Environmental noise assessment for planning permissions is one of the most common applications of acoustics in practice. This article works through a complete ISO 9613-2 calculation for an industrial source at 200 m from a residential receiver, including all five attenuation terms, octave-band arithmetic, and a BS 4142 compliance assessment.
The Scenario
Source: Industrial fan unit on roof of factory building
- Source height above ground: h_s = 6.0 m
- Octave-band sound power levels (Lw, dB re 1 pW):
| Octave Band (Hz) | 63 | 125 | 250 | 500 | 1k | 2k | 4k |
|---|---|---|---|---|---|---|---|
| Lw (dB) | 100 | 98 | 95 | 92 | 88 | 84 | 79 |
Receiver: Ground floor bedroom window of residential dwelling
- Receiver height above ground: h_r = 1.5 m
- Source-to-receiver distance: d = 200 m
- Terrain: Flat, grass ground (ground factor G = 1.0)
- Intervening barrier: 2.4 m high solid brick wall, located 80 m from source (120 m from receiver)
- Atmospheric conditions: Temperature 10°C, relative humidity 70% (representative UK conditions)
- Wind: Downwind propagation (worst-case, per ISO 9613-2 long-term average method)
Step 1 — Geometrical Divergence (A_div)
For a point source radiating into a hemisphere (half-space, ground level), the geometrical spreading attenuation is:
A_div = 20 × log₁₀(d) + 11 dB
Where d is the source-to-receiver distance in metres.
A_div = 20 × log₁₀(200) + 11 = 20 × 2.301 + 11 = 46.02 + 11 = 57.0 dB
This term is frequency-independent — the same 57.0 dB applies at all octave bands.
The additional "+11" combines the 4π spherical radiation factor and the hemispherical correction (+3 dB for ground reflection). For directional sources, a directivity correction DC would be added, but we assume omnidirectional radiation here.
Step 2 — Atmospheric Absorption (A_atm)
Atmospheric absorption depends on frequency, temperature, and humidity. At 10°C, 70% RH, the ISO 9613-1 attenuation coefficients α (dB/km) are:
| Band (Hz) | 63 | 125 | 250 | 500 | 1k | 2k | 4k |
|---|---|---|---|---|---|---|---|
| α (dB/km) | 0.1 | 0.4 | 1.0 | 1.9 | 3.7 | 9.7 | 32.8 |
Atmospheric absorption over 200 m path: A_atm = α × d/1000
| Band (Hz) | α (dB/km) | d = 0.200 km | A_atm (dB) |
|---|---|---|---|
| 63 | 0.1 | × 0.200 | 0.02 |
| 125 | 0.4 | × 0.200 | 0.08 |
| 250 | 1.0 | × 0.200 | 0.20 |
| 500 | 1.9 | × 0.200 | 0.38 |
| 1k | 3.7 | × 0.200 | 0.74 |
| 2k | 9.7 | × 0.200 | 1.94 |
| 4k | 32.8 | × 0.200 | 6.56 |
Atmospheric absorption is small at 200 m except at 4k Hz. At distances over 1 km, this term becomes significant at 2k–4k Hz.
Step 3 — Ground Effect (A_gr)
ISO 9613-2 ground effect accounts for destructive interference between the direct sound path and the ground-reflected path. The formula depends on the ground factor G (0 = hard ground like concrete; 1.0 = soft ground like grass), source height h_s, receiver height h_r, and distance d.
This is the most complex term in ISO 9613-2. The standard defines ground effect through intermediate quantities involving the mean height h_m of the propagation path.
Mean path height: h_m = (h_s + h_r) / 2 for a flat path ≈ (6.0 + 1.5) / 2 = 3.75 m
For the general method with flat soft ground (G = 1.0):
Ground factor for source region: G_s = G × q_s where q_s depends on h_s and d_s (distance from source to barrier or receiver if no barrier):
Without barrier, d_s = d = 200 m.
The simplified ISO 9613-2 ground correction for propagation over uniform ground uses the octave-band values:
For G = 1.0 (soft ground), the ground attenuation A_gr per octave band (simplified tabulation):
| Band (Hz) | 63 | 125 | 250 | 500 | 1k | 2k | 4k |
|---|---|---|---|---|---|---|---|
| A_gr (dB) soft ground | −3.0 | −3.0 | −3.0 | −3.0 | 0 | 0 | 0 |
| A_gr (dB) hard ground | −3.0 | −3.0 | −3.0 | −3.0 | −3.0 | −3.0 | −3.0 |
(Negative A_gr means amplification — ground reflection adds constructive interference at these conditions. The standard uses a more complex formula; this simplified tabulation is adequate for conceptual calculations.)
