Decibel
The decibel (dB) is a logarithmic unit used to express the ratio of a measured sound intensity to a reference intensity, typically the threshold of human hearing (I₀ = 10⁻¹² W/m²). Because the human ear responds to sound over an enormous range of intensities (about 10¹² to 1), a logarithmic scale compresses this range into a manageable 0–140 dB scale. The decibel is used extensively in acoustics, telecommunications, electronics, and audio engineering.
Key Formula
L = 10 × log10(I / I0)
LaTeX: L = 10 \log_{10}\left(\frac{I}{I_0}\right)
| Symbol | Meaning | Unit |
|---|---|---|
| L | Sound level | dB |
| I | Sound intensity being measured | W/m² |
| I_0 | Reference intensity (threshold of hearing = 10⁻¹² W/m²) | W/m² |
Worked Example
Problem
A factory machine produces a sound intensity of 10⁻⁴ W/m². Calculate the sound level in decibels. (I₀ = 10⁻¹² W/m²)
Solution
Step 1: Use L = 10 × log₁₀(I / I₀). Step 2: Compute the ratio: I / I₀ = 10⁻⁴ / 10⁻¹² = 10⁸. Step 3: L = 10 × log₁₀(10⁸) = 10 × 8 = 80 dB.
Answer
Sound level = 80 dB
Decibel Scale Reference Points
| Sound Source | dB Level | Intensity (W/m²) | Ratio to I₀ | Perceived Loudness |
|---|---|---|---|---|
| Threshold of hearing | 0 dB | 10⁻¹² | 10⁰ | Just audible |
| Library whisper | 30 dB | 10⁻⁹ | 10³ | Very quiet |
| Normal conversation | 60 dB | 10⁻⁶ | 10⁶ | Comfortable |
| Heavy traffic | 90 dB | 10⁻³ | 10⁹ | Loud, tiring |
| Jackhammer | 110 dB | 10⁻¹ | 10¹¹ | Very loud, harmful |
| Jet takeoff | 130 dB | 10 | 10¹³ | Pain threshold |
Interactive Tools
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Related Terms
Sound Intensity
Sound intensity is the power carried by a sound wave per unit area perpendicular to the direction of propagation, measured in watts per square metre (W/m²). It quantifies how much acoustic energy passes through a given surface each second and decreases with the square of the distance from a point source — the inverse square law. Sound intensity is the physical basis for the decibel scale and is central to audiology, architectural acoustics, and occupational noise exposure standards.
Acoustic Resonance
Acoustic resonance occurs when an object or air column vibrates at its natural frequency in response to an external sound source at that same frequency, resulting in a dramatic amplification of the sound. The phenomenon arises when standing waves are set up within the resonating object, with nodes and antinodes at fixed positions. Acoustic resonance is exploited in all musical instruments — strings, pipes, and percussion — as well as in architectural acoustics, industrial machinery fault detection, and medical imaging.
Ultrasound
Ultrasound refers to sound waves with frequencies above the upper limit of human hearing, typically above 20,000 Hz (20 kHz), extending to several gigahertz in specialised applications. Because of its high frequency and corresponding short wavelength, ultrasound can resolve fine structural details and is strongly absorbed or reflected by tissue boundaries, making it invaluable in medical diagnostics (obstetric scans, echocardiography), industrial non-destructive testing, sonar navigation, and the sonication used in cleaning and chemical processing.
From "deci-" (one-tenth, Latin "decimus") + "bel", named in honour of Alexander Graham Bell (1847–1922) by engineers at Bell Telephone Laboratories in the 1920s. The bel itself was the original unit; the decibel (1/10 of a bel) proved more practical.