---
title: "Outdoor Sound Propagation"
description: "Atmospheric absorption of sound (ISO 9613-1) and the ISO 9613-2 general method for outdoor sound propagation from a point source: geometrical divergence, atmospheric absorption, ground effect, barrier screening and the meteorological correction, with a per-term octave-band attenuation breakdown."
references:
  - type: book
    authors: ["Salomons, E. M."]
    year: 2001
    title: "Computational atmospheric acoustics"
    publisher: "Kluwer Academic Publishers"
    doi: "10.1007/978-94-010-0660-6"
    note: "ISBN 978-1-4020-0390-5. The wave-based theory (parabolic equation, fast field program, refraction and turbulence) that quantifies what the favourable-condition assumption and the K_met factor approximate."
  - type: book
    authors: ["Attenborough, K.", "Van Renterghem, T."]
    year: 2021
    title: "Predicting outdoor sound"
    edition: "2nd ed."
    publisher: "CRC Press"
    doi: "10.1201/9780429470806"
    note: "Ground impedance models, the spherical-wave reflection coefficient behind the ground dip of section 2, and the meteorological effects on barriers."
  - type: article
    authors: ["Maekawa, Z."]
    year: 1968
    title: "Noise reduction by screens"
    journal: "Applied Acoustics"
    volume: 1
    issue: 3
    pages: "157-173"
    doi: "10.1016/0003-682X(68)90020-0"
    note: "The screen-attenuation chart against Fresnel number that Eq. (14)'s diffraction term descends from."
  - type: report
    authors: ["Kephalopoulos, S.", "Paviotti, M.", "Anfosso-Lédée, F."]
    year: 2012
    title: "Common noise assessment methods in Europe (CNOSSOS-EU)"
    number: "EUR 25379 EN"
    institution: "Publications Office of the European Union"
    doi: "10.2788/31776"
    url: "https://publications.jrc.ec.europa.eu/repository/handle/JRC72550"
    note: "The common EU framework contrasted with ISO 9613-2 in section 3."
  - type: standard
    organization: "International Organization for Standardization"
    year: 1993
    title: "Acoustics — Attenuation of sound during propagation outdoors — Part 1: Calculation of the absorption of sound by the atmosphere"
    designation: "ISO 9613-1:1993"
    url: "https://www.iso.org/standard/17426.html"
    note: "The implemented pure-tone attenuation coefficient of section 1: alpha (Eq. (5)) with the oxygen and nitrogen relaxation frequencies (Eq. (3)/(4)), the Annex B humidity conversion and the exact Table 1 midbands (Eq. (6), Note 5)."
  - type: standard
    organization: "International Organization for Standardization"
    year: 1996
    title: "Acoustics — Attenuation of sound during propagation outdoors — Part 2: General method of calculation"
    designation: "ISO 9613-2:1996"
    url: "https://www.iso.org/standard/20649.html"
    note: "The implemented attenuation chain of section 2: the downwind receiver level (Eq. (3)/(4)) assembled from geometrical divergence (Eq. (7)), atmospheric absorption (Eq. (8)), the ground effect (Eq. (9), Table 3) with its A-weighted alternative (Eq. (10)/(11)), barrier screening (Eqs. (12)-(17)) and the meteorological correction (Eq. (21)/(22)). A revised edition was published in 2024; this module implements the 1996 method."
  - type: standard
    organization: "International Organization for Standardization"
    year: 2003
    title: "Acoustics — Measurement of sound absorption in a reverberation room"
    designation: "ISO 354:2003"
    url: "https://www.iso.org/standard/34545.html"
    note: "Only the clause 8.1.2.1 conversion m = alpha/(10 lg e) behind air_attenuation_m; the reverberation-room method itself is covered in the Room Acoustics guide."
---

import Video from '../../../components/Video.astro';
import ThemeImage from '../../../components/ThemeImage.astro';
import ReportPreview from '../../../components/ReportPreview.astro';

Predicting the noise a distant source delivers to a receiver outdoors is a
book-keeping exercise: start from the source **sound power** and subtract, band
by band, every mechanism that attenuates the sound on its way: spherical
spreading, the air itself, the ground, and any barrier in between. This page
covers the two parts of ISO 9613 that supply those terms: **ISO 9613-1**, the
pure-tone atmospheric absorption coefficient $\alpha$, and **ISO 9613-2**, the
general method that assembles $\alpha$ with divergence, ground and screening
into an octave-band prediction. The atmospheric coefficient also closes the loop
with room acoustics, since ISO 354 defers its air power-attenuation coefficient
$m$ entirely to $\alpha$.

