Combining Sound Levels

Decibels are logarithms and logarithms do not add. Two identical machines each producing 60 dB together produce 63 dB, not 120 — a fact that sounds like a trick and is simple arithmetic. Convert each level back to relative energy with $10^{L/10}$, add the energies, take $10\log_{10}$ of the sum. Two equal energies is a doubling, and $10\log_{10}2 = 3.01$ dB. That is the whole answer, and it does not depend on the level: two 40 dB sources give 43, two 100 dB sources give 103.

The addition table is worth memorising, because it makes noise-control priorities obvious without any calculator. Levels equal: add 3.0 dB. One decibel apart: add 2.5. Three apart: add 1.8. Six apart: add 1.0. Ten apart: add 0.4. Fifteen apart: add 0.1, which is nothing. The practical rule follows directly — a source 10 dB below its neighbour is not worth silencing, because removing it entirely would buy less than half a decibel, which nobody can hear. Fix the loudest thing, then look again. And the reverse: $n$ identical sources make $L + 10\log_{10}n$, so ten of them add 10 dB, and halving the number of machines running buys 3 dB.

Run the same equation backwards and it subtracts a background reading from a measurement, which is how a machine's own contribution is separated from the ambient. This is legitimate, and it is fragile. When the combined reading and the background are 10 dB apart the correction is a reliable 0.4 dB. When they are 3 dB apart the correction is 3 dB and every error in either reading is amplified into the answer. When they are within about 1 dB, the arithmetic is meaningless. The standard convention is to accept the correction above a 10 dB difference, apply it with caution between 3 and 10, and above 3 dB report only that the source level is not greater than the measurement — then find a way to measure with the background off.

One caution about what is being added. Simple energy addition assumes the sources are uncorrelated — separate machines, independent noise. Two loudspeakers reproducing the same signal are correlated, and their pressures add rather than their energies, giving up to 6 dB where they arrive in phase and cancellation where they do not. And a single-figure A-weighted level is itself a summation across frequency bands done exactly this way, which is why A-weighted levels from different sources can be combined but octave-band data must be combined band by band before weighting.