Sound intensity

Mathematical definition

Sound intensity, denoted I, is defined by

I = p v

where

p is the sound pressure;

v is the particle velocity.

Both I and v are vectors, which means that both have a direction as well as a magnitude. The direction of sound intensity is the average direction in which energy is flowing.

The average sound intensity during time T is given by

⟨ I ⟩ = 1 T ∫ 0 T p ( t ) v ( t ) d t . Also,Intensity of Sound = 2π²n²A²ρvWhere,n is frequency of sound, A is the Amplitude of sound wave, v is velocity of sound, and ρ is density of medium in which sound is traveling

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

I ( r ) = P A ( r ) = P 4 π r 2 ,

where

P is the sound power;

A(r) is the area of a sphere of radius r.

Thus sound intensity decreases as 1/r² from the centre of the sphere:

I ( r ) ∝ 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 dB, and defined by

L I = 1 2 ln ( I I 0 )   N p = log 10 ( I I 0 )   B = 10 log 10 ( I I 0 )   d 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 commonly used reference sound intensity in air is

I 0 = 1   p W / m 2 .

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 if they are not accepted by the SI.

The reference sound intensity I₀ is defined such that a progressive plane wave has the same value of sound intensity level (SIL) and sound pressure level (SPL), since

I ∝ p 2 .

The equality of SIL and SPL requires that

I I 0 = p 2 p 0 2 ,

where p₀ = 20 μPa is the reference sound pressure.

For a progressive spherical wave,

p v = z 0 ,

where z₀ is the characteristic specific acoustic impedance. Thus,

I 0 = p 0 2 I p 2 = p 0 2 p v p 2 = 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), the SIL can be taken as being equal to the SPL. This fact is exploited to measure sound power in anechoic conditions.

Measurement

One method of sound intensity measurement involves the use of two microphones located close to each other, normal to the direction of sound energy flow. A signal analyser is used to compute the crosspower between the measured pressures and the sound intensity is derived from (proportional to) the imaginary part of the crosspower.