## 1. Atmospheric absorption (ISO 9613-1)

Air is not lossless. A propagating tone bleeds energy into shear viscosity and
heat conduction (the *classical and rotational* losses that grow as $f^2$) and
into the **vibrational relaxation** of the oxygen and nitrogen molecules, each an
energy store that resonates near a humidity- and temperature-dependent
relaxation frequency. ISO 9613-1:1993, Eq. (5) collects all of this into the
pure-tone attenuation coefficient $\alpha$, in decibels per metre:

$$
\alpha = 8.686\ f^2 \Big[ 1.84\times10^{-11} \big(p_a/p_r\big)^{-1} \big(T/T_0\big)^{1/2}
       + \big(T/T_0\big)^{-5/2} \big( 0.01275\ \tfrac{e^{-2239.1/T}}{f_{rO} + f^2/f_{rO}}
       + 0.1068\ \tfrac{e^{-3352.0/T}}{f_{rN} + f^2/f_{rN}} \big) \Big],
$$

with the oxygen and nitrogen relaxation frequencies $f_{rO}$, $f_{rN}$
(Eq. (3)/(4)), the reference conditions $T_0 = 293.15$ K and $p_r = 101.325$ kPa
(Clause 4.2), and the molar water-vapour concentration $h$ obtained from the
relative humidity by the Annex B psychrometric conversion. Two features dominate
the shape: at low frequency $\alpha \propto f^2$, so it rises steeply; and near
each relaxation frequency the matching term peaks and rolls off. Between them
$\alpha$ climbs about two decades from 50 Hz to 10 kHz, and, because humidity
sets $f_{rO}$, sweeping the humidity moves a relaxation peak across the band, so
the driest air is not always the least absorbing.

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/air_absorption_alpha.svg" alt="ISO 9613-1 pure-tone atmospheric attenuation coefficient alpha in dB/km against frequency on log-log axes, for four temperature and humidity combinations, showing the f-squared low-frequency growth and the humidity-dependent relaxation roll-off" width="80%" loading="eager" />

*The dry 20 °C / 10 % curve absorbs most at mid frequencies, but the humid
curves overtake it below ~200 Hz: the relaxation signature.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import environmental

freqs = np.geomspace(50.0, 10000.0, 400)
fig, ax = plt.subplots()
for temp, rh in [(20.0, 50.0), (20.0, 10.0), (0.0, 70.0), (30.0, 80.0)]:
    ax.loglog(freqs, environmental.air_attenuation(freqs, temp, rh) * 1000.0,
              label=f"{temp:g} °C, {rh:g} % RH")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Attenuation coefficient alpha [dB/km]")
ax.legend()
plt.show()
```

</details>

```python
import numpy as np
from phonometry import environmental

bands = [63, 125, 250, 500, 1000, 2000, 4000, 8000]   # octave-band centres [Hz]

# Pure-tone attenuation coefficient alpha [dB/m] at 20 °C, 50 % RH, one atmosphere
alpha = environmental.air_attenuation(bands, temperature=20.0, relative_humidity=50.0)
print(np.round(alpha * 1000.0, 2))          # in dB/km, as Table 1 tabulates
# [  0.12   0.44   1.31   2.73   4.66   9.89  29.67 105.29]

# Reproduce an ISO 9613-1 Table 1 cell exactly (10 °C, 70 %, 1 kHz)
cell = environmental.air_attenuation(1000.0, 10.0, 70.0, exact_midband=True) * 1000.0
print(round(float(cell), 2))                # 3.66  (dB/km, Table 1)

