Decibels Calculator

The Decibel Formula

The decibel is a dimensionless logarithmic ratio, not an absolute unit. For power quantities (watts, milliwatts): dB = 10 × log₁₀(P₂ / P₁). For amplitude quantities (voltage, pressure, sound pressure): dB = 20 × log₁₀(V₂ / V₁). The factor of 20 arises because power is proportional to the square of amplitude — so doubling voltage quadruples power, a +6 dB change in amplitude corresponds to +6 dB in voltage but +6 dB in power means a factor of 4.

Reference Levels and dB Variants

The suffix after "dB" identifies the reference level. dBm references 1 milliwatt — widely used in RF engineering and fibre optics (0 dBm = 1 mW, 30 dBm = 1 W). dBW references 1 watt. dBSPL references 20 micropascals (20 µPa), the approximate threshold of human hearing at 1 kHz — this is the standard for acoustic measurements. dBu and dBV reference voltage levels used in audio electronics (0.775 V and 1 V respectively).

Human Hearing Range

The threshold of human hearing is 0 dBSPL (20 µPa). Normal conversation sits around 60 dBSPL. Prolonged exposure above 85 dBSPL risks hearing damage according to OSHA guidelines. A loud rock concert reaches 110–120 dBSPL, and the pain threshold is approximately 130 dBSPL. Because the scale is logarithmic, 120 dBSPL represents a sound pressure 1 million times greater than the hearing threshold (10⁶ = 120 dB in the amplitude formula), illustrating why the dB scale is so useful for expressing the enormous dynamic range of human hearing.

Worked Examples

Example 1: Amplifier power ratio

A 1 W amplifier is replaced with a 100 W amplifier feeding the same speaker. The electrical power ratio is 100:1. Converting to dB: 10 × log₁₀(100 / 1) = 10 × 2 = +20 dB. Because perceived loudness roughly doubles every +10 dB, the new amp sounds about 4 times louder — far less than the 100× increase in raw power suggests.

Example 2: Adding two equal noise sources

Two HVAC units each measure 70 dBSPL at a property line. What is their combined level when both run?

Example 3: Distance attenuation from a point source

A speaker measures 90 dBSPL at 1 metre. Assuming free-field propagation (no reflections), what level remains at 10 metres? Sound pressure falls as 1/r for a point source, so L(r) = L(r₀) − 20 × log₁₀(r / r₀) = 90 − 20 × log₁₀(10) = 90 − 20 = 70 dB. Every doubling of distance drops sound pressure by 6 dB in free field. In a reverberant room, the actual drop is much smaller because reflected sound fills the space.

Example 4: Converting dBm to watts

A radio transmitter is rated 30 dBm. In watts: P = 1 mW × 10^30/10 = 1 mW × 1000 = 1 W. Likewise, 0 dBm = 1 mW, 20 dBm = 100 mW, 40 dBm = 10 W, and 50 dBm = 100 W. dBm is ubiquitous in RF link budgets because adding dB values is easier than multiplying raw powers.

Common Pitfalls

• Confusing 10log and 20log. Use 10log for power (watts, milliwatts). Use 20log for amplitude (voltage, sound pressure, current). Mixing them gives answers wrong by a factor of 2 in dB.
• Treating dB as an absolute unit. dB is always a ratio. Quoting "40 dB" with no reference is meaningless; you need dBm, dBSPL, dBV, or similar.
• Adding dB linearly. Two 60 dB sources do not sum to 120 dB. Convert to linear, add, convert back — or use the shortcut that doubling equal incoherent sources adds 3 dB.
• Applying the inverse-square law indoors. Free-field 6 dB-per-doubling assumes no reflections. Inside a room, the reverberant field flattens distance attenuation beyond the critical distance — often only 1–2 metres from the source.
• Assuming +10 dB equals "twice as loud." Perceived loudness is an auditory perception, not a physical quantity. The rough rule is +10 dB ≈ 2× perceived loudness, but individual perception varies and depends on frequency content.

Frequently Asked Questions

Why use logarithms for sound and signal levels?

The dynamic range of human hearing spans from 20 µPa (threshold) to around 20 Pa (pain threshold) — a factor of one million. Expressing ratios this large in linear units is unwieldy, but on a logarithmic dB scale the same range compresses neatly into 0 to 120 dB. The ear itself responds roughly logarithmically, so the scale matches perception.

How loud is "dangerous"?

OSHA sets permissible exposure at 90 dBA for 8 hours per day, halving the allowable exposure for every 5 dB above that. NIOSH is more conservative at 85 dBA and 3 dB per doubling. Brief exposure at 130 dBA or above (fireworks, gunshots) can cause immediate permanent damage. Hearing protection is rated in dB of attenuation (NRR).

Do dB values from different references compare directly?

No. 30 dBm (RF power, referenced to 1 mW) is 1 W; 30 dBSPL (sound, referenced to 20 µPa) is a whisper-level sound pressure. They share the dB scale but measure different physical quantities against different references, so comparing them directly is a category error.

What is dBA and why is it weighted?

The A-weighting curve adjusts a flat sound-pressure measurement to match the frequency sensitivity of human hearing, which rolls off significantly below 500 Hz and above 6 kHz. A reading in dBA emphasises the frequencies people actually find loud. Most noise regulations and hearing-damage guidelines are expressed in dBA because that is closer to perceived loudness than unweighted dB.

How does active noise cancelling affect dB?

An ANC headphone generates an inverted copy of the noise waveform. When the inverted signal is phase-locked to the real noise, they interfere destructively and the net pressure falls. A typical consumer ANC headphone achieves 20–35 dB of reduction in the 100–500 Hz band where most aircraft and HVAC rumble lives, but much less at higher frequencies where passive isolation takes over.

Related Calculators

• → Frequency Calculator — wavelength and period for any acoustic or RF signal.
• → Impedance Calculator — voltage/current ratios drive the 20log form of dB.
• → Power Calculator — convert raw watts into dBm or dBW.
• → Ohm's Law — the foundation for voltage-to-power conversions.

View all Science Calculators →

Disclaimer

This calculator is provided for educational and informational purposes only. While we strive for accuracy, users should verify all calculations independently. We are not responsible for any errors, omissions, or damages arising from the use of this calculator.