For our path (soft ground, h_m = 3.75 m, d = 200 m):
ISO 9613-2 gives the ground effect formula:
A_gr = 4.8 − (2h_m/d) × [17 + (300/d)]
At 500 Hz: 4.8 − (2 × 3.75/200) × [17 + 300/200] = 4.8 − 0.0375 × [17 + 1.5] = 4.8 − 0.694 = 4.11 dB
The octave-band ground attenuation using the full ISO 9613-2 method:
| Band (Hz) | 63 | 125 | 250 | 500 | 1k | 2k | 4k |
|---|---|---|---|---|---|---|---|
| A_gr (dB) | −3.0 | −3.0 | −3.0 | 4.1 | 4.1 | 4.1 | 4.1 |
At frequencies below 250 Hz: Ground effect provides amplification (−3 dB = +3 dB level increase) because wavelengths are long relative to source/receiver heights and coherent reflection adds constructively.
At 500 Hz and above: Ground attenuation of +4.1 dB reduces the level.
Step 4 — Barrier Attenuation (A_bar)
Barrier: 2.4 m high solid brick wall
- Source to barrier: d_sb = 80 m
- Barrier to receiver: d_br = 120 m
- Source height: h_s = 6.0 m
- Receiver height: h_r = 1.5 m
- Barrier height: h_b = 2.4 m
Line of sight between source and receiver at the barrier location:
Height of source-receiver line at barrier (linear interpolation): h_los = h_s − (h_s − h_r) × (d_sb / d) = 6.0 − (6.0 − 1.5) × (80/200) = 6.0 − 4.5 × 0.40 = 6.0 − 1.80 = 4.20 m
Barrier height = 2.4 m < Line-of-sight height of 4.20 m → Barrier does NOT break line of sight.
The barrier top is 1.8 m below the source-receiver line. The barrier provides some attenuation through diffraction, but it is not screening the source.
Step 4b — Fresnel number N
The path length difference δ determines the Fresnel number:
δ = (d_s + d_r) − d
where d_s is distance from source to top of barrier and d_r is distance from barrier top to receiver.
d_s = √(d_sb² + (h_s − h_b)²) = √(80² + (6.0−2.4)²) = √(6400 + 12.96) = √6412.96 = 80.08 m
d_r = √(d_br² + (h_b − h_r)²) = √(120² + (2.4−1.5)²) = √(14400 + 0.81) = √14400.81 = 120.003 m
δ = 80.08 + 120.003 − 200 = 0.083 m
This small positive path length difference confirms the barrier is in the "shadow zone fringe" — it has a small positive Fresnel number.
Fresnel number N = 2δ/λ where λ = c/f:
| Band (Hz) | f | λ = 340/f (m) | N = 2×0.083/λ |
|---|---|---|---|
| 63 | 63 | 5.40 | 0.031 |
| 125 | 125 | 2.72 | 0.061 |
| 250 | 250 | 1.36 | 0.122 |
| 500 | 500 | 0.68 | 0.244 |
| 1k | 1000 | 0.34 | 0.488 |
| 2k | 2000 | 0.17 | 0.976 |
| 4k | 4000 | 0.085 | 1.953 |
Barrier attenuation for thin barrier (Maekawa formula, from ISO 9613-2):
A_bar = 10 × log₁₀(3 + 20N) for N > 0
| Band (Hz) | N | 3 + 20N | A_bar (dB) |
|---|---|---|---|
| 63 | 0.031 | 3.62 | 5.6 |
| 125 | 0.061 | 4.22 | 6.3 |
| 250 | 0.122 | 5.44 | 7.4 |
| 500 | 0.244 | 7.88 | 8.9 |
| 1k | 0.488 | 12.76 | 11.1 |
| 2k | 0.976 | 22.52 | 13.5 |
| 4k | 1.953 | 42.06 | 16.2 |
The barrier provides only modest attenuation (6–16 dB) because it does not intercept the line of sight. A taller barrier breaking the line of sight would increase N dramatically and yield 15–25+ dB of attenuation.
Step 5 — Calculate Octave-Band Receiver Level
Receiver level per octave band:
L_p = Lw − A_div − A_atm − A_gr − A_bar
(A_gr is subtracted when positive, added when negative — ground effect formula gives attenuation A_gr; negative values mean amplification)
| Band (Hz) | Lw | A_div | A_atm | A_gr | A_bar | L_p (dB) |
|---|---|---|---|---|---|---|
| 63 | 100 | 57.0 | 0.02 | −3.0 | 5.6 | 100−57.0−0.02+3.0−5.6 = 40.4 |
| 125 | 98 | 57.0 | 0.08 | −3.0 | 6.3 | 98−57.0−0.08+3.0−6.3 = 37.6 |
| 250 | 95 | 57.0 | 0.20 | −3.0 | 7.4 | 95−57.0−0.20+3.0−7.4 = 33.4 |
| 500 | 92 | 57.0 | 0.38 | 4.1 | 8.9 | 92−57.0−0.38−4.1−8.9 = 21.6 |
| 1k | 88 | 57.0 | 0.74 | 4.1 | 11.1 | 88−57.0−0.74−4.1−11.1 = 15.1 |
| 2k | 84 | 57.0 | 1.94 | 4.1 | 13.5 | 84−57.0−1.94−4.1−13.5 = 7.5 |
| 4k | 79 | 57.0 | 6.56 | 4.1 | 16.2 | 79−57.0−6.56−4.1−16.2 = −4.9 (→ 0) |
Step 6 — A-Weighting and Final Level
Apply A-weighting corrections:
| Band (Hz) | L_p (dB) | A-weighting | L_pA (dB) |
|---|---|---|---|
| 63 | 40.4 | −26.2 | 14.2 |
| 125 | 37.6 | −16.1 | 21.5 |
| 250 | 33.4 | −8.6 | 24.8 |
| 500 | 21.6 | −3.2 | 18.4 |
| 1k | 15.1 | 0 | 15.1 |
| 2k | 7.5 | +1.2 | 8.7 |
| 4k | 0 | +1.0 | 1.0 |
A-weighted sum: L_Aeq = 10 × log₁₀(10^1.42 + 10^2.15 + 10^2.48 + 10^1.84 + 10^1.51 + 10^0.87 + 10^0.10)
= 10 × log₁₀(26.3 + 141.3 + 301.9 + 69.2 + 32.4 + 7.4 + 1.3)
= 10 × log₁₀(579.8) = 27.6 dBA
Predicted receiver level: 27.6 dBA
The dominant contributing bands are 125 Hz (21.5 dBA) and 250 Hz (24.8 dBA). The low-frequency dominance is due to the ground effect amplification at these bands (−3.0 dB correction = amplification) and the relatively high source power at 63–250 Hz.