# Feed real conditions into the ISO 354 power attenuation coefficient m [1/m]
m = environmental.air_attenuation_m([1000.0, 4000.0], temperature=20.0, relative_humidity=50.0)
print(np.round(m, 5))                        # [0.00107 0.00683]
```

`air_attenuation` returns dB/m (Table 1 prints dB/km, i.e. $\times 1000$); it is
fully vectorized over `frequencies` with scalar temperature, humidity and
pressure. Passing `exact_midband=True` snaps each requested frequency onto the
exact one-third-octave midband $f_m = 1000 \cdot 10^{k/10}$ (Eq. (6), Note 5)
used to compute Table 1, so the library reproduces every tabulated point to under
0.4 %, the standard's own three-significant-figure precision, far inside its
stated $\pm 10$ % accuracy (Clause 7.1). Inputs outside the tabulated ranges
(−20…+50 °C, 10…100 % RH, 50…10 000 Hz, or above the 200 kPa validity envelope)
still compute but raise an `AtmosphericAbsorptionWarning`; non-physical inputs
(non-positive frequency, humidity outside 0…100 %, sub-absolute-zero
temperature) raise `ValueError`. `air_attenuation_m` composes $\alpha$ with the
ISO 354 conversion $m = \alpha/(10 \lg e)$ so an ISO 354 caller can feed real
atmospheric conditions straight into `absorption_area` /
`absorption_coefficient` instead of hand-entering $m$.

### `air_attenuation()` / `air_attenuation_m()` parameters

| Parameter | Type / shape | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `frequencies` | scalar or 1D array | Hz | > 0 | Vectorized; 50–10 000 Hz tabulated |
| `temperature` | float | °C | default `20.0` | −20…+50 tabulated; outside warns |
| `relative_humidity` | float | % | default `50.0` | 10…100 tabulated; `[0, 100]` allowed |
| `pressure` | float | kPa | default `101.325` | ≤ 200 valid (Clause 7); above warns |
| `exact_midband` | bool | — | default `False` | Snap to $f_m = 1000\cdot10^{k/10}$ (reproduces Table 1) |

`air_attenuation` returns $\alpha$ in dB/m; `air_attenuation_m` returns
$m = \alpha/(10 \lg e)$ in 1/m for ISO 354.

### A plottable result: `atmospheric_attenuation()`

For a figure or a quick look, `atmospheric_attenuation()` wraps the same
coefficient in a small `AtmosphericAttenuation` result. It carries the frequency
grid, the coefficient $\alpha$ and the atmospheric conditions, and exposes
`.plot()` for the classic $\alpha$-versus-frequency curve (drawn in dB/km, the
Table 1 unit, on a linear ordinate over a logarithmic frequency axis). Passing a
`distance` also records the total attenuation $A = \alpha\,d$, in decibels, over
that path as `total_attenuation`, the ISO 9613-2 $A_{atm}$ of Eq. (8).

```python
from phonometry import environmental

res = environmental.atmospheric_attenuation(
    [63, 125, 250, 500, 1000, 2000, 4000, 8000],
    temperature=20.0, relative_humidity=50.0,
)
res.plot()   # alpha in dB/km against frequency (needs matplotlib)
```

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/atmospheric_attenuation.svg" alt="ISO 9613-1 pure-tone atmospheric attenuation coefficient alpha in dB/km against frequency, on a linear decibel ordinate over a logarithmic frequency axis, for the reference 20 degrees Celsius and 50 percent relative humidity atmosphere, produced by the AtmosphericAttenuation result plot method" width="80%" loading="lazy" />

*On a linear decibel ordinate the coefficient stays near zero up to about
1 kHz and then climbs steeply, passing 20 dB/km near 3 kHz and reaching over
150 dB/km by 10 kHz at the reference 20 °C / 50 % RH atmosphere.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import environmental

# One line: the coefficient curve straight from the result.
res = environmental.atmospheric_attenuation(
    np.geomspace(50.0, 10000.0, 400), temperature=20.0, relative_humidity=50.0,
)
res.plot()
plt.show()

# Or by hand from air_attenuation (dB/m, so scale by 1000 for dB/km):
freqs = np.geomspace(50.0, 10000.0, 400)
fig, ax = plt.subplots()
ax.semilogx(freqs, environmental.air_attenuation(freqs, 20.0, 50.0) * 1000.0)
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Attenuation coefficient alpha [dB/km]")
plt.show()
```

