Sound power

Mathematical definition

Sound power, denoted P, is defined by $${\displaystyle P=\mathbf{f}\cdot \mathbf{v}=Ap\mathbf{u}\cdot \mathbf{v}=Apv}$$ where

f is the sound force of unit vector u;

v is the particle velocity of projection v along u;

A is the area;

p is the sound pressure.

In a medium, the sound power is given by $${\displaystyle P=\frac{A{p}^2}{\rho c}\cos \theta ,}$$ where

A is the area of the surface;

ρ is the mass density;

c is the sound velocity;

θ is the angle between the direction of propagation of the sound and the normal to the surface;

p is the sound pressure.

For example, a sound at SPL = 85 dB or p = 0.356 Pa in air (ρ = 1.2 kg⋅m⁻³ and c = 343 m⋅s⁻¹) through a surface of area A = 1 m² normal to the direction of propagation (θ = 0°) has a sound energy flux P = 0.3 mW.

This is the parameter one would be interested in when converting noise back into usable energy, along with any losses in the capturing device.

Relationships with other quantities

Sound power is related to sound intensity: $${\displaystyle P=AI,}$$ where

A stands for the area;

I stands for the sound intensity.

Sound power is related sound energy density: $${\displaystyle P=Acw,}$$ where

c stands for the speed of sound;

w stands for the sound energy density.

Sound power level

Sound power level (SWL) or acoustic power level is a logarithmic measure of the power of a sound relative to a reference value.
Sound power level, denoted L_W and measured in dB, is defined by: $${\displaystyle L_W=\frac{1}{2}\ln \left(\frac{P}{P_0}\right)\mathrm{N}\mathrm{p}={\log }_{10}\left(\frac{P}{P_0}\right)\mathrm{B}=10{\log }_{10}\left(\frac{P}{P_0}\right)\mathrm{d}\mathrm{B},}$$ where

P is the sound power;

P₀ is the reference sound power;

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 power in air is $${\displaystyle P_0=1\mathrm{p}\mathrm{W}.}$$ The proper notations for sound power level using this reference are L_{W/(1 pW)} or L_W (re 1 pW), but the suffix notations dB SWL, dB(SWL), dBSWL, or dB_SWL are very common, even if they are not accepted by the SI.

The reference sound power P₀ is defined as the sound power with the reference sound intensity I₀ = 1 pW/m² passing through a surface of area A₀ = 1 m²: $${\displaystyle P_0=A_0{I}_0,}$$ hence the reference value P₀ = 1 pW.

Relationship with sound pressure level

The generic calculation of sound power from sound pressure is as follows: $${\displaystyle L_W=L_p+10{\log }_{10}\left(\frac{A_S}{A_0}\right)\mathrm{d}\mathrm{B},}$$ where A_S defines the area of a surface that wholly encompasses the source. This surface may be any shape, but it must fully enclose the source.

In the case of a sound source located in free field positioned over a reflecting plane (i.e. the ground), in air at ambient temperature, the sound power level at distance r from the sound source is approximately related to sound pressure level (SPL) by $${\displaystyle L_W=L_p+10{\log }_{10}\left(\frac{2\pi r^2}{A_0}\right)\mathrm{d}\mathrm{B},}$$ where

L_p is the sound pressure level;

A₀ = 1 m²;

2πr² defines the surface area of a hemisphere; and

r must be sufficient that the hemisphere fully encloses the source.

Derivation of this equation: $${\displaystyle \begin{array}{cc}{L}_W& =\frac{1}{2}\ln \left(\frac{P}{P_0}\right)\\ & =\frac{1}{2}\ln \left(\frac{AI}{A_0{I}_0}\right)\\ & =\frac{1}{2}\ln \left(\frac{I}{I_0}\right)+\frac{1}{2}\ln \left(\frac{A}{A_0}\right).\end{array}}$$ For a progressive spherical wave,

z₀ = p/v is the characteristic specific acoustic impedance;

A = 4πr² is the surface area of a sphere.

Consequently, $${\displaystyle I=pv=\frac{p^2}{z_0},}$$ and since by definition I₀ = p₀²/z₀, where p₀ = 20 μPa is the reference sound pressure, $${\displaystyle \begin{array}{cc}{L}_W& =\frac{1}{2}\ln \left(\frac{p^2}{p_0^2}\right)+\frac{1}{2}\ln \left(\frac{4\pi r^2}{A_0}\right)\\ & =\ln \left(\frac{p}{p_0}\right)+\frac{1}{2}\ln \left(\frac{4\pi r^2}{A_0}\right)\\ & =L_p+10{\log }_{10}\left(\frac{4\pi r^2}{A_0}\right)\mathrm{d}\mathrm{B}.\end{array}}$$

The sound power estimated practically does not depend on distance. The sound pressure used in the calculation may be affected by distance due to viscous effects in the propagation of sound unless this is accounted for.