Sound - Attenuation and the Directivity Coefficient

For a continuing sound source the sound level in a room is the sum of direct and reverberant sound. The sound pressure for a receiver can be expressed as

L_p = L_N + log(D / (4 π r²) + 4 / R)                                        (1)

where

L_p = received sound pressure level (dB)

L_N = sound power level from source (dB)

D = directivity coefficient

R = room constant (m² Sabine)

π = 3.14 .....

r = distance from source (m)

Directivity coefficient - D

The figure below can be used to approximate the Directivity coefficient - D - for typical locations of the receiver and the sound source

(1) can be transformed to express the difference between the received sound pressure level and emitted sound power level - the attenuation - as: 

L_p - L_N =  10 log(D / (4 π r²) + 4 / R)                                                                 (2)

Room Sound Attenuation Calculator

The attenuation (L_p - L_N) can also be estimated from the diagram below by modifying (2) to be expressed as the ratio between "distance r between source and receiver" and "square of the directivity coefficient D" - and the total room sound absorption in m² Sabine.

Example - Sound Attenuation in a Room 

For a room the acoustic property total room sound absorption is 1000 m² Sabine.  For a listener in the middle of the room with a directivity constant 1 and at distance 1 m from the sound source - the ratio

r / D^1/2 = 1 / (1 m)^1/2

          = 1

From the diagram above the attenuation can estimated to approximately

10 dB