</details>

## 2. General method of calculation (ISO 9613-2)

ISO 9613-2:1996 predicts the octave-band level at a receiver **downwind** of a
point source (or under the equivalent moderate temperature inversion, Clause 5).
The equivalent-continuous downwind level is

$$
L_{fT}(DW) = L_W + D_c - A, \qquad A = A_{div} + A_{atm} + A_{gr} + A_{bar} + A_{misc},
$$

(Eq. (3)/(4)) where $L_W$ is the octave-band sound power level, $D_c$ the
directivity correction (directivity index plus a solid-angle index
$D_\Omega$), and $A$ the total attenuation. The library implements the four
general terms of Clause 7; the informative $A_{misc}$ (foliage, industrial
sites, housing; Annex A) and reflections off vertical obstacles (Clause 7.5) are
**not implemented**: the standard treats them as informative and they are left
to the caller.

The geometry of the screening term fixes the vocabulary: the diffraction edge
splits the source-to-receiver distance into $d_{ss}$ and $d_{sr}$, and the extra
path length over the edge is $z = d_{ss} + d_{sr} - d$. When the line of sight
passes *above* the top edge, the standard gives $z$ a negative sign
(`Barrier(line_of_sight_clear=True)`) and Eq. (14) still applies with
$K_{met} = 1$: $D_z$ falls continuously from $10 \lg 3 \approx 4.8$ dB at
grazing incidence to zero as the clearance deepens, never below zero.

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_outdoor_geometry.svg" alt="ISO 9613-2 source-barrier-receiver geometry: a point source at height hs, a barrier whose top edge splits the path into dss and dsr, and a receiver at height hr, with the blocked direct ray and the diffracted ray over the edge, the path difference z and the Dz formula" width="92%" />

### The four attenuation terms

* **Geometrical divergence** $A_{div} = 20 \log_{10}(d/d_0) + 11$ dB, $d_0 = 1$ m
  (Eq. (7)): spherical spreading from a point source. Exactly 51 dB at 100 m,
  +6 dB per distance doubling. The $+11$ ($= 10 \lg 4\pi$) sets the level at the
  1 m reference distance.
* **Atmospheric absorption** $A_{atm} = \alpha\ d$ (Eq. (8)) with $\alpha$ the
  ISO 9613-1 coefficient, negligible at low frequency and dominant at 8 kHz over
  long paths. $\alpha$ is evaluated at the *exact* base-10 midband behind each
  nominal band label (7 943.3 Hz for "8 kHz"), the convention behind the
  ISO 9613-2 Table 2 coefficients (the nominal-frequency evaluation runs
  ~1.3 % high at 8 kHz). The ISO 9613-2 functions default to 20 °C and 70 %
  relative humidity (one of the Table 2 reference atmospheres the standard
  tabulates) while `air_attenuation` itself defaults to the ISO 9613-1
  usual 50 %.
* **Ground effect** $A_{gr} = A_s + A_r + A_m$ (Eq. (9)) sums a source, receiver
  and middle region, each from the Table 3 functions $a'/b'/c'/d'$ and its
  ground factor $G$ (0 = hard/reflective, 1 = porous/absorbing). A **negative**
  $A_{gr}$ is a net *gain* from constructive ground reflection.
* **Screening** by a barrier is the diffraction insertion loss
  $D_z = 10 \log_{10}\big[ 3 + (C_2/\lambda)\ C_3\ z\ K_{met} \big]$ (Eq. (14)),
  capped at 20 dB (single edge) or 25 dB (double edge). For a top-edge barrier
  the ground effect of the screened path folds into it,
  $A_{bar} = D_z - A_{gr} \ge 0$ (Eq. (12), Note 13); for a lateral barrier
  $A_{bar} = D_z$ and the ground term is retained (Eq. (13)).

