Decibel Calculator

Calculate sound level, intensity, or pressure ratio using L = 10·log(I/I₀) with reference I₀ = 10⁻¹² W/m².

🔊 Sound📐 L = 10log(I/I₀)🎵 Decibels
Intensity (I) W/m²

I₀ = 10⁻¹² W/m² (threshold of hearing)

⚠️ Intensity must be positive.

What Is Decibels?

The decibel (dB) scale measures sound intensity logarithmically. The sound pressure level is: L = 10·log₁₀(I/I₀) dB, where I is the intensity in W/m² and I₀ = 10⁻¹² W/m² is the threshold of human hearing. This logarithmic scale compresses a range of 10¹² (a trillion) into 0–120 dB.

The logarithmic scale reflects human perception: the ear perceives loudness logarithmically. A 10 dB increase doubles the perceived loudness (and means 10× the intensity). A 3 dB increase doubles the intensity but increases perceived loudness by only about 23%. Two identical sound sources together increase level by only 3 dB (10·log₁₀(2) ≈ 3).

Common reference points: 0 dB = threshold of hearing; 40 dB = quiet library; 60 dB = normal conversation; 85 dB = prolonged exposure causes hearing damage; 110 dB = rock concert; 120 dB = threshold of pain; 140 dB = jet engine at 30 m. Every 10 dB represents a 10× intensity change.

For pressure levels: L_p = 20·log₁₀(P/P₀), where P₀ = 20 μPa (reference pressure). Since intensity ∝ pressure², the formula becomes L_p = 10·log(P/P₀)² = 20·log(P/P₀) — same numeric result. For electronics, dBm, dBV, and dBu measure power/voltage relative to defined references.

Formula Reference Table

Solve ForFormulaNotes
Sound pressure levelL = 10·log₁₀(I/I₀) dBI₀ = 10⁻¹² W/m²
Intensity from dBI = I₀ · 10^(L/10)I₀ = 10⁻¹² W/m²
Pressure levelL_p = 20·log₁₀(p/p₀) dBp₀ = 20 μPa
Combining two sourcesL_total = 10·log(10^(L1/10) + 10^(L2/10))Intensities add, not dB values
Distance attenuationL₂ = L₁ − 20·log(r₂/r₁)Inverse square law in free field
Perceived loudness+10 dB ≈ doubled loudnessHuman perception (Stevens' Law)

3 Worked Examples

Example 1
Library vs. Concert

A library at I = 10⁻⁸ W/m². A rock concert at I = 0.1 W/m². Find dB levels.

  • Library: L = 10·log(10⁻⁸/10⁻¹²) = 10·log(10⁴) = 40 dB
  • Concert: L = 10·log(0.1/10⁻¹²) = 10·log(10¹¹) = 110 dB
  • Ratio: 110 − 40 = 70 dB → 10⁷× more intense (10 million times)
✓ Library = 40 dB; Concert = 110 dB (10 million times more intense)
Example 2
Two Identical Sources

Two machines, each 80 dB. Find combined level.

  • I₁ = I₂ = 10⁻⁴ W/m²; I_total = 2 × 10⁻⁴
  • L = 10·log(2×10⁻⁴/10⁻¹²) = 10·log(2×10⁸)
  • L = 10·(log2 + 8) = 10·(0.301 + 8) = 83.0 dB
  • Two identical sources: +3 dB, not +80 dB
✓ Combined level = 83 dB (just 3 dB more)
Example 3
Distance Attenuation

Jackhammer at 100 dB at 1 m. Level at 10 m?

  • L₂ = L₁ − 20·log(r₂/r₁) = 100 − 20·log(10/1)
  • L₂ = 100 − 20 = 80 dB
  • At 100 m: L = 100 − 40 = 60 dB (normal conversation level)
✓ At 10 m: 80 dB; At 100 m: 60 dB

Real-World Applications

🎵
Music Production
Sound engineers work in dB because human hearing is logarithmic. Mixing consoles show faders in dB; 0 dBFS is full scale digital. A −6 dB fader means half amplitude (−20 dB = one-tenth amplitude, 10% level).
🔕
Noise Regulation
OSHA limits occupational noise exposure: 90 dB for 8 hr/day, 95 dB for 4 hr, 100 dB for 2 hr (each 5 dB doubles permissible exposure time). Ear protection required above 85 dB.
📻
Radio & Electronics
RF signal strength is measured in dBm (relative to 1 mW). A cellular base station transmits at +43 dBm; a phone receives at −90 to −110 dBm — a difference of over 130 dB, spanning 10 trillion times in power.
✈️
Airport Noise
Aircraft noise is measured in dB(A) (A-weighted for human hearing) and logged with distance. Airport neighbors track cumulative noise exposure (CNEL) to assess health impacts. Noise contours determine land use around airports.
🩺
Hearing Tests
Audiometry tests hearing threshold at each frequency using dB HL (hearing level). Normal hearing: 0–25 dB HL. Mild loss: 26–40 dB. Moderate: 41–55 dB. Severe: 71–90 dB. Profound loss > 90 dB HL. Hearing aids amplify by 20–60 dB.

