Sound intensity and decibels

dBsound levelinverse square law soundloudness

Sound power spread over area, how it falls with distance, and why the ear's response forces a logarithmic scale.

Sound Intensity (I = P/A)

I = \frac{P}{A}

Sound intensity is the acoustic power passing through each square meter of surface.

Inverse-Square Law for Sound

I = \frac{P}{4\pi r^{2}}

A point source's intensity falls with the square of distance as its power spreads over an expanding sphere.

Decibel Sound Level

\beta = 10 \log_{10}\!\left(\frac{I}{I_0}\right)

Sound level in decibels compares an intensity to the threshold of hearing, I₀ = 10⁻¹² W/m².

Speed of Sound in Air

v = 331.3 + 0.606\, T_C

The speed of sound in dry air grows about 0.6 m/s for every degree Celsius above freezing.

How they fit together

Intensity is power divided by the area it spreads across, so a point source obeys the inverse-square law: double the distance and intensity falls to a quarter. The decibel scale exists because human hearing spans roughly twelve orders of magnitude of intensity, from the threshold of audibility to the threshold of pain, and no linear scale can carry that range usefully.

The trap is treating decibels as though they add arithmetically. Two identical machines are not twice as loud in decibels — doubling intensity adds about 3 dB, not 100%. And a dB figure means nothing without its reference: sound pressure level is quoted against 20 μPa in air, intensity level against 1 pW/m². Quoting “85 dB” with no reference is a number without units.