`outdoor_propagation_attenuation` assembles the four terms into an
`OutdoorAttenuation` result whose per-band arrays sum, band by band, to
`a_total`, so the divergence, atmospheric, ground and barrier contributions stay
separable. The stacked breakdown makes the frequency character obvious: the
barrier helps most at high frequency (short wavelength) until it saturates at the
cap, atmospheric absorption bites only at 8 kHz, and the ground dip lives in the
mid bands.

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/outdoor_attenuation_breakdown.svg" alt="ISO 9613-2 per-octave-band attenuation breakdown as a stacked bar of Adiv, Aatm, Agr and Abar with the total A overlaid, for a 200 m path over porous ground with a 4 m barrier" width="80%" />

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import environmental

bands = np.array([63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0, 8000.0])
barrier = environmental.Barrier(source_to_edge=101.0, edge_to_receiver=101.0)
att = environmental.outdoor_propagation_attenuation(
    200.0, 1.5, 1.5, bands, ground_source=1.0, ground_middle=1.0,
    ground_receiver=1.0, barrier=barrier, temperature=15.0,
    relative_humidity=70.0,
)

# One line — the same stacked breakdown with the total overlaid:
att.plot()

# By hand:
x = np.arange(len(bands))
fig, ax = plt.subplots()
# Separate positive and negative baselines: a negative term (Agr is a net
# gain at 63 Hz here) stacks below zero, and the signed heights sum to a_total.
pos_bottom = np.zeros(len(bands))
neg_bottom = np.zeros(len(bands))
for term, label in [(att.a_div, "Adiv — divergence"),
                    (att.a_atm, "Aatm — atmospheric"),
                    (att.a_gr, "Agr — ground"),
                    (att.a_bar, "Abar — barrier")]:
    ax.bar(x, term, bottom=np.where(term >= 0.0, pos_bottom, neg_bottom),
           label=label)
    pos_bottom += np.maximum(term, 0.0)
    neg_bottom += np.minimum(term, 0.0)
ax.plot(x, att.a_total, "D-", color="black", label="A — total")
ax.set_xticks(x)
ax.set_xticklabels([f"{b:g}" for b in bands])
ax.set_xlabel("Octave-band centre frequency [Hz]")
ax.set_ylabel("Attenuation A [dB]")
ax.legend()
plt.show()
```

</details>

```python
import numpy as np
from phonometry import environmental

bands = [63, 125, 250, 500, 1000, 2000, 4000, 8000]   # octave-band centres [Hz]

# A point source and receiver 1.5 m high, 200 m apart over porous ground
# (G = 1), screened midway by a barrier that raises the path over its top edge
# (dss = dsr ~ 101 m). Geometry feeds the pathlength-difference equations.
barrier = environmental.Barrier(source_to_edge=101.0, edge_to_receiver=101.0)
att = environmental.outdoor_propagation_attenuation(
    200.0, source_height=1.5, receiver_height=1.5, frequencies=bands,
    ground_source=1.0, ground_middle=1.0, ground_receiver=1.0,
    barrier=barrier, temperature=15.0, relative_humidity=70.0,
)
print(np.round(att.a_div, 1))     # [57. 57. 57. 57. 57. 57. 57. 57.]  divergence
print(np.round(att.a_gr, 2))      # [-4.65  2.34 13.79  9.76  1.3  -0.   -0.   -0.  ]
print(np.round(att.a_bar, 2))     # [13.78  8.89  0.    6.69 18.01 20.   20.   20.  ]
print(np.round(att.a_total, 1))   # [66.2 68.3 71.  73.9 77.1 78.8 82.3 96. ]