Step 7 — BS 4142 Assessment
Background noise at receiver: From a BS 4142 survey (not modelled here), the measured LA90 background noise level at the receptor is 35 dBA (suburban residential area, daytime).
Specific source level: 27.6 dBA
Tonal character check: The fan has a strong tonal component at its blade-pass frequency (6 blades × 480 rpm / 60 = 48 Hz). Because the tone is clearly audible, a +5 dB tonality penalty applies per BS 4142.
Rating level: L_rating = L_specific + 5 = 27.6 + 5.0 = 32.6 dBA
BS 4142 assessment:
Difference = L_rating − L_background = 32.6 − 35.0 = −2.4 dB
A rating level below the background noise level by −2.4 dB indicates low likelihood of adverse impact under BS 4142. The industrial source does not cause a noise problem at the residential receptor at this distance.
Sensitivity: What if the Barrier Weren't There?
Without the barrier (A_bar = 0 at all bands), recalculate L_p:
| Band (Hz) | L_p without barrier (dB) | L_pA |
|---|---|---|
| 63 | 46.0 | 19.8 |
| 125 | 43.9 | 27.8 |
| 250 | 39.4 | 30.8 |
| 500 | 30.5 | 27.3 |
| 1k | 26.2 | 26.2 |
| 2k | 20.9 | 22.1 |
| 4k | 11.3 | 12.3 |
L_Aeq without barrier = 10 × log₁₀(10^1.98 + 10^2.78 + 10^3.08 + 10^2.73 + 10^2.62 + 10^2.21 + 10^1.23) = 10 × log₁₀(95.5 + 602.6 + 1202 + 537 + 417 + 162 + 17.0) = 10 × log₁₀(3033) = 34.8 dBA
Rating level without barrier: 34.8 + 5 = 39.8 dBA vs background 35.0 dBA = +4.8 dB
Without the barrier: Difference = +4.8 dB → "may be of concern, likely to be noticeable" per BS 4142. A formal planning objection would likely succeed.
The barrier provides 27.6 − 34.8 = −7.2 dBA reduction — modest because it does not intercept the line of sight. A 4.5 m barrier (breaking line of sight by approximately 0.3 m) would provide approximately 12–14 dBA additional attenuation, bringing the rating level well below background.
Summary
| Term | 63 Hz | 125 Hz | 250 Hz | 500 Hz | 1k Hz | 2k Hz | 4k Hz |
|---|---|---|---|---|---|---|---|
| Lw | 100 | 98 | 95 | 92 | 88 | 84 | 79 |
| A_div | 57.0 | 57.0 | 57.0 | 57.0 | 57.0 | 57.0 | 57.0 |
| A_atm | 0.0 | 0.1 | 0.2 | 0.4 | 0.7 | 1.9 | 6.6 |
| A_gr | −3.0 | −3.0 | −3.0 | +4.1 | +4.1 | +4.1 | +4.1 |
| A_bar | 5.6 | 6.3 | 7.4 | 8.9 | 11.1 | 13.5 | 16.2 |
| L_p (dB) | 40.4 | 37.6 | 33.4 | 21.6 | 15.1 | 7.5 | 0 |
| L_pA (dB) | 14.2 | 21.5 | 24.8 | 18.4 | 15.1 | 8.7 | 1.0 |
Total L_Aeq = 27.6 dBA. BS 4142 rating level = 32.6 dBA. Background = 35.0 dBA. Assessment: LOW IMPACT.
For more complex propagation scenarios involving terrain, multiple barriers, or reflective facades, use specialist software (SoundPLAN, CadnaA, Predictor) implementing the full ISO 9613-2 algorithm. For straightforward path calculations like this example, the AcousPlan calculator provides point-to-point propagation estimates suitable for early-stage planning assessments.