Propagation of Sound Outdoors - Attenuation vs. Distance

The energy in the propagation direction of the sound is inversely proportional to the increasing surface area the sound propagates through and can be expressed as

L_p = L_N + 10 log [(Q / 4 π) (1 / r₁²+ 1 / r₂²+ ...+1 / r_n²) + 4 / R]                             (1)

or modified for a single source:

L_p = L_N + 10 log (Q / (4 π r²) + 4 / R)                                (1b)

where

L_p = sound pressure level (dB)

L_N  = sound power level source in decibel (dB)

Q = directivity coefficient (1 uniform spherical, 2 uniform half spherical (single reflecting surface), 4 uniform radiation over 1/4 sphere (two reflecting surfaces, corner)

r = distance from source  (m)

R = room constant (m²)

Single Sound Source - Spherical Propagation

With uniform spherical radiation (1) can be modified to express the sound pressure level from a single sound source as

L_p = L_N + 10 log (1 / (4 π r²) 

   = L_N - 10 log (4 π r²)                                   (2)

since

Q  = 1

R ≈ ∞

(2) can also be expressed as: 

L_p = L_N - 20 log (r) + K'                            (2b)

where 

K' = -11 (single sound source and spherical radiation) 

Single Sound Source - Hemi-Spherical Propagation

When the sound source propagates hemi-spherically with the source near ground, the constant can be set to 

• K' = - 8

Note! When the distance - r - from a power source doubles, the sound pressure level decreases with 6 dB. This relationship is also known as the inverse square law.

Other factors that affects the radiation of the sound are the direction of the source, barriers between the source and the receiver and atmospheric conditions. Equation (1) can be modified to: 

L_p = L_N - 20 log r + K' + DI - A_a - A_b                                 (3)

where

DI = directivity index

A_a = attenuation due to atmospheric conditions

A_b = attenuation due to barriers

Linear Sound Source

With a linear sound source, like a road or high-way with heavy traffic, (1) can be summarized (integrated) to express the sound pressure as

L_p = L_N - 10 log (4 π r)                                   (4)

Note! When the distance - r - from a linear power source doubles, the sound pressure level decreases with 3 dB.

Temperature ^oC
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^oF

Length m
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naut miles

Area m²
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Volume m³
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Weight kg_f
N
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Velocity m/s
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knots

Pressure Pa
bar
mm H₂O
kg/cm²
psi
inches H₂O

Flow m³/s
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