# Predicted receiver level from an octave-band sound power Lw = 95 dB
lw = np.full(len(bands), 95.0)
lp = environmental.predicted_receiver_level(
    lw, 200.0, 1.5, 1.5, bands, 1.0, 1.0, 1.0,
    barrier=barrier, temperature=15.0, relative_humidity=70.0,
)
print(np.round(lp, 1))            # [28.8 26.7 24.  21.1 17.9 16.2 12.7 -1. ]
```

`predicted_receiver_level` composes $L_{fT}(DW) = L_W + D_c - A$ with
$D_c = $ `directivity_index` $+ $ `d_omega`. Pass `c0=` to subtract the
meteorological correction $C_{met}$ (Eq. (21)/(22)) band by band for a long-term
average; note that the standard applies $C_{met}$ to the A-weighted level, so the
per-band form here is a **convenience**, not a literal reading of Clause 8.

### Ground: the alternative A-weighted method

When only the A-weighted receiver level matters and the sound travels over
porous or mostly-porous ground (and is not a pure tone), 7.3.2 offers a simpler
closed form $A_{gr} = 4.8 - \frac{2 h_m}{d}\left(17 + \frac{300}{d}\right) \ge 0$ (Eq. (10), negative
results clamped to zero), paired with the solid-angle index $D_\Omega$
(Eq. (11)) that must then be added to $D_c$. These are exposed as
`ground_attenuation_alternative` and `directivity_omega`, but they are **not
auto-wired** into `outdoor_propagation_attenuation` (which always uses the
general per-region method of 7.3.1); combine them by hand when the alternative
method is appropriate.

