EN-02Environmental & Occupational Noise

Sound Power ⇄ Sound Pressure

Predict sound pressure level from sound power, directivity and room absorption. Find the critical distance.

Source

Sound power level L_W

W = 0.01 watts acoustic

Directivity Q

Distance r

Environment

Total room surface S

m²

Mean absorption coefficient ᾱ

Sound pressure level

86.4dB

Direct field only

78.0dB

Room constant R

106m²

Critical distance

2.05m

Source position

Level vs distance

Direct fieldDirect + reverberantReverberant field

§ Theory of operation

Power is the cause, pressure the effect

Sound power level $L_W$ is a property of the source, independent of where you measure. Sound pressure level $L_p$ depends on distance, the source's surroundings and the room. Manufacturers declare $L_W$ precisely so that $L_p$ can be predicted for any installation.

Free field

L_p = L_W + 10\,\log_{10}\!\left(Q\,/\,4\pi r^2\right)

The directivity factor Q accounts for reflecting planes close to the source: a machine standing on a hard floor radiates into a hemisphere, doubling the intensity (+3 dB). In a corner the energy is confined to one eighth of a sphere (+9 dB).

In a room: direct + reverberant field

\begin{aligned} L_p &= L_W + 10\,\log_{10}\!\left(Q/4\pi r^2 + 4/R\right) \\ R &= S\,\bar{\alpha}\,/\,(1 - \bar{\alpha}) \qquad (\text{room constant, }\mathrm{m}^2) \end{aligned}

Close to the source the direct field dominates and the level falls 6 dB per doubling of distance. Further away, the diffuse reverberant field takes over and the level becomes constant. They are equal at the critical distance:

r_c = \sqrt{Q\,R\,/\,16\pi} \approx 0.14\,\sqrt{Q\,R}

Adding absorption only helps receivers beyond $r_c$ ; a worker standing right next to a machine gains little from an acoustic ceiling. Doubling the absorption reduces the reverberant level by 3 dB.

Limitations

The diffuse-field model assumes a reasonably proportioned room with evenly distributed absorption. In long, flat or very absorbent spaces (open-plan offices, factories with low ceilings) the reverberant level keeps decaying with distance.