Sound Intensity Calculator

Sound intensity is acoustic power flowing through unit area, measured in watts per square meter, and unlike pressure it is a vector with a direction of energy flow. A point source radiating freely spreads its power over an expanding sphere, so intensity at radius r is the source power divided by 4 pi r squared. That geometry, not any property of air, produces the inverse square law: doubling the distance quarters the intensity, a drop of 6 dB, so 100 dB at 1 m becomes 94 dB at 2 m and 88 dB at 4 m. Intensity level in decibels is 10 times the base-10 logarithm of intensity over the reference 10 to the minus 12 watts per square meter. Converting to pressure uses the characteristic impedance of air, about 413 rayls at 20 degrees C. Independent sources add as energies, not as decibel numbers, so N identical machines raise the level by 10 log10 N: two give 3 dB, eight give about 9 dB, ten give 10 dB. Exposure thresholds follow from the resulting level. NIOSH recommends 85 dBA for 8 hours with a 3 dB exchange rate, halving the permitted time for each 3 dB increase, so 88 dBA allows 4 hours, 94 dBA one hour and 100 dBA fifteen minutes; the enforceable OSHA limit is more permissive at 90 dBA with a 5 dB exchange rate. The spherical model's main limitation is that it assumes free field. Indoors, reflected energy dominates beyond the critical distance and the level flattens out instead of continuing to fall, and a source on a hard floor radiates into a hemisphere, adding about 3 dB.

The energy view of sound was established by 19th-century physics. John William Strutt, Lord Rayleigh, set out the wave theory in The Theory of Sound, published in two volumes in 1877 and 1878, and introduced the disc that bears his name as an early means of measuring acoustic particle velocity rather than pressure. Wallace Clement Sabine, working at Harvard between 1895 and 1900, turned acoustics into a quantitative engineering discipline by relating reverberation time to room volume and absorption, which is also the origin of the critical distance concept that limits where the inverse square law applies indoors. The decibel machinery came from telephony. Bell Telephone Laboratories engineers formalized the transmission unit in the early 1920s and it was renamed the decibel in 1928 in honor of Alexander Graham Bell, after which the 10 to the minus 12 watts per square meter and 20 micropascal reference values were fixed to anchor the scale near audibility. Direct two-microphone intensity measurement, which resolves the pressure gradient to obtain particle velocity and hence the direction of energy flow, became practical in the 1970s through work by researchers including Frank Fahy in the United Kingdom and Finn Jacobsen and colleagues in Denmark, and was codified in the ISO 9614 series for determining sound power from intensity measurements. NIOSH published its recommended noise exposure criteria in 1972 and revised them in 1998, moving to the 85 dBA, 3 dB exchange rate basis used here.