Decibel Level Addition Calculator
Why Decibels Cannot Be Added Linearly
One of the most common misconceptions in acoustics is that sound levels in decibels can be added like ordinary numbers. A room with two equally loud sources does not produce twice the decibel level—it produces only about 3 decibels more. This counterintuitive result stems from the logarithmic nature of the decibel scale itself. The decibel was designed to match human hearing perception, where our ears perceive loudness logarithmically rather than linearly. Understanding why and how to properly combine sound levels is essential for audio engineers, architects, occupational safety professionals, environmental consultants, and anyone designing acoustic environments.
The decibel scale compresses an enormous range of sound pressures—from the quietest audible sound at 20 micropascals to potentially ear-damaging levels above 200 pascals—into a manageable numerical range from 0 to about 140 dB. This compression is achieved through logarithmic mathematics. Because of this logarithmic relationship, simply adding decibel values produces incorrect results that overestimate the combined sound. For example, a sound at 70 dB plus an identical 70 dB sound does not yield 140 dB but rather approximately 73 dB. This calculator demonstrates the correct method for combining decibel levels from multiple sources.
The Mathematics of Decibel Addition
The decibel is formally defined as:
where is the sound pressure level in decibels, is the measured pressure, and is the reference pressure (20 micropascals for sound in air). To properly combine multiple decibel levels, we must first convert each back to linear pressure values, sum those pressures, and then convert back to decibels.
When combining sound sources with levels , , and so forth, the combined level is:
This formula first converts each decibel value to its linear equivalent by raising 10 to the power of (dB/10), then sums all these linear values, and finally converts the sum back to decibels by multiplying by 10 and taking the logarithm base 10. This is the mathematically correct procedure for combining independent sound sources that contribute pressure simultaneously.
Worked Example: Combining Two Identical Sources
Consider an audio test in which a loudspeaker produces 70 dB SPL in a room. An engineer then adds a second identical loudspeaker playing the same signal in phase. Using the correct decibel addition method:
First, convert each level:
Sum the two sources: 10,000,000 + 10,000,000 = 20,000,000
Convert back to dB: dB
The result is 73.01 dB, not 140 dB or even 71 dB. The 3 dB increase is sometimes called the "3 dB rule"—every time you double the acoustic power, the decibel level increases by 3 dB. This fundamental principle appears repeatedly in acoustic design and noise control. When four identical sources are added, the increase is another 3 dB (reaching 76 dB from four 70 dB speakers), and so on. Understanding this rule helps predict how adding more speakers or machines will affect overall sound levels.
Comparing Different Sound Level Combinations
The following table illustrates how various combinations of sound sources add together using the logarithmic formula rather than simple arithmetic:
| Source 1 (dB) | Source 2 (dB) | Source 3 (dB) | Incorrect Linear Sum | Correct Logarithmic Sum | Error from Linear Addition |
|---|---|---|---|---|---|
| 60 | 60 | — | 120 | 63.01 | +56.99 dB |
| 70 | 70 | — | 140 | 73.01 | +66.99 dB |
| 75 | 72 | — | 147 | 77.40 | +69.60 dB |
| 80 | 80 | 80 | 240 | 84.77 | +155.23 dB |
| 65 | 65 | 65 | 195 | 71.39 | +123.61 dB |
This table vividly demonstrates how catastrophically wrong linear addition becomes. Someone who naively adds 70 dB + 70 dB = 140 dB might mistakenly conclude the combined level is dangerously loud—when in fact it is only 73 dB. Conversely, underestimating the combined effect of three sources by using crude rules can lead to insufficient acoustic treatment in noise-critical environments. The correct logarithmic method is essential for accurate acoustic design.
Real-World Applications
HVAC engineers must combine noise from multiple building systems: a 70 dB supply fan, a 68 dB return fan, and 65 dB from corridor noise outside a mechanical room. Using this calculator, they find the combined level is approximately 73.5 dB—above acceptable office background noise limits. Architects then specify vibration isolation, ductwork lining, and structural decoupling to reduce each source, iteratively bringing the combined level into compliance.
Occupational safety professionals assess workplace noise exposure by combining sound levels from machinery, compressed air, and human activity. A factory floor with a 85 dB lathe, an 82 dB drill press, and an 80 dB compressor combines to approximately 86.7 dB. Workers in this environment exceed the OSHA 8-hour time-weighted average limit and require hearing protection and engineering controls.
In residential settings, neighbors complain about noise from HVAC systems, refrigeration units, and foot traffic in adjacent apartments. A 55 dB HVAC system, a 52 dB refrigerator, and 50 dB ambient noise combine to about 57.5 dB in a bedroom—which might still exceed WHO guidance for sleep. Proper isolation design reduces each component, and this calculator verifies that the combined effect meets standards.
Why Simple Rules Often Fail
Many sound meter manuals or simplistic guides suggest "if sources differ by more than 10 dB, just use the louder value." While there is truth in this—a 80 dB source combined with a 60 dB source yields approximately 80.4 dB—the rule breaks down when sources are close in level. When two sources differ by 3 dB or less, the contribution of the quieter source is substantial. When sources are within 1 dB, a competent acoustic analysis must include both. The calculator eliminates guesswork by computing the exact logarithmic sum.
Accounting for Phase and Directionality
The formulas used in this calculator assume incoherent sound addition, meaning the sources are uncorrelated and their pressure waves do not interfere constructively or destructively in a predictable way. This is the standard assumption for environmental noise and most practical acoustic scenarios. However, in specialized cases—such as loudspeaker arrays, coherent musical tones, or laboratory measurement rigs—sources can be in phase and their pressures add directly, requiring different mathematics. For typical noise assessment, environmental consulting, and building acoustics, the incoherent method used here is appropriate.
Frequency-Weighted Measurements and A-Weighting
Sound level meters often display A-weighted decibels (dBA) rather than linear dB SPL. A-weighting mimics human hearing by de-emphasizing very low and very high frequencies. When combining A-weighted levels, use the same logarithmic formula: convert each dBA value to linear, sum them, and convert back. However, be aware that A-weighted combination can be misleading if sources have very different spectral content. For most environmental and occupational noise, A-weighted addition is acceptable, but full-spectrum analysis may be warranted for specialized applications.
Using the Calculator
Enter the sound pressure levels of each source in decibels. You need at least two sources; additional sources up to five can be included by filling the optional fields. The calculator applies the logarithmic addition formula and displays the combined level immediately. This result represents the total sound pressure from all independent sources acting simultaneously. Use this value to assess whether the combined noise meets regulatory limits, design thresholds, or comfort standards. The calculation is transparent and can be verified with any scientific calculator or decibel addition reference.
Limitations and Practical Considerations
This calculator assumes steady-state sound levels and linear superposition of independent sources. Real-world noise is often fluctuating, with peaks and valleys. Sound meters measure time-averaged levels, and different weighting and time constants may be required for regulatory compliance. Measurement distance, reflection from surfaces, and outdoor versus indoor propagation all affect actual observed levels. The calculator provides the mathematical sum of the sources provided; interpreting that result in context of regulatory standards, architectural design, and actual field conditions requires professional judgment and often field measurement verification.
Additionally, this tool combines the root-mean-square (RMS) acoustic pressure of each source. If you have peak levels, impulse levels, or time-history data, those require specialized analysis beyond this calculator. For most general noise assessment in offices, factories, and residences, the RMS-based logarithmic addition is appropriate and this calculator provides the correct mathematical result.