Decibel Calculator: dB to Intensity (W/m²), Pressure (Pa)

A sound wave in air is a longitudinal pressure disturbance: a pattern of compressions and rarefactions travelling at v = 343 m/s in 20°C air. The instantaneous acoustic pressure p(x, t) is what your eardrum (and a microphone diaphragm) actually feels. Sound pressure is small compared to atmospheric pressure: a normal conversation produces RMS pressures of about 20 mPa on top of a 101 kPa background, which is one part in five million. The ear is doing something extraordinary to detect that.

SPL is defined as 20 × log₁₀(p_rms / p_0) with p_0 = 20 μPa. The factor of 20 (not 10) is because pressure is a field quantity, not a power quantity. Intensity is proportional to pressure squared, and 10 × log(p²/p_0²) = 20 × log(p/p_0). Get the wrong factor and your numbers come out off by a factor of two in dB, which propagates through every later step. The ISO 1683 reference 20 μPa is the average threshold of human hearing at 1 kHz for young adults with undamaged ears; it's what 0 dB SPL means by definition.

Phase shows up in two diagnostic places. First, in interference between sources, which is the next-to-last section. Second, in time-weighting on the meter: "fast" weighting integrates over 125 ms, "slow" over 1000 ms, and impulse modes use 35 ms. A short bang reads higher on impulse than on slow because the slow average dilutes the peak across a longer window. OSHA-compliance measurements use slow A-weighting; that's the convention you stick to unless the standard tells you otherwise.

v = fλ holds in air at all audible frequencies, with v ≈ 343 m/s at 20°C. λ ranges from about 17 m (20 Hz, the bass-rumble end of audibility) to about 1.7 cm (20 kHz, the upper limit for young ears). What changes when you cross from air into water is v: it jumps to roughly 1480 m/s in fresh water at 20°C and 1500 m/s in sea water. Frequency is conserved across the boundary because the source is still oscillating at the same rate, so wavelength has to jump by the same factor v changes by. A 1 kHz tone with λ = 0.343 m in air arrives in water with λ ≈ 1.48 m. A diver hears it at the same pitch but with very different spatial structure.

Sound speed in a gas follows v = √(γRT/M), where γ ≈ 1.4 for diatomic air, R is the gas constant, T is absolute temperature, and M is molar mass. The room-temperature linearisation v ≈ 331 + 0.6T (T in °C) is good across normal weather. v doesn't depend on pressure (to first order), only on temperature and molecular composition. Helium has a much smaller M, so v_He ≈ 1007 m/s; that's why a balloon-helium voice sounds high. The vocal folds vibrate at the same f, but the resonant frequencies of the vocal tract scale with v/L, so they all shift up by a factor of about 3.

Intensity falls off with distance. For a small source radiating into free space, I ∝ 1/r². In dB: subtract 20 × log₁₀(r₂/r₁) when you double the distance. Doubling r reduces intensity by a factor of 4 and SPL by 6 dB. Tripling r reduces SPL by 9.5 dB. This is the inverse-square law, and it only holds in free field (no walls, far from the source compared to its size). Indoors the reverberant field eventually dominates and the level stops dropping; that's the room constant kicking in. Outdoor measurements past about ten times the source dimension can usually trust 1/r² to within a dB.

Speed-of-sound corrections almost never change SPL by more than a fraction of a dB for typical conditions, so for compliance and educational work you can use 343 m/s and ρc = 413 Pa·s/m without flinching. For high-altitude or extreme-temperature work (jet-engine test cells, aerospace acoustics), the corrections add up and the references in Pierce's "Acoustics" walk through them carefully.

Sound is a longitudinal compression wave in a fluid or solid. Light is a transverse electromagnetic wave. A wave on a string is a transverse mechanical wave. v = fλ is shared. The source of v differs.

The dB scale itself is not unique to acoustics. Decibels are used for any power-ratio measurement: signal-to-noise in electronics (with reference 1 mW for dBm), antenna gain over an isotropic radiator (dBi), optical power (dBm in fibre optics). What's unique to acoustics is the choice of reference (20 μPa for pressure, 10⁻¹² W/m² for intensity) and the A-weighting convention that approximates the ear's frequency response. Apply the wrong reference and you get the right number on the wrong scale; combining dB readings across domains is a common rookie mistake.

