Decibel Calculator

A decibel is a ratio, not an amount

L = 10 log₁₀(I/I₀) never reports an intensity. It reports how many times larger one intensity is than a reference, compressed logarithmically. The reference I₀ is fixed by convention at 1 × 10⁻¹² W/m², the faintest sound a healthy young ear detects, so a decibel figure for airborne sound is meaningful without anyone restating the reference. Change I₀ and the same physical sound gets a different number, which is why the reference field here starts at the standard threshold and is never the one solved for.

The logarithm is there because hearing spans an absurd range. From the threshold to the onset of pain is about twelve orders of magnitude, and taking the log turns that into a scale running from 0 to roughly 120. It also means 0 dB is the reference, not silence. A sound quieter than the threshold has a negative level: 1 × 10⁻¹³ W/m² is −10 dB, and it is perfectly real. True silence has no decibel value at all, because the logarithm of zero is undefined, so the scale is unbounded below.

Worked example

The default intensity is 1 µW/m², about the level of ordinary conversation, measured against the standard threshold.

• Convert both to SI: I = 1 × 10⁻⁶ W/m², I₀ = 1 × 10⁻¹² W/m²
• Form the ratio: I/I₀ = 10⁻⁶ / 10⁻¹² = 1 × 10⁶
• Take the logarithm: log₁₀(10⁶) = 6
• L = 10 × 6 = 60 dB

A conversation is therefore a million times as intense as the quietest audible sound, and the scale renders that as the tidy number 60. Step back to twice the distance from the speaker and the reading falls to 53.98 dB, because intensity spreads over four times the area and 10 log₁₀(4) = 6.0206 dB.

What the round numbers actually are

Two shortcuts are worth knowing exactly. A tenfold rise in intensity is 10 log₁₀(10) = 10 dB, exact by construction. A doubling is 10 log₁₀(2) = 3.0103 dB, close to 3 but not equal to it, and the difference accumulates if you chain several doublings by hand. Perceived loudness follows neither cleanly: ten decibels sounds roughly twice as loud, while the 3 dB from doubling a sound source is barely detectable. That is why twice the traffic on a road changes the measured level so little.

The distance rule follows from geometry rather than acoustics. Intensity from a point source falls with the square of distance, so every doubling of distance costs 6.0206 dB and every halving gains the same. Four times the distance costs 12.04 dB, which is why a measurement is meaningless without the range it was taken at.

Exposure limits use the same arithmetic. The usual convention allows about 85 dB for an eight-hour day, and since 3 dB doubles the intensity, every 3 dB above halves the permitted time. At 100 dB that is fifteen decibels over, five halvings, so 8 h ÷ 2⁵ = 0.25 h, about fifteen minutes. Above roughly 120 dB the damage stops being cumulative and can happen at once. It is also permanent, because the hair cells in the cochlea do not regrow.

Common mistakes

• Adding decibels arithmetically. Two 60 dB sources do not make 120 dB. Add the intensities, not the levels: doubling gives 63.01 dB.
• Treating 0 dB as silence. It is the threshold of hearing, 1 × 10⁻¹² W/m². Negative levels exist and describe sounds below that threshold.
• Using the 20 log formula for intensity. The factor of 20 belongs to amplitude quantities such as sound pressure. Intensity is a power quantity and takes the factor of 10, and mixing them doubles or halves the answer.
• Quoting a level without a distance. A jet engine at 30 m and the same engine at 300 m differ by 20 dB. Without the range the number describes nothing.

Converting units first? Use the intensity conversion table.