Common Mistakes to Avoid

⚠️
Adding dB values directly

Two 70 dB sources are NOT 140 dB — they are 73 dB. dB values cannot be added arithmetically; intensities add. Use L_total = 10·log(10^(L1/10) + 10^(L2/10)) for combining.

⚠️
Using 10 vs 20 multiplier

Intensity: L = 10·log(I/I₀). Pressure: L_p = 20·log(p/p₀). The 20 arises because intensity ∝ p². Using 20 for intensity gives double the correct dB level.

⚠️
Forgetting the reference level

dB is always relative to a reference. Sound: I₀ = 10⁻¹² W/m². Electronics: dBm = relative to 1 mW. Without knowing the reference, a dB value is meaningless.

⚠️
Confusing 3 dB and 10 dB changes

3 dB = 2× intensity (barely audibly louder). 10 dB = 10× intensity (perceived as twice as loud). A 1 dB change is barely perceptible; 3 dB is just noticeable; 10 dB is a large perceptible change.

⚠️
Ignoring directivity and room effects

L = 10·log(I/I₀) assumes a point source in free field (anechoic room). In real rooms, reflections increase level (room gain); directional speakers have higher level on-axis; barriers reduce level in shadow zones.

Frequently Asked Questions

Why is the dB scale logarithmic? ▾
Human hearing spans 10¹² in intensity (0–120 dB) and perceives loudness logarithmically (Stevens' Law). A logarithmic scale compresses this vast range into a manageable 120 dB range and matches human perception. Doubling sound intensity adds 3 dB; making it feel twice as loud requires 10 dB.
How do you combine two different dB levels? ▾
Convert each to intensity (I = 10⁻¹² × 10^(L/10)), add intensities, then convert back: L_total = 10·log(10^(L1/10) + 10^(L2/10)). Two 80 dB sources: L = 10·log(2×10⁸) = 83 dB. A 90 dB and 80 dB source: L = 10·log(10⁹ + 10⁸) = 10·log(1.1×10⁹) = 90.4 dB — the louder source dominates.
What is the dB(A) scale? ▾
dB(A) applies A-weighting: frequency-dependent correction matching human hearing sensitivity (less sensitive at low and very high frequencies). dB(A) better predicts perceived loudness than unweighted dB. Occupational and environmental noise regulations use dB(A).
What causes the inverse square law for sound? ▾
Sound spreads outward in all directions from a point source — the surface area of a sphere grows as r². The same total power is spread over 4πr², so intensity I = P/(4πr²). Doubling distance → 4× area → 4× less intensity = −6 dB.
What is the threshold of pain? ▾
Approximately 120–130 dB (I ≈ 1 W/m²). Above this, the acoustic energy is sufficient to cause immediate physical discomfort and rapid hearing damage. Brief exposures above 140 dB can cause permanent hearing loss in seconds.
What is infrasound and why is it dangerous? ▾
Infrasound: sound below 20 Hz. High-intensity infrasound (120+ dB) is used as a non-lethal deterrent and in some industrial machinery. It can cause resonance in body cavities (chest, sinuses, eyeballs) at specific frequencies, causing disorientation and nausea. Natural sources include earthquakes, volcanoes, and large waterfalls.
How do noise-canceling headphones work? ▾
Active noise cancellation: microphone measures ambient noise, DSP generates an inverted (antiphase) sound wave, added to audio output. Constructive interference of noise + antiphase = near-zero amplitude = silence. Effective for steady low-frequency noise (10–1,000 Hz); less effective for sudden transients.
What is NIHL (Noise-Induced Hearing Loss)? ▾
Prolonged exposure to sounds > 85 dB causes permanent hair cell damage in the cochlea. The equal-energy rule: 85 dB for 8 hr = 88 dB for 4 hr = 91 dB for 2 hr = ... (3 dB per halving of time). NIHL is cumulative, irreversible, and preventable with hearing protection.

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