Sound intensity in decibels:
turning watts per square metre into dB

Why sound levels are quoted on a logarithmic scale

Human ears cope with an enormous spread of sound power. The faintest tone a healthy young listener can notice carries roughly one trillionth of a watt through each square metre, while a loud rock concert pushes about a watt through the same area. That is a trillion-to-one ratio, and writing it out in watts per square metre would be clumsy.

The decibel scale squeezes that spread into a handful of readable numbers by taking a logarithm. It also matches how hearing behaves: we judge loudness by ratios, not by differences. Going from 1 to 2 units of intensity sounds like the same step as going from 100 to 200, and a log scale reflects that.

Sound intensity itself is power per unit area, measured in watts per square metre (W/m²). The decibel figure is not a different physical quantity; it is the same intensity re-expressed relative to an agreed starting point, which is why the formula always needs a reference value.

The formula and its reference intensity

Take the intensity you are measuring, divide it by the reference intensity, apply the base-10 logarithm and multiply by ten. The reference used for airborne sound is 10⁻¹² W/m², a conventional stand-in for the threshold of hearing near 1 kHz.

To go the other way, undo the logarithm: intensity equals the reference multiplied by ten raised to the power of L divided by ten. This is the step people forget when they need to add sounds together, because decibels cannot be added directly.

Worked example: a sound of one nanowatt per square metre

Suppose a meter reports an intensity of 0.000000001 W/m², which is 10⁻⁹ W/m². Work out its level against the standard reference.

Notice that you can read the answer straight from the exponents: 10⁻⁹ is three powers of ten above 10⁻¹², and each power of ten is worth 10 dB. For intensities that are not clean powers of ten you need a calculator for the logarithm. For instance, 85 dB corresponds to 10^8.5 times the reference, which is about 3.2 × 10⁻⁴ W/m².

A thirty-decibel sound is comparable to a quiet rural bedroom at night or a soft whisper at a short distance. It is audible but easily masked by almost any background noise.

What changes of 3 dB and 10 dB really mean

Because the scale is logarithmic, a change in decibels is a multiplication in intensity. Learn a few anchor points and you can estimate without a calculator.

The 3 dB figure comes from 10 × log₁₀(2) = 3.01. Doubling the intensity, for example by switching a second identical speaker on, adds only about 3 dB. Perceived loudness is subjective, so the last column is only a rule of thumb; listeners usually describe a 10 dB rise as roughly doubling the loudness.

Combining several sources and moving away from them

Two sources add in intensity, never in decibels. Two machines that each produce 60 dB at your position do not give 120 dB. Each has an intensity of 10⁻⁶ W/m², so together they give 2 × 10⁻⁶ W/m², and 10 × log₁₀(2 × 10⁶) comes to 63 dB. Adding a second equal source raises the level by about 3 dB, and a tenth equal source raises it by 10 dB.

Distance works the other way. For a small source in open air, intensity falls with the square of the distance, so doubling the distance cuts intensity to a quarter. That is a drop of about 6 dB each time the distance doubles. Walls, ground reflections and wind change this in real rooms, so treat 6 dB as a free-field idealisation.

• Equal sources: add 10 × log₁₀(N) dB for N identical sources.
• Doubling the distance in open space: subtract about 6 dB.
• Unequal sources: convert each level to intensity, add the intensities, then convert back.

Familiar sounds placed on the scale

Anchoring the formula to sounds you know makes the numbers easier to trust. The table pairs typical levels with the intensity each one implies, found from I = 10⁻¹² × 10^(L ÷ 10). The descriptions are broad, since real sounds vary with distance and surroundings.

Between the first and last rows the intensity grows by a factor of a trillion, yet the level only climbs from 0 to 120. That compression is exactly what the logarithm was chosen for.

Prolonged exposure to levels from about 85 dB upward is associated with hearing damage, and permitted exposure times shorten rapidly as the level rises. Exact limits depend on the regulator and the year, so check the current workplace or health guidance rather than treating this table as a safety rule.

Common slips and where the formula stops applying

• Forgetting the factor of ten. With amplitude or pressure ratios the multiplier is 20, not 10, because power goes with the square of amplitude. Intensity ratios use 10.
• Using the wrong reference. Underwater acoustics uses a different pressure reference, so decibel values from water and from air are not directly comparable.
• Taking a logarithm of the raw intensity without dividing by the reference first, which leaves units inside the log.
• Averaging decibel readings arithmetically. Average the intensities and convert, or use an equivalent continuous level from your meter.

The formula gives a physical level, not a safety limit. Hearing-protection rules are written around exposure level and time, and the thresholds differ between regulators, so check the current occupational guidance that applies to your workplace.