```python
from phonometry import environmental

# Alternative ground term (mean path height hm = 2 m, d = 200 m)
print(round(environmental.ground_attenuation_alternative(200.0, 2.0), 2))          # 4.43 dB
# Its companion solid-angle index (add to Dc when using Eq. (10))
print(round(environmental.directivity_omega(1.5, 1.5, 200.0), 2))                  # 3.01 dB
# Long-term meteorological correction (C0 = 2 dB) to subtract from LAT(DW)
print(round(environmental.meteorological_correction(200.0, 1.5, 1.5, 2.0), 2))     # 1.7 dB
```

### The image source behind the ground effect

Every entry of Table 3 is an engineering fit to one physical picture: the
receiver hears two copies of the source, the direct ray $r_1$ and a reflection
that arrives exactly as if it were radiated by an **image source** mirrored
below the ground plane. The two copies interfere according to the path
difference $\delta = r_2 - r_1$ (about $2 h_s h_r / d$ for a grazing far-field
path): in phase they add up to $+6$ dB; at $\delta = \lambda/2$ over a rigid
plane they cancel into a sharp dip.

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_ground_reflection.svg" alt="A point source at height hs and a receiver microphone at height hr over a hatched ground plane; a straight direct ray r1 connects them and a reflected ray bounces at the specular point with equal grazing angles; the reflection is unfolded as a dashed straight ray r2 from a ghosted image source mirrored below the ground, and the annotations give the path difference delta = r2 minus r1, the interference phase 2 pi delta over lambda plus the reflection phase, the in-phase gain of up to +6 dB and the deep dip at half a wavelength over hard ground" width="92%" />

Over acoustically hard ground ($G = 0$) the reflection coefficient stays close
to $+1$ in every octave, the long-wavelength sum is fully constructive, and the
general method duly returns $A_{gr} = -3$ dB in each band. Porous ground turns
the reflection coefficient complex and angle-dependent (for a point source
near grazing incidence, the spherical-wave coefficient of the Chien-Soroka
solution, with a ground-wave term no plane-wave picture captures): part of the
reflection flips phase, and the destructive notch lands in the 250 to 1000 Hz
octaves. That notch is precisely what the $a'$ to $d'$ height-and-distance
functions of Table 3 parameterize, with $G$ blending the hard and porous
behaviours. The same two-ray geometry carries over to aircraft lateral
attenuation and to every ray-based outdoor model; Salomons and Attenborough &
Van Renterghem develop the full theory that the engineering fit compresses.

A negative $A_{gr}$ (a net gain) is plain interference: the ground-reflected
wave adds to the direct one. Below, a 400 Hz source 1.5 m over rigid ground
builds the lobe pattern of that interference, and the level sampled on an arc
converges to the two-path image-source model, dips included.

<Video
  name="anim_fdtd_ground_effect"
  width={2400}
  height={1350}
  title="Ground-effect interference over rigid ground (2D FDTD)"
  description="A 2D FDTD simulation of a 400 Hz point source 1.5 metres above rigid ground. The direct and ground-reflected wavefronts interfere and a lobe pattern forms, the ghosted image source below the ground explains the geometry, and the level sampled on an 8 metre arc converges to the two-path image-source model with its predicted nulls."
/>

### `Barrier` and the screening term

The `Barrier` dataclass describes the diffraction geometry directly, which is
the cleanest match to Eq. (14)/(16)/(17). Single diffraction (a thin screen)
leaves `edge_separation=None` ($C_3 = 1$); giving the edge spacing `e` selects
double (thick-barrier) diffraction with the $C_3$ factor of Eq. (15) and the
25 dB cap. `ground_reflections_by_image=True` switches $C_2$ from 20 to 40
(reflections handled by image sources), and `lateral=True` selects vertical-edge
diffraction (Eq. (13), $K_{met}=1$, ground term retained).

The simulation below shows why $D_z$ grows with frequency: against the same
2.5 m screen, a 100 Hz wavefront (λ ≈ 3.4 m) diffracts over the edge and fills
the shadow zone, while at 500 Hz the shadow is deep and sharp. A barrier only
works when the wavelength is short next to the path difference.

<Video
  name="anim_fdtd_barrier"
  width={2400}
  height={1350}
  title="Barrier diffraction at 100 Hz and 500 Hz (2D FDTD)"
  description="A 2D FDTD simulation of a point source behind a thin 2.5 metre rigid barrier on reflecting ground, at 100 Hz and 500 Hz side by side. The long wavelength diffracts over the edge and fills the shadow zone with an insertion loss near 8 dB, while the short wavelength is cast into a deep clean shadow of about 17 dB."
/>

### `outdoor_propagation_attenuation()` parameters

| Parameter | Type / shape | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `distance` | float | m | > 0 | Straight-line source–receiver distance $d$ |
| `source_height` / `receiver_height` | float | m | ≥ 0 | $h_s$, $h_r$ above ground |
| `frequencies` | 1D array | Hz | default 8 octaves 63–8000 | `DEFAULT_FREQUENCIES` |
| `ground_source` / `ground_middle` / `ground_receiver` | float | — | `[0, 1]`, default `0.0` | Ground factor $G$ (0 hard, 1 porous) |
