Sound Intensity Level (Decibels)

Equivalent forms

L = 20\,\log_{10}\!\left(\frac{p}{p_0}\right)

I = I_0\,10^{L/10}

A single logarithm compresses a trillion-to-one range of loudness into a friendly 0-120 scale.

Symbol	Name	SI unit	Dimension	Range
$I$	Sound intensity Acoustic power per unit area	W/m^2	M T^-3	1e-12 – 1
$I_0$	Reference intensity Threshold of hearing	W/m^2	M T^-3	1e-12 – 1e-12
$L$	Sound leveloutput Intensity level in decibels	dB	1	0 – 130

Unit systems

SI:: I in W/m^2, L in dB
CGS:: I in erg/(s cm^2)
Imperial:: L in dB (dimensionless)

Where it holds

Standard reference

I_0 = 1e-12\,\mathrm{W/m}^2

for airborne sound. Perceived loudness (phons/sones) additionally depends on frequency via the equal-loudness contours; dB SIL is purely physical.

Discovery

Bell Telephone Laboratories · 1924

The 'bel', named for Alexander Graham Bell, was devised at Bell Labs to quantify transmission loss in telephone lines. The decibel (one tenth of a bel) proved a more practical unit and spread across acoustics and electronics.

Try this

Why does a 10x louder sound only add '10 dB' instead of multiplying the number by ten?

Human hearing spans a trillion-to-one range of intensities. The decibel scale tames that with a logarithm: each factor of 10 in intensity is just +10 dB.

Research status: stable

Common misconceptions

+3 dB is a doubling of intensity (power), not of loudness; perceived loudness roughly doubles every +10 dB. Decibels add for cascaded gains, they do not multiply.

Derivation

Define level as the logarithm of an intensity ratio in bels:

L_bel = \log 10(I/I_{0})

.

Multiply by 10 for decibels:

L = 10 \log 10(I/I_{0})

.

Since intensity scales as pressure squared

(I \sim p^2)

,

L = 20 \log 10(p/p_{0})

, explaining the factor 20 for pressure.

Limiting cases

I = I_0

⟶

L = 0\,\mathrm{dB}

— the threshold of hearing

I = 1\,\mathrm{W/m}^2

⟶

L = 120\,\mathrm{dB}

— the threshold of pain

Intensity of an Electromagnetic Wave→Acoustic Impedance→Doppler Effect→Beat Frequency→