Intermediate

Sound Intensity Level

Loudness in decibels relative to threshold.

Formula

\beta = 10\log_{10}\frac{I}{I_0}

Variables

IIntensity

I_0Reference (1e-12 W/m²)

How it works

Sound intensity level converts a physical intensity into a logarithmic scale measured in decibels, which matches how our ears perceive changes in loudness. The factor of 10 appears because intensity is a power per area quantity, and the logarithm compresses the enormous range of audible intensities into manageable numbers. The reference intensity, 1.0 × 10^-12 W/m², is chosen to represent the approximate threshold of hearing for a normal young ear.

When to use it

Use this formula when you need to express or compare sound strength in decibels, such as for speakers, traffic noise, classroom acoustics, or hearing safety. It is also useful whenever a problem gives intensity in W/m² and asks for the corresponding sound level.

Worked example, step by step

A machine produces a sound intensity of 3.0 × 10^-6 W/m². Use β = 10 log10(I / I0) Substitute the values: β = 10 log10((3.0 × 10^-6) / (1.0 × 10^-12)) Simplify the ratio: β = 10 log10(3.0 × 10^6) Split the logarithm: β = 10[log10(3.0) + log10(10^6)] Calculate: β = 10(0.4771 + 6) β = 10(6.4771) β = 64.771 dB Final answer: β ≈ 64.8 dB

Common mistakes to avoid

Using 20 log10 instead of 10 log10; 20 log10 is for field quantities like pressure, not intensity
Forgetting to divide by the reference intensity I0 before taking the logarithm
Putting the intensity in the wrong units, such as watts instead of watts per square metre

Did You Know?

The decibel scale is logarithmic, so a 10 dB rise is actually ten times the sound intensity.

sound decibel intensity

Sound Intensity Level

Formula

Variables

How it works

When to use it

Worked example, step by step

Common mistakes to avoid

Did You Know?

Share this formula

More in Optics & Waves

Wave Speed

Wave Period

Snell's Law

Refractive Index

Critical Angle

Thin Lens Equation