Sound Power Level to Sound Pressure Level

Sound power and sound pressure are constantly confused and are not the same kind of thing. Sound POWER, $L_W$, is a property of the source alone: the acoustic watts it emits, referenced to a picowatt. It does not depend on the room, the distance or the microphone. Sound PRESSURE, $L_p$, is what a meter reads at a particular place, referenced to 20 μPa, and it depends on all three. A manufacturer's rated sound power is comparable between machines; a "62 dB" quoted with no distance is not a specification, it is a rumour.

Getting from one to the other in the free field is geometry. The power spreads over a sphere of area $4\pi r^2$, so the intensity falls as $1/r^2$ and the level drops by $10\log_{10}(4\pi r^2)$. At one metre in free space that is 11 dB, giving the shortcut $L_p \approx L_W - 11$ at 1 m. The directivity factor $Q$ accounts for surfaces near the source that stop the sound spreading in every direction: 1 in free space, 2 sitting on a hard floor or against a wall (a hemisphere), 4 in a wall-floor junction, 8 tucked into a corner. Each of those doublings adds 3 dB, which is the argument against putting a noisy unit in a corner when the middle of the wall was available.

Indoors, the formula only describes the region near the machine. Beyond a certain distance the reverberant field — everything that has already bounced — dominates, and the level stops falling no matter how far you walk. That distance, the reverberation radius, can be as little as a metre or two in a hard-surfaced plant room. It explains a common frustration: moving a workstation further from a machine achieves nothing measurable, while adding absorption to the room lowers the reverberant level everywhere at once. Outdoors, or in a room treated well enough to behave like a free field, the distance term earns its keep.

Determining a machine's sound power properly is a standardised measurement — the ISO 3740 series — using a hemispherical or spherical array of microphone positions, or an enclosing intensity scan, or a comparison against a calibrated reference source. Backing one out of a single pressure reading in a real room, as the reverse of this formula allows, is an estimate and should be labelled as one; the room contributed to that reading and this equation has not been told about the room.