Sound Intensity Level (dB)
Sound intensity expressed in decibels relative to the hearing threshold.
Formula
Variables
Example
I = 10⁻⁶ W/m²: L = 10 log(10⁶) = 60 dB
How it works
This formula converts a physical sound intensity into a logarithmic scale called decibels, which matches how humans perceive loudness: equal ratios in intensity feel more similar than equal differences. The reference intensity I_0 = 10^-12 W/m² is chosen to represent the approximate threshold of human hearing, so the result tells you how many "power levels" above that baseline the sound is. The factor of 10 is used because decibels are defined for power-like quantities such as intensity.
When to use it
Use this when you know a sound’s intensity in watts per square metre and want its level in decibels. It appears in acoustics, noise measurement, audio engineering, and any problem comparing sound strengths on a perceptual scale.
Worked example, step by step
Given I = 3.0 × 10^-8 W/m² and I_0 = 1.0 × 10^-12 W/m² Step 1: Form the ratio I / I_0 = (3.0 × 10^-8) / (1.0 × 10^-12) = 3.0 × 10^4 Step 2: Take the base-10 logarithm: log10(3.0 × 10^4) = log10(3.0) + 4 ≈ 0.4771 + 4 = 4.4771 Step 3: Multiply by 10: L = 10 × 4.4771 = 44.771 dB Final answer: L ≈ 44.8 dB
Common mistakes to avoid
- Using ln instead of log10
- Forgetting that the formula uses an intensity ratio, not an absolute intensity
- Using 20 log10(I/I0) instead of 10 log10(I/I0) for intensity
Did You Know?
Because it is logarithmic, a 10 dB rise sounds roughly twice as loud but carries ten times the power.
Share this formula
More in Music & Acoustics
View allEqual-Temperament Note Frequency
IntermediateFrequency of a note n semitones from a reference pitch.
Octave Frequency Ratio
BasicAn octave above a note has exactly double its frequency.
Combining Two Sound Levels
AdvancedTotal level when two sound sources play together.
Wave Speed
BasicSound speed equals frequency times wavelength.
Vibrating String Frequency
AdvancedFundamental pitch of a string from length, tension and mass.
Harmonic Series
IntermediateOvertones of a note are whole-number multiples of the fundamental.