Noise work goes wrong in one place more than any other: somebody adds two decibels together. So before a single relation, the scale itself.

A decibel is ten times the logarithm of an energy ratio. Everything follows from that one sentence. Ten times the energy is +10 dB. A hundred times is +20 dB. And the line to carry out of this lesson: double the energy and you gain 3 dB, because $10\log_{10} 2 = 3.01$, and it does not matter whether you started at 40 dB or at 110. Two 80 dB machines make 83, never 160.

Four quantities live in this chapter and they are constantly confused. Take each in turn.

Sound power level, $L_W$ — read aloud L sub W — is what the machine emits, in decibels referred to one picowatt. It belongs to the machine alone: no distance, no room, no listener. That is why a manufacturer can honestly print it on a sheet without knowing where you will stand.

Sound pressure level, $L_p$, is what a meter reads at a stated position, in decibels referred to 20 micropascals — the quietest pressure a young ear can detect. It is a reading, not a rating, and it changes the moment the meter moves. The bridge between the two is geometry: $L_p = L_W + 10\log_{10}\!\left(\dfrac{Q}{4\pi r^{2}}\right)$, where $r$ is the distance from the source in metres and $Q$ is the directivity factor — 1 for a source in free air, 2 sitting on hard ground, 4 in the angle of two walls. That term is always negative, because power spreads out over a growing surface.

Total absorption, $A$, is an area, in square metres of sabins. It measures how much sound a room's surfaces swallow, and it goes up when material is added. Nothing wearing decibels is an absorption.

Noise reduction coefficient, NRC, is a bare number between 0 and 1: one material's absorption averaged over four octave bands. It says how well a surface soaks sound up — and never how well it stops sound crossing to the next room. Absorbing is not blocking. A ceiling tile at NRC 0.90 will quieten an open office and will not keep one conversation out of the next.

And when two sources do have to be combined, the only honest route is back through the energy: $L_t = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right)$, where $L_1$ and $L_2$ are the two levels at the same point and $L_t$ is the total. Undo the logarithm, add, take it again. There is no shortcut that survives contact with a meter.