Sound - Abatement vs. the Distance from Source

The sound pressure from a source is reduced with distance from source.

Spherical Distance

Sound pressure in spherical distance from a noise source can be calculated as:

p² = ρ c N / (4 π r²)              (1)

where

p = sound pressure (Pa, N/m²)

ρ = density of air (kg/m³)

c = speed of sound (m/s)

N = sound power (W)

π = 3.14

r = distance from source (m)

Half Spherical Distance

Sound pressure in half spherical distance from a source can be expressed as:

p² = ρ c N / (4 π r² / 2)

    = 2 ρ c N / (4 π r²)                (2)

A more generic expression for sound pressure in distance from source can be formulated to:

p² = D ρ c N / (4 π r²)              (3)

where

D = directivity coefficient (1 spherical, 2 half spherical)

The directivity coefficient depends on several parameters - the position and direction of the source, the room and the surrounding area, etc.

The Sound Pressure Level - L_p - can be expressed logarithmic in decibels as:

L_p = 20 log (p / p_ref)

     = 20 log ((D ρ c N / (4 π r²))^1/2 / p_ref)

      = 20 log (1 / r (D ρ c N / (4 π))^1/2 / p_ref)                 (4)

where

L_p = sound pressure level (dB)

p_ref = 2 10^-5 - reference sound pressure (Pa)

Note! - a doubling of the distance from a sound source - will reduce the sound pressure level - L_p - with 6 decibels.

Sound Pressure Level Calculator

Example - Sound Pressure from a Wood Planer

The sound power generated from a wood planer is estimated to 0.01 W. The sound pressure in distance 10 m from the planner can be calculated as

L_p = 20 log ((D ρ c N / (4 π r²))^1/2 / p_ref )

   = 20 log (2 (1 kg/m³) (331.2 m/s) (0.01 W) / (4 π (10 m)²))^1/2 / (2 10^-5 Pa))

   = 71 dB

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