Level-domain engineering worksheet
Acoustic Level Summation
Choose the physical model before adding decibels: incoherent pressure contributions, independent source sound powers, or coherent same-frequency pressure vectors.
Documented components
All fields start blank. Use each physical component once; do not count shared background repeatedly.
Conditional result
Model 1 — mutually incoherent pressure contributions
For compatible mean-square sound-pressure levels from mutually incoherent simultaneous components at one receiver, the cross terms average to zero:
Every Lᵢ must be the same descriptor: for example, all LAeq,15min at one receiver over the same interval, or all levels in the same one-third-octave band. Two equal incoherent components add exactly 10 log₁₀(2) = 3.0103 dB.
Model 2 — independent source sound powers
Independent acoustic powers add linearly, so compatible sound-power levels use the same 10 log₁₀ expression:
The output remains a sound-power level. It is not a receiver sound-pressure level and does not include directivity, distance, propagation, reflections or source interaction. If operating sources change one another’s emission, separately measured powers are not an independent-source model.
Model 3 — coherent same-frequency pressure components
For stable coherent tones at the same frequency and receiver, convert each pressure level to a relative amplitude and add complex pressure vectors:
Lp,Σ = 20 log₁₀|pΣ/p₀|
Two equal in-phase components add 6.0206 dB; equal quadrature components add 3.0103 dB; ideal equal components 180° apart cancel. Real acoustic phase varies with frequency and position, so this mode is not valid for broadband levels, unstable phase, partial coherence or spatial averaging.
Compatibility and exclusion table
| Question | Required answer | If not established |
|---|---|---|
| Same physical quantity and reference? | Yes: Lp with Lp, or Lw with Lw | Do not sum |
| Same weighting/band and time descriptor? | হ্যাঁ | Reprocess the underlying data first |
| Same receiver and simultaneous pressure basis? | Yes for pressure modes | Do not present the result as a receiver total |
| Shared background counted once? | Removed, negligible, or explicitly isolated | Inputs can double-count energy |
| Correlation/coherence known? | Supported by the selected model | Cross-spectral information is required |
| Occupational exposure decision intended? | No—this is level summation only | Use a duration-based framework-specific exposure assessment |
Sources and standards boundary
Equation classification: general logarithmic level relations derived from linear energy/power addition or coherent pressure superposition. They are not labelled as an ISO compliance calculation.
- OSHA Technical Manual, Section III, Chapter 5, Appendix B.3 — official 10 log₁₀ summation of compatible pressure, intensity or power levels.
- NIOSH Industrial Noise Control Manual — random-noise assumption and simultaneous-source energy addition.
- NIST Special Publication 811 — 20 log₁₀ field-quantity and 10 log₁₀ power-quantity level definitions and the requirement to state the reference.
- Boston University, The Decibel Scale — public instructional derivation distinguishing coherent amplitude addition from incoherent intensity addition.
- ISO 80000-8:2020, Edition 2 + Amendment 1:2025 — current names, symbols, definitions and units for acoustical quantities.
- IEC 61672-1:2013, Edition 2 — distinguishes frequency-weighted, time-weighted, time-averaged and exposure sound-level measurements.
Official records checked 13 July 2026. Partial-coherence cross terms, background-correction procedures and closed standard clauses are not guessed here.
Questions that prevent common errors
Can I mix dB and dBA?
No. “dB” alone does not identify the measured quantity or weighting. All inputs must have the same explicitly recorded descriptor.
Can I add measurements made with each machine on while the same background remains?
Not directly. Each reading then contains the background, so adding them counts that energy repeatedly. Use source-only contributions obtained through a justified isolation/background method, or measure the simultaneous total.
Can I add sequential tasks to obtain daily exposure?
Not with this simultaneous-source worksheet. Sequential exposure periods require duration-weighted energy integration or the formula specified by the governing exposure framework.
Why can a coherent result be below every component?
Coherent pressures can interfere destructively. The result depends on relative phase at the specific receiver and frequency; it is not an energy sum.
Is there an 85 dBA hearing-protection line?
No. A source-level sum is not automatically an eight-hour occupational exposure. OSHA, NIOSH and EU action/limit quantities differ and require duration and framework-specific assessment.
