formula.expert

Noise at the workstation

Occupational noise · levels, combination, exposure

A shop compressor measures 98 dB at 1 m. The assembly bench sits 4 m from it, beside a grinder that contributes 84 dB of its own at the bench. Site policy follows the 3 dB exchange rate against an 85 dB criterion over an 8-hour shift. Find the compressor's level at the bench, the combined level the worker actually stands in, and the longest that position can be worked in a day.

plan viewcompressorL₁r₁r₂benchL_gT

Every number in this problem is editable — change any value below and the whole chain recalculates.

Given
  • L₁ = 98 dB — Compressor at 1 m
  • r₁ = 1 m — Reference distance
  • r₂ = 4 m — Bench distance
  • L_g = 84 dB — Grinder at the bench
  • L_c = 85 dB — Criterion level
  • q = 3 dB — Exchange rate
Determine
  1. (a)the compressor's level at the bench
  2. (b)the combined level at the worker's ear
  3. (c)the allowable hours at that position
Step 1 of 3(a) · solve for Level at the far distance

Distance is the cheapest hearing protection ever invented: every doubling costs the source 6 dB, so four metres instead of one takes the compressor from 98 to 86 dB before anyone reaches for a muff. Free-field spreading is the model — a hard-walled shop returns some of it as reverberation.

L1L2r1r2
Rearranged for L₂
L2=L1−20log⁡10 ⁣(r2r1)L_2 = L_1 - 20\log_{10}\!\left(\frac{r_2}{r_1}\right)
Your values, in your units
L2=(98 dB)−20log⁡10 ⁣((4 m)(1 m))L_2 = \left( 98\ \text{dB} \right) - 20\log_{10}\!\left(\frac{\left( 4\ \text{m} \right)}{\left( 1\ \text{m} \right)}\right)
Answer
L2=85.959 dBL_2 = 85.959\ \text{dB}

Carried onward at full precision, not this rounded figure.

Open the Distance Attenuation from a Point Source solver →

Step 2 of 3(b) · solve for Combined level

Decibels refuse ordinary addition — 86 plus 84 is not 170 but 88.1, because what adds is energy, not level. Two nearly-equal sources give at most 3 dB over the louder; the grinder matters here exactly because it is CLOSE to the compressor's level, not because it is loud.

L1L2Lt
Rearranged for Lt
Lt=10log⁡10 ⁣(10L1/10+10L2/10)L_t = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right)
85.959 dBcarried from step 1
Your values, in your units
Lt=10log⁡10 ⁣(10(85.9588 dB)/10+10(84 dB)/10)L_t = 10\log_{10}\!\left(10^{\left( 85.9588\ \text{dB} \right)/10} + 10^{\left( 84\ \text{dB} \right)/10}\right)
Answer
Lt=88.099 dBL_t = 88.099\ \text{dB}

Carried onward at full precision, not this rounded figure.

Open the Combining Sound Levels solver →

Step 3 of 3(c) · solve for Allowable exposure time

The exchange rate turns level into time: 3 dB over criterion halves the day. At 88.1 dB the position supports 3.9 hours — half a shift. Past that, the choice is engineering controls, rotation, or protection with a real rating, in that order of preference.

LLcqT
Rearranged for T
T=82(L−Lc)/qT = \frac{8}{2^{(L - L_c)/q}}
88.099 dBcarried from step 2
Your values, in your units
T=82((88.0992 dB)−(85 dB))/(3 dB)T = \frac{8}{2^{(\left( 88.0992\ \text{dB} \right) - \left( 85\ \text{dB} \right))/\left( 3\ \text{dB} \right)}}
Answer
T=234.56 minT = 234.56\ \text{min}

Carried onward at full precision, not this rounded figure.

Open the Allowable Noise Exposure Time solver →

Answer

Therefore the compressor arrives at the bench at 86.0 dB, the grinder lifts the field to 88.1 dB, and the position is good for 3.9 hours a day — half a shift, decided by a machine four metres away and a grinder that only added two decibels.

Why this order

The three steps are the three habits of noise arithmetic. Inverse square first: level falls 6 dB per doubling of distance from a point source, which makes layout the most cost-effective control in the shop — moving a bench three metres did more here than a 20-dollar earmuff rated optimistically. Energy addition second: levels combine through their antilogs, so two equal sources gain 3 dB and a source 10 dB below the other adds essentially nothing — the loudest machine owns the room. Exposure last, because the dose limit is a time-level trade, not a ceiling: 88 dB is not forbidden, it is rationed.

The trap is the exchange rate itself. Under the 3 dB rule used by NIOSH and most of the world, 88.1 dB allows 3.9 hours; under OSHA's 5 dB rule the same bench computes to 5.2 hours. Same worker, same noise, forty percent more allowed exposure — know which rule your jurisdiction writes down before quoting anybody a number.

Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.