Light intensity uses the W/m² unit too, but the reference for "dB" in optics is application-specific (often 1 mW absolute, called dBm). There's no equivalent of dB SPL in optics because the eye's response to light isn't logarithmic in the same way the ear's is to sound; the photometric system uses lumens and lux instead of a logarithmic dB. The lesson: dB by itself is dimensionless, and you can't compare two dB values unless you know the reference each was measured against.

Quick Reference: Common Sound Levels

Values compiled from NIOSH, EPA, and acoustic measurement literature. dBA indicates A-weighted measurements that approximate human hearing response.

Decibels can't be added arithmetically because they represent logarithmic ratios. When combining N uncorrelated sound sources, convert each to intensity, sum the intensities linearly, then convert back. Two equal sources at 80 dB give 83 dB, not 160. Three equal sources at 80 dB give 84.8 dB. Ten give 90 dB. A hundred give 100 dB. The pattern: +10 × log₁₀(N) dB above one source for N identical incoherent sources.

Coherent sources (multiple speakers fed the same signal) add differently. Two perfectly in-phase sources at certain points give +6 dB; out-of-phase sources at the same point give cancellation. Real PA arrays use this on purpose: line arrays steer their main lobe by introducing controlled phase delays between elements, so the high-energy region points where the audience is and the level falls off rapidly toward the stage. The same physics that makes a two-slit interference pattern controls the directivity of a multi-driver speaker array.

For a worker rotating between machine workstations during a shift, you don't add the levels in parallel. You compute the dose as a fraction of allowed time at each station and sum the fractions. OSHA's allowed time is T = 8 / 2^((L − 90)/5) hours; total dose D = Σ (t_i / T_i). D > 1 means the 8-hour TWA exceeds 90 dBA and the worker is over the PEL. The math here is not parallel addition (the worker only experiences one station at a time), it's serial accumulation across the shift.

When two pure tones at slightly different frequencies are played together, the listener hears beats: a single tone at the average frequency that swells and fades at the difference frequency. Two tuning forks at 440 and 442 Hz give a 2 Hz beat, which a piano tuner uses to bring the strings into unison. The math: y_1 + y_2 = 2A cos(Δω t / 2) sin(ω̄ t), where ω̄ is the average and Δω the difference. The cos term is the slow envelope; the sin term is the fast carrier.

In a room, two loudspeakers playing the same signal create a standing-wave pattern at any frequency where the path-length difference to the listener is an integer or half-integer number of wavelengths. At 1 kHz in air, λ ≈ 34 cm, so moving your head by 17 cm can shift you from a peak to a null. This is the "comb filter" effect that wrecks stereo imaging in an untreated room. The fix is dense early reflections (carpet, acoustic panels) that smear the null pattern across many delays, plus careful speaker placement so the main listening position isn't sitting on a deep null.

For occupational measurements, phase isn't directly diagnostic, but instabilities in the meter reading often point to coherent-source effects. A measurement that drifts ±1 dB as you move the meter by 30 cm at 1 kHz is reading interference, not source level. The fix is averaging over a measurement track or using a directional microphone aimed at the source under test.

Human ears don't respond equally to all frequencies. The A-weighting curve standardised in IEC 61672-1 approximates the ear's frequency response at moderate sound levels (around 40 phon). A-weighted measurements (dBA) are required for occupational and most environmental noise regulations. The curve heavily attenuates frequencies below 500 Hz, slightly boosts 2 to 4 kHz where the ear is most sensitive, and rolls off above 8 kHz. Below are the IEC values at standard octave-band centres.

Values from IEC 61672-1:2013 Table 2. A-weighting (dBA) for occupational exposure and most regulatory work. C-weighting (dBC) for peak impact noise and hearing-protector selection. Z-weighting (unweighted, dBZ) for acoustic research and building acoustics analysis.

US occupational noise exposure is regulated by OSHA (mandatory) and guided by NIOSH recommendations (best practice). The agencies use different exchange rates and exposure limits, with NIOSH being more protective.

OSHA PEL: 90 dBA TWA, action level 85 dBA, 5-dB exchange rate, hearing-conservation program required at 85+ dBA. NIOSH REL: 85 dBA TWA, 3-dB exchange rate (equal-energy), based on 8% excess risk criterion. At 100 dBA, OSHA allows 2 hours; NIOSH allows 15 minutes.