Physics · Classical Mechanics · Oscillations & Waves
Sound Intensity Calculator
Calculates sound intensity, intensity level in decibels, and sound pressure from acoustic power and distance.
Calculator
Formula
I is sound intensity in W/m², P is acoustic power in watts, r is distance from the source in metres, L_I is the intensity level in decibels (dB), I_0 = 10^{-12} W/m² is the reference intensity (threshold of human hearing), p is the RMS sound pressure in pascals, \rho is the air density (≈ 1.225 kg/m³ at 20 °C), and c is the speed of sound in air (≈ 343 m/s at 20 °C).
Source: ISO 1683:2015 — Acoustics: Preferred reference values for acoustical and vibratory levels; Kinsler et al., Fundamentals of Acoustics, 4th ed.
How it works
Sound intensity quantifies the power carried by a sound wave per unit area perpendicular to the direction of propagation. When a source radiates power P uniformly in all directions — a so-called omnidirectional or monopole source — that power spreads over the surface area of an expanding sphere. At distance r from the source, the sphere has area 4πr², so the intensity falls off as 1/r². This relationship, known as the inverse square law, is fundamental to architectural acoustics, environmental noise modelling, and speaker system design. Doubling the distance from a point source reduces the intensity by a factor of four, corresponding to a 6 dB drop in level.
The intensity level L_I is defined on a logarithmic scale relative to the international reference intensity I₀ = 10⁻¹² W/m², which corresponds approximately to the threshold of human hearing at 1 kHz. The formula L_I = 10 log₁₀(I / I₀) compresses the enormous dynamic range of audible sound — spanning twelve orders of magnitude in intensity — into a manageable decibel scale from 0 dB (threshold of hearing) to around 194 dB (near a rocket engine). The RMS sound pressure p is related to intensity through the specific acoustic impedance of the medium: p = √(ρcI), where ρ is air density and c is the speed of sound. At standard conditions (20 °C, 1 atm), ρc ≈ 420 Pa·s/m (rayl), and I₀ corresponds to p₀ = 20 μPa, which is the reference pressure for the more commonly reported sound pressure level (SPL).
Practical applications of this calculator include setting safe exposure limits for workers near industrial machinery (OSHA permissible levels), predicting noise levels from wind turbines or construction sites at receptor locations, designing home theatre systems for optimal listening distance, and validating acoustic simulation software outputs against measured data. Sound intensity measurements are also used in near-field acoustic holography and sound power determination per ISO 9614 standards.
Worked example
Suppose a small loudspeaker has an acoustic power output of P = 0.01 W (10 mW) and a listener is seated r = 5 m away. Air conditions are standard: ρ = 1.225 kg/m³ and c = 343 m/s.
Step 1 — Sound Intensity: The surface area of a sphere of radius 5 m is 4π(5)² = 4π × 25 ≈ 314.16 m². Dividing the power by this area: I = 0.01 / 314.16 ≈ 3.18 × 10⁻⁵ W/m².
Step 2 — Intensity Level: L_I = 10 log₁₀(3.18 × 10⁻⁵ / 10⁻¹²) = 10 log₁₀(3.18 × 10⁷) = 10 × 7.502 ≈ 75.0 dB. This is roughly equivalent to the loudness of a normal conversation at close range.
Step 3 — RMS Sound Pressure: ρc = 1.225 × 343 ≈ 420.2 Pa·s/m. Then p = √(420.2 × 3.18 × 10⁻⁵) = √(0.01336) ≈ 0.1156 Pa, which is about 5,780 times the 20 μPa reference pressure — consistent with the ~75 dB level.
If the listener moves to 10 m, the distance doubles, so the intensity drops by a factor of 4 to ≈ 7.96 × 10⁻⁶ W/m² and the level falls by exactly 6 dB to ≈ 69.0 dB, confirming the inverse square law.
Limitations & notes
This calculator assumes a point source radiating uniformly into a free field (anechoic, unbounded environment). Real-world sound fields deviate significantly from this model: reflections from walls, floor, and ceiling in enclosed spaces create reverberant energy that sustains sound levels well above the free-field prediction at large distances; directional sources (such as horns or arrays) concentrate energy in specific directions, meaning the 4πr² denominator must be replaced by a directivity-corrected solid angle; near-field effects close to extended sources (where r is comparable to source dimensions or wavelength) invalidate the point-source approximation; and atmospheric absorption at high frequencies (above ~2 kHz) causes additional attenuation beyond 1/r² at long distances, which is not accounted for here. Air density and speed of sound vary with temperature and humidity — users should adjust these inputs accordingly. For indoor or reverberant environments, use the Sabine or Eyring room acoustics models instead of the free-field formula.
Frequently asked questions
What is the difference between sound intensity and sound pressure level?
Sound intensity (W/m²) is a vector quantity describing energy flow per unit area through a surface, while sound pressure level (dB SPL) is a scalar logarithmic measure of the RMS pressure fluctuation relative to 20 μPa. In a free plane wave they are directly related through the acoustic impedance ρc, but in reverberant or complex fields they can differ substantially. Intensity is direction-dependent and useful for locating noise sources; SPL is what microphones and sound level meters typically report.
Why does sound level drop by 6 dB every time the distance doubles?
Because intensity follows the inverse square law (I ∝ 1/r²), doubling r reduces I by a factor of 4. On the decibel scale, 10 log₁₀(1/4) = −6.02 dB, which rounds to −6 dB. This 6 dB per doubling rule is a practical rule of thumb for point sources in free-field conditions, and it helps engineers quickly estimate noise levels at different receptor distances without repeating full calculations.
What is the reference intensity I₀ = 10⁻¹² W/m² based on?
The reference intensity I₀ = 10⁻¹² W/m² corresponds to the approximate threshold of human hearing at 1 kHz for a young, healthy adult with no hearing damage. It was standardised internationally (ISO 1683) to provide a universal baseline for the decibel scale. At this intensity in air, the RMS pressure equals the reference pressure p₀ = 20 μPa, ensuring that intensity level and sound pressure level values agree for plane waves in a free field.
How do I account for a directional sound source?
For a directional source, replace the omnidirectional formula I = P/(4πr²) with I = P·Q/(4πr²), where Q is the directivity factor (dimensionless). A source placed flush against a flat reflective surface has Q = 2 (hemisphere); in a corner it can reach Q = 8. Directivity index DI = 10 log₁₀(Q) in dB. The calculator above assumes Q = 1 (free field, all directions equally), so for mounted or directional sources, multiply the computed intensity by the appropriate Q value.
At what sound intensity level does hearing damage occur?
According to OSHA and NIOSH guidelines, prolonged exposure to levels above 85 dB(A) increases the risk of noise-induced hearing loss. Instantaneous exposure above 140 dB can cause immediate, permanent damage. At 120–130 dB (the pain threshold), even brief exposure is harmful. The intensity corresponding to 85 dB is I = I₀ × 10^(85/10) ≈ 3.16 × 10⁻⁴ W/m², roughly 316 billion times the hearing threshold intensity.
Last updated: 2025-01-15 · Formula verified against primary sources.