Sound Intensity and Intensity Level (Li): Directional Acoustics Explained
Pair this piece with the decibel calculator and the sound power level explainer to connect directional measurements, total energy emission, and regulatory metrics.
Use the content alongside the ISO 80000-8 overview and sound pressure level guide for consistent terminology across acoustic diagnostics and reporting.
Overview
Sound intensity I is the vector rate of acoustic energy flow per unit area (W·m⁻²), defined as the time average of the instantaneous product of acoustic pressure p(t) and particle velocity u(t):
I = ⟨p(t) · u(t)⟩
The corresponding intensity level, Li, is:
L_i = 10 log10(|I| / I_0) dB, I_0 = 1 pW·m^-2
ISO 80000-8 defines these quantities and units, distinguishing intensity from pressure-only measures by its directional and energetic meaning.
Historical Context
While pressure has long been straightforward to measure, direct measurement of intensity became practical with the advent of phase-accurate dual-microphone (p-p) and pressure–velocity (p-u) probes and digital signal processing. This enabled energy-flow mapping, source identification on complex machinery, and robust sound-power determination in challenging environments.
Conceptual Foundations
Active vs. reactive intensity
The active component represents net energy flow; the reactive component oscillates without net transport (e.g., near sources and in standing waves). Only the active part contributes to radiated power:
P = ∬_S I_active · dS
Relationships in plane waves
In a plane progressive wave, pressure and particle velocity are in phase and related by the characteristic impedance Z0 = ρ0 c:
|I| = p_rms^2 / (ρ_0 c)
implying a direct connection between SPL and intensity in the far field. In reactive or near-field regions, this relation breaks down due to phase differences and evanescent components.
Specific acoustic impedance
z = p/u (Pa·s·m⁻¹) characterizes the local relationship between pressure and velocity; deviations from Z0 indicate standing waves, boundary layers, or resonant behavior.
Measurement Techniques
p-p (two-microphone) intensity probes
Two phase-matched microphones separated by a small distance estimate the pressure gradient and infer particle velocity via Euler’s equation, enabling computation of intensity. Accurate phase calibration, spacing selection relative to wavelength, and phase-mismatch correction are critical.
p-u probes
Combine a pressure microphone with a particle-velocity sensor (e.g., hot-wire or MEMS-based). These offer extended low-frequency performance and compact probe heads, at the cost of additional calibration demands.
Signal processing
Narrow-band or 1/3-octave processing yields vector intensity spectra, enabling frequency-by-frequency diagnosis of energy pathways. Time averaging, transient capture, and order tracking support rotating machines and nonstationary phenomena.
Good Practice and Uncertainty
- Calibration of microphone sensitivities, phase, and velocity sensor transfer functions is mandatory.
- Field indicators (pressure–intensity index, field non-uniformity) quantify suitability of measurements in the presence of background or strong reflections.
- Scanning protocols (raster grids, conformal paths) and surface closure strategies ensure proper integration of I · dS.
- Environmental control (temperature, flow, and turbulence) reduces bias; windscreens and flow shields mitigate convective terms that can contaminate intensity.
Applications
Sound-power determination
Intensity integration over a measurement surface yields sound power even in high background noise or partially reverberant spaces, where SPL-based methods struggle. This is invaluable for in-situ product testing.
Source localization and ranking
Vector intensity arrows plotted over machinery surfaces reveal leak paths, panel radiation hot spots, and dominant edges on fans, compressors, or enclosures—guiding targeted design fixes.
Building and environmental acoustics
Intensity maps diagnose flanking transmission and façade leakage, distinguishing airborne from structure-borne contributions. Outdoors, intensity supports directional measurements separated from ambient noise.
Electroacoustics and transducers
Loudspeaker characterization benefits from intensity-derived radiation efficiency and directivity estimates, capturing true radiated power beyond pressure-only metrics.
Why Intensity and Li Matter
Intensity unifies magnitude and direction of acoustic energy, connecting field measurements directly to physical power. The associated level Li (dB re 1 pW·m⁻²) communicates results compactly across decades of dynamic range. By formalizing symbols, units, and reference values, ISO 80000-8 ensures that intensity-based diagnostics, power quantification, and design decisions are traceable, comparable, and physically grounded across laboratories and industries.
Tools and Further Reading
Related article
Sound Pressure Level (Lp): Definition, Measurement, and Applications
Relate vector intensity back to pressure-based level reporting in the far field.
Related article
Sound Power Level (Lw): Definition, Methods, and Use Cases
See how intensity integrations close the loop on total sound power determination.
Related article
ISO 80000-8: Quantities and Units of Acoustics
Confirm symbol sets, reference values, and derived quantity relationships for acoustic fields.
Related article
The Decibel (dB): Logarithmic Quantities, and Ratio Levels
Align logarithmic expressions for intensity levels with broader ratio conventions.
Calculator
Noise Exposure Limit Calculator
Estimate safe exposure times once intensity measurements are converted into A-weighted SPL.
Calculator
Decibel to Power Percentage Calculator
Relate Li results to linear energy flow changes when optimizing acoustic treatments.
Calculator
Logarithm Base Conversion
Double-check logarithmic manipulations used in intensity spectra reporting.