Noise from a machine to the property line

sound power to sound pressurenoise assessmentwill the neighbours complainnoise exposure limit

From a nameplate sound power to the level at the fence and the hours a worker may stand beside it, through whatever wall is in between.

Sound Power Level to Sound Pressure Level

L_p = L_W + 10\log_{10}\!\left(\frac{Q}{4\pi r^{2}}\right)

A machine's sound POWER is a property of the machine; its sound PRESSURE is what a meter reads at a given place. This converts one to the other in a free field, using the distance and the directivity of wherever the machine is sitting.

Distance Attenuation from a Point Source

L_2 = L_1 - 20\log_{10}\!\left(\frac{r_2}{r_1}\right)

Move twice as far from a point source in the open and the level drops 6 dB — every time, regardless of the starting level. Ten times the distance is 20 dB. This is the cheapest noise control there is, when there is room for it.

Combining Sound Levels

L_t = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right)

Decibels do not add. Two 60 dB machines running together measure 63 dB, not 120 — the energies add and the logarithm is taken again at the end. This is the most misused arithmetic in noise control, and the fix is one line long.

Mass Law Transmission Loss

TL = 20\log_{10}\!\left(\frac{\pi m f}{\rho_0 c}\right) - 5

For a single limp panel, blocking sound is almost entirely about weight: transmission loss climbs about 6 dB every time the surface density doubles, and another 6 dB every time the frequency doubles. It is the reason there is no light way to stop bass.

Composite Transmission Loss

TL_c = 10\log_{10}\!\left(\frac{S_w + S_d}{S_w\,10^{-TL_w/10} + S_d\,10^{-TL_d/10}}\right)

A wall with a door or a window in it, rated as one partition. The two paths add by AREA AND TRANSMISSION, never by decibels, and the answer is the most sobering arithmetic in building acoustics: the weak element sets the rating almost by itself.

Allowable Noise Exposure Time

T = \frac{8}{2^{(L - L_c)/q}}

How long a person may be exposed to a steady noise level before reaching a full daily dose. The answer depends entirely on which exchange rate the jurisdiction uses, and the two in common use disagree by hours.

How they fit together

Start by separating the two quantities that get called the same thing, because almost every wrong answer in this field starts here. Sound power is what a machine emits and it is a property of the machine alone — it does not change if you move, and it is what a nameplate or a test certificate quotes. Sound pressure is what a meter reads at a place, and it depends on distance, direction and the room. Sound power level to sound pressure level is the bridge, and the directivity factor Q in it is the usual slip: a source sitting on a hard floor radiates into a hemisphere rather than a sphere and is 3 dB louder than the free-field figure, and one in a corner is 9 dB louder.

From there, distance attenuation moves the level out to the receiver at 6 dB per doubling, and combining sound levels handles more than one machine. That second one is worth doing by formula every time rather than by instinct, because the instinct is wrong: two identical machines are 3 dB louder than one, not twice as loud, and a 90 dB source next to a 80 dB source is 90.4 dB. The practical consequence is that silencing the quieter half of a plant buys you almost nothing measurable, and the whole budget belongs on the loudest item.

The barrier steps are where money gets spent. Mass law gives the transmission loss of a solid partition, and it climbs only 6 dB per doubling of surface density or frequency, which is the arithmetic reason a heavier wall is such a poor deal and why real construction reaches for separated leaves and cavities instead. Composite transmission loss is the one that governs the outcome in practice: it combines a good wall with a weak element, and the weak element wins far more easily than seems fair. A 45 dB wall with 1% of its area at 20 dB — one poor door, one unsealed penetration — performs at about 39 dB, so six of the decibels you paid for leave through a hundredth of the surface. Treat the gap before thickening the wall. Allowable exposure time then converts the level at the operator into the regulated answer, and its exchange rate is a jurisdiction question rather than a physics one: 3 dB in most of the world and in the ACGIH guidance, 5 dB in the US OSHA rule. At 95 dB those two give two hours and four hours. Both are correct, for different readers, and you must state which you used.