Formula: dB = 10 * log10(I / I_ref) where I_ref = 10^-12 W/m^2
Worked example โ 110 dB at 1m = 90 dB at 10m (0.001 W/m^2)
Formula
dB = 10 * log10(I / I_ref) where I_ref = 10^-12 W/m^2
Sound level in dB equals 10 times the log of intensity divided by the reference intensity (threshold of hearing). Sound pressure: dB = 20*log10(P/P_ref). Inverse square law: intensity drops 6 dB per distance doubling.
Worked Examples
Example 1: Concert Sound Level at Distance
Problem:A concert speaker produces 110 dB at 1 meter. What is the level at 10 meters?
Solution:dB at new distance = original dB - 20*log10(new_dist/original_dist) dB = 110 - 20*log10(10/1) dB = 110 - 20 = 90 dB Intensity = 10^-12 * 10^(90/10) = 10^-3 W/m^2
Result:110 dB at 1m = 90 dB at 10m (0.001 W/m^2)
Example 2: Combining Office Noise Sources
Problem:An office has 5 identical printers each producing 65 dB. What is the combined level?
Solution:Combined dB = single source dB + 10*log10(N) Combined = 65 + 10*log10(5) Combined = 65 + 6.99 = 71.99 dB Combined intensity = 5 * 3.162e-6 = 1.581e-5 W/m^2
Result:5 printers at 65 dB each = 72 dB combined
Frequently Asked Questions
What is the decibel scale and how does it measure sound?
The decibel (dB) scale measures sound intensity logarithmically relative to the threshold of human hearing (10^-12 W/m^2). The formula is dB = 10 * log10(I / I_ref). Because it is logarithmic, every 10 dB increase represents a tenfold increase in intensity. Normal conversation is about 60 dB, a lawn mower about 90 dB, and a jet engine about 140 dB. The scale compresses the enormous range of human hearing (from 0 dB to 140 dB) into manageable numbers.
How does sound intensity decrease with distance?
Sound intensity follows the inverse square law in free space: intensity decreases proportionally to 1/r^2, where r is distance from the source. This means doubling the distance reduces the intensity by a factor of 4, which corresponds to a 6 dB decrease. A speaker producing 90 dB at 1 meter will produce about 84 dB at 2 meters and 78 dB at 4 meters. Indoors, reflections from walls reduce this attenuation effect.
How do you add sound levels from multiple sources?
Sound levels in decibels cannot be added directly because the scale is logarithmic. Instead, convert each level to intensity, add the intensities, then convert back to dB. For N identical sources, the combined level is original dB + 10*log10(N). Two identical 80 dB sources produce 83 dB (not 160 dB). Ten identical sources add 10 dB. This is why doubling speakers only adds 3 dB rather than doubling the perceived loudness.
What sound levels are dangerous to hearing?
Occupational safety standards (OSHA and NIOSH) set exposure limits based on duration. At 85 dB, the recommended maximum exposure is 8 hours. For every 3 dB increase, the safe exposure time halves: 88 dB for 4 hours, 91 dB for 2 hours, 94 dB for 1 hour, and so on. At 100 dB (power tools), damage can occur in just 15 minutes. Sound above 120 dB causes pain, and 140 dB can cause immediate permanent hearing damage.
What is the difference between sound intensity level and sound pressure level?
Intensity level compares power flow per unit area against 10^-12 W/m^2 using 10*log10. Sound pressure level compares pressure amplitude against 20 micropascals using 20*log10. In a free progressive wave in air at ordinary conditions the two references were chosen so the numbers agree, but close to a source, inside a duct or in a reverberant room, intensity and pressure decouple and the two levels diverge.
How does acoustic impedance connect pressure to intensity?
For a plane wave, intensity is I = p^2/(rho*c), where rho*c is the characteristic acoustic impedance of the medium. In air near 20 degrees C, rho is about 1.2 kg/m^3 and c about 343 m/s, giving rho*c close to 413 Pa*s/m. That single quantity is what lets a converter move between pascals and watts per square metre, and it shifts with air temperature, humidity and altitude.
What do A-weighting and C-weighting do to a measured level?
Weighting filters shape the spectrum before the level is summed so the meter responds more like an ear. A-weighting rolls off low frequencies heavily and is reported as dBA, the basis of most occupational noise rules. C-weighting stays much flatter at low frequency and suits peak and impulse measurement. An unweighted reading and a dBA reading of the same bass-heavy source can differ by many decibels.
Why does a 10 dB rise sound about twice as loud when intensity has risen tenfold?
Loudness perception is compressive: the ear responds roughly to the cube root of intensity, so subjective loudness roughly doubles for each 10 dB even though 10 dB is a factor of ten physically. A 3 dB change doubles intensity but is only just noticeable, which is why adding a second identical loudspeaker is far less dramatic than people expect from doubling the power.
Which reference pressure applies underwater, and why does it matter?
Underwater acoustics references 1 micropascal, not the 20 micropascals used in air. That alone shifts a level by 20*log10(20), about 26 dB, and the far higher acoustic impedance of water changes the pressure-to-intensity relationship again. Underwater and airborne decibel figures are therefore not directly comparable, and any quoted level should always name its reference.
How do I subtract background noise from a measured level?
Because decibels are logarithmic, convert both the total and the background to intensity, subtract, then convert back: L_source = 10*log10(10^(L_total/10) - 10^(L_bg/10)). If the total exceeds the background by 10 dB or more the correction is under half a decibel and can be ignored. If the gap is under about 3 dB the background dominates and the source level cannot be resolved reliably.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator ยท Editorial policy
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