| `barrier` | `Barrier` or None | — | default `None` | Screening obstacle |
| `temperature` / `relative_humidity` / `pressure` | float | °C / % / kPa | 20 / 70 / 101.325 | Passed to $A_{atm}$ |
| `projected_distance` | float or None | m | default $\sqrt{d^2-(h_s-h_r)^2}$ | Ground-plane $d_p$ |

Returns an `OutdoorAttenuation` with `a_div`, `a_atm`, `a_gr`, `a_bar`,
`a_total` and `d_omega`, all one value per band; its `.plot()` draws the
stacked per-band breakdown with the total overlaid (the figure above).

### `Barrier` fields

| Field | Type | Units | Default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `source_to_edge` | float | m | — | $d_{ss}$, source to first edge |
| `edge_to_receiver` | float | m | — | $d_{sr}$, (last) edge to receiver |
| `parallel_distance` | float | m | `0.0` | Component $a$ parallel to the edge |
| `edge_separation` | float or None | m | `None` | $e$; given ⇒ double diffraction (25 dB cap) |
| `ground_reflections_by_image` | bool | — | `False` | `True` ⇒ $C_2 = 40$ |
| `lateral` | bool | — | `False` | `True` ⇒ vertical-edge diffraction (Eq. (13)) |
| `line_of_sight_clear` | bool | — | `False` | `True` ⇒ the sight line passes above the top edge: the path difference takes a negative sign and Kmet = 1 (text after Eq. (16)) |

## 3. Scope, assumptions and pitfalls

### What "favourable propagation conditions" means

ISO 9613-2 does not predict the level under the weather of the moment. Every
equation assumes conditions **favourable to propagation** (Clause 5): wind
blowing from source to receiver (within about 45° of the connecting line, at
roughly 1 to 5 m/s measured 3 to 11 m above ground), or the moderate
ground-based temperature inversion of a clear, calm night, which curves sound
rays downward the same way. Downward refraction closes the acoustic shadow
zones that upwind or neutral atmospheres would form, so the predicted
$L_{fT}(DW)$ is close to the highest level the geometry can deliver: over a
year the actual level is usually lower and rarely meaningfully higher. The
choice is deliberate. Complaints arrive on the still nights when a distant
plant is clearly audible, not on the gusty afternoons when it vanishes, and a
method that predicts the audible case protects the assessment. The long-term
average is recovered by subtracting $C_{met}$ (Eq. (21)/(22)), whose $C_0$
encodes how often the wind actually favours the path (about $+3$ dB when half
the time is favourable, values above 2 dB already exceptional, Notes 20/22).
The stated $\pm 1$ to $\pm 3$ dB accuracy (Table 5) holds under favourable
conditions, for broadband sources, up to 1000 m; beyond that the standard
makes no accuracy claim at all.

### Barrier pitfalls beyond the animation

The screening term is the easiest one to over-trust; four fine points decide
whether a real barrier delivers its computed $D_z$:

* **The caps are physical, not editorial.** $D_z$ is capped at 20 dB for
  single and 25 dB for double diffraction no matter how tall the wall,
  because atmospheric turbulence scatters sound into the shadow zone and sets
  a ceiling that extra height cannot buy back. An insertion loss beyond
  roughly 20 dB is enclosure territory, not screen territory. Eq. (14) itself
  is a smoothed engineering curve in the tradition of Maekawa's screen chart,
  an empirical fit over three decades of Fresnel number $N = 2z/\lambda$.
* **$K_{met}$ quietly erodes distant barriers.** The meteorological factor
  (Eq. (18)) discounts the screening because the same downward-curved rays
  that make conditions favourable also pass over the top edge. Within 100 m
  of source-receiver distance $K_{met} \approx 1$ (Note 17), but for a long
  path with a small path difference it can strip several decibels off a
  barrier that looks generous on the section drawing.
* **Double diffraction is a modest bonus.** A thick obstacle (two edges
  separated by $e$) raises $C_3$ from 1 toward 3 (Eq. (15)), worth at most
  about $10 \lg 3 \approx 4.8$ dB extra plus the higher 25 dB cap. A building
  modelled as a double edge only earns that bonus if both edges really are
  continuous and the roof between them is closed.
* **The ground effect is spent, not kept.** For a top-edge barrier the
  standard folds the screened path's ground effect into the diffraction:
  $A_{bar} = D_z - A_{gr}$ (Eq. (12), Note 13). Over porous ground that was
  already providing 5 to 10 dB of $A_{gr}$ in the mid bands, the *net* gain
  of building the barrier is correspondingly smaller than its nominal $D_z$;
  the two effects do not stack.

An obstacle must also qualify as a barrier at all (Clause 7.4): surface
density at least 10 kg/m², a closed surface without large gaps, and a
horizontal extent normal to the path larger than the wavelength. A slatted
fence or a short container screens far less than Eq. (14) promises.

### ISO 9613-2 or CNOSSOS-EU?

Two frameworks dominate outdoor noise prediction in Europe, and they answer
different questions:

* **ISO 9613-2** is a general *engineering attenuation* method: given the
  octave-band sound power of any source you can decompose into point sources,
  it returns the favourable-condition receiver level. Source emission is out
  of scope; the sound power comes from measurement (the ISO 3740 family) or
