Acoustics / sound level arithmetic

Decibel arithmetic

Add two sound sources properly, convert between power and amplitude decibels, and apply inverse-square distance loss. Decibels cannot be added directly — this engine adds the underlying powers instead.

Δ between sources5.0 dBSum gain+1.19 dBQuantityPower

Combined level

86.2 dB

Incoherent sum of both sources

Distance attenuation

-20.0 dB

From 1.00 m to 10.00 m

Level at distance

65.0 dB

Source 1 after geometric spreading

Power ratio

3.162e+8×

10^(L/10)

Decibel reference table

What a given level change means as a power ratio and as an amplitude ratio.

86.2 dB combined

Decibel reference table
Change	Power ratio	Amplitude ratio	Perception
3 dB	2.00×	1.41×	Barely audible
6 dB	3.98×	2.00×	Clearly audible
10 dB	10.00×	3.16×	Twice as loud
20 dB	100.00×	10.00×	Four times as loud
40 dB	1.000e+4×	100.00×	Very large
Sum of the two sources	4.162e+8×	2.040e+4×	+1.19 dB over the louder

Two equal sources add 3 dB, not double the level. Adding a source 10 dB quieter changes the total by only 0.4 dB.

Level at increasing distance

Point-source spreading costs 6 dB per doubling of distance.

Sum of two sources vs their difference

The closer the two levels, the more the sum exceeds the louder one.

Canonical equations

The exact relationships implemented by this engine.

L = 10log₁₀(P/P₀) · energy quantities

L = 20log₁₀(A/A₀) · pressure, voltage, amplitude

L = 10log₁₀(10^L₁ᐟ¹⁰ + 10^L₂ᐟ¹⁰) · uncorrelated sources

ΔL = −20log₁₀(r₂/r₁) · inverse square for a point source

Show model notation and computationOpen the symbolic form, the variable definitions, and the step-by-step derivation when you need the detail.

A clear path from inputs to results.

Decibels are logarithms, which means they cannot be added. Two 85 dB sources make 88 dB, not 170. This engine converts between linear ratios and decibels in both the power and amplitude conventions, sums sources properly, and applies the 6 dB per doubling that governs distance loss.

Step-by-step methodWorked exampleAssumptions explained

How to use

Enter the level of the first source.

Enter the level of the second source.

Set the reference and receiver distances.

Choose whether you are working with power quantities (10 log) or amplitude quantities (20 log).

Read the combined level and the attenuation.

Check the reference table for what a 3, 6 or 10 dB change actually means perceptually.

The formula

L = 10\log_{10}\frac{P}{P_0}, \qquad L = 20\log_{10}\frac{A}{A_0}, \qquad L_{\Sigma} = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right), \qquad \Delta L = -20\log_{10}\frac{r_2}{r_1}

Reference: IEC 60027-3 (logarithmic quantities); ISO 9613-1 for atmospheric attenuation.

Worked example

85 dB and 80 dB sources, 1 m to 10 m (default inputs)

Inputs

Source 1 at 85 dB.
Source 2 at 80 dB.
Reference distance 1 m; receiver 10 m.

Calculation path

1
Convert to power ratios: 10^8.5 = 3.16×10⁸ and 10^8.0 = 1.0×10⁸.
2
Sum: 4.16×10⁸, which is 86.19 dB.
3
The louder source alone is 85 dB, so the second adds only 1.19 dB.
4
Distance loss: −20 log₁₀(10) = −20 dB.
5
Level at 10 m from source 1: 85 − 20 = 65 dB.

Result

Combined 86.19 dB (+1.19 over the louder) · 20 dB attenuation · 65 dB at 10 m.

Interpreting the result

Adding a source 5 dB quieter buys 1.19 dB — barely perceptible. Two equal sources give exactly +3.01 dB, and ten equal sources give +10 dB, which is roughly a doubling of perceived loudness. The distance result is the more useful one in practice: every doubling of distance costs 6 dB, so moving from 1 m to 10 m removes three-quarters of the sound energy.

Assumptions & limitations

Sources are incoherent — uncorrelated, so their powers add and their amplitudes do not.

Free-field propagation with no reflections. Indoors, reverberation flattens the distance loss considerably.

Point source, so the inverse-square law applies. A line source such as a highway loses only 3 dB per doubling.

Atmospheric absorption is excluded. Over long distances it adds several dB at high frequencies.

No barriers, ground effects or directivity are modelled.

Frequently asked questions

Why can’t I just add decibels?

Because they are logarithms of ratios. Adding 85 and 80 would correspond to multiplying the underlying powers, which is not what happens when two sources operate simultaneously. You must convert to linear power, add, then convert back.

How much louder is twice the sound?

A 10 dB increase is perceived as roughly twice as loud, and it is ten times the acoustic power. A 3 dB increase is double the power but only just noticeable.

What is dB(A)?

A-weighted decibels, filtered to approximate the ear’s sensitivity at moderate levels. It is the standard for environmental noise limits, and it is not the same as the unweighted dB computed here.

What is the difference between 10 log and 20 log?

Power quantities (intensity, acoustic power, electrical power) use 10 log. Amplitude quantities (pressure, voltage, displacement) use 20 log, because power is proportional to amplitude squared.

Does distance really cost 6 dB per doubling?

In a free field with a point source, yes — the energy spreads over a sphere whose area quadruples, so intensity falls by four, which is 6 dB. Indoors or near a reflective surface, less.

Can decibels be negative?

Yes, when the quantity is below the reference. Sound pressure level in air references 20 µPa, so quieter-than-reference pressures give negative dB.