Sound intensity

Inverse-square law

For a spherical sound wave, the intensity in the radial direction as a function of distance r from the centre of the sphere is given by $${\displaystyle I(r)=\frac{P}{A(r)}=\frac{P}{4\pi r^2},}$$ where

• P is the sound power;
• A(r) is the surface area of a sphere of radius r.

Thus sound intensity decreases as 1/r² from the centre of the sphere: $${\displaystyle I(r)\propto \frac{1}{r^2}.}$$

This relationship is an inverse-square law.

Sound intensity level

Sound intensity level (SIL) or acoustic intensity level is the level (a logarithmic quantity) of the intensity of a sound relative to a reference value.

It is denoted L_I, expressed in nepers, bels, or decibels, and defined by $${\displaystyle L_I=\frac{1}{2}\ln \left(\frac{I}{I_0}\right)\mathrm{N}\mathrm{p}={\log }_{10}\left(\frac{I}{I_0}\right)\mathrm{B}=10{\log }_{10}\left(\frac{I}{I_0}\right)\mathrm{d}\mathrm{B},}$$ where

• I is the sound intensity;
• I₀ is the reference sound intensity;
  • 1 Np = 1 is the neper;
  • 1 B = ⁠1/2⁠ ln(10) is the bel;
  • 1 dB = ⁠1/20⁠ ln(10) is the decibel.

The international standard reference sound intensity is $${\displaystyle I_0=1\mathrm{p}\mathrm{W}/{\mathrm{m}}^2=1\cdot {10}^{-12}\mathrm{W}/{\mathrm{m}}^2.}$$

being approximately the lowest sound intensity hearable by an undamaged human ear under room conditions. The proper notations for sound intensity level using this reference are L_{I /(1 pW/m²)} or L_I (re 1 pW/m²), but the notations dB SIL, dB(SIL), dBSIL, or dB_SIL are very common, even though they are not accepted with the SI.

The reference sound intensity I₀ is defined such that a progressive plane wave in air has the same value of sound intensity level (SIL) and sound pressure level (SPL), since $${\displaystyle I\propto p^2.}$$

The equality of SIL and SPL requires that $${\displaystyle \frac{I}{I_0}=\frac{p^2}{p_0^2},}$$ where p₀ = 20 μPa is the reference sound pressure in air.

For a progressive spherical wave, $${\displaystyle \frac{p}{c}=z_0,}$$ where z₀ is the characteristic specific acoustic impedance. Thus, $${\displaystyle I_0=\frac{p_0^{2}I}{p^2}=\frac{p_0^{2}pc}{p^2}=\frac{p_0^2}{z_0}.}$$

In air at ambient temperature, z₀ = 410 Pa·s/m, hence the reference value I₀ = 1 pW/m².

In an anechoic chamber which approximates a free field (no reflection) with a single source, measurements in the far field in SPL can be considered to be equal to measurements in SIL. This fact is exploited to measure sound power in anechoic conditions.

Measurement

Sound intensity is defined as the time averaged product of sound pressure and acoustic particle velocity. Both quantities can be directly measured by using a sound intensity p-u probe comprising a microphone and a particle velocity sensor, or estimated indirectly by using a p-p probe that approximates the particle velocity by integrating the pressure gradient between two closely spaced microphones.

Pressure-based measurement methods are widely used in anechoic conditions for noise quantification purposes. The bias error introduced by a p-p probe can be approximated by $${\displaystyle {\hat{I}}_n^{p-p}\simeq I_n-\frac{{\phi }_{\text{pe}}{p}_{\text{rms}}^2}{k\mathrm{\Delta }r\rho c}=I_n\left(1-\frac{{\phi }_{\text{pe}}}{k\mathrm{\Delta }r}\frac{p_{\text{rms}}^2/\rho c}{I_r}\right),}$$ where ${\displaystyle I_n}$ is the “true” intensity (unaffected by calibration errors), ${\displaystyle {\hat{I}}_n^{p-p}}$ is the biased estimate obtained using a p-p probe, ${\displaystyle p_{\text{rms}}}$ is the root-mean-squared value of the sound pressure, ${\displaystyle k}$ is the wave number, ${\displaystyle \rho }$ is the density of air, ${\displaystyle c}$ is the speed of sound and ${\displaystyle \mathrm{\Delta }r}$ is the spacing between the two microphones (assuming measurement in air). This expression shows that phase calibration errors are inversely proportional to frequency and microphone spacing and directly proportional to the ratio of the mean square sound pressure to the sound intensity. If the pressure-to-intensity ratio is large then even a small phase mismatch will lead to significant bias errors. In practice, sound intensity measurements cannot be performed accurately when the pressure-intensity index is high, which limits the use of p-p intensity probes in environments with high levels of background noise or reflections.

On the other hand, the bias error introduced by a p-u probe can be approximated by $${\displaystyle {\hat{I}}_n^{p-u}=\frac{1}{2}\mathrm{Re}\left\{P{\hat{V}}_n^{\ast }\right\}=\frac{1}{2}\mathrm{Re}\left\{P{V}_n^{\ast }{e}^{-j{\phi }_{\text{ue}}}\right\}\simeq I_n+{\phi }_{\text{ue}}{J}_n,}$$ where ${\displaystyle {\hat{I}}_n^{p-u}}$ is the biased estimate obtained using a p-u probe, ${\displaystyle P}$ and ${\displaystyle V_n}$ are the Fourier transform of sound pressure and particle velocity, ${\displaystyle J_n}$ is the reactive intensity and ${\displaystyle {\phi }_{\text{ue}}}$ is the p-u phase mismatch introduced by calibration errors. Therefore, the phase calibration is critical when measurements are carried out under near field conditions, but not so relevant if the measurements are performed out in the far field. The “reactivity” (the ratio of the reactive to the active intensity) indicates whether this source of error is of concern or not. Compared to pressure-based probes, p-u intensity probes are unaffected by the pressure-to-intensity index, enabling the estimation of propagating acoustic energy in unfavorable testing environments provided that the distance to the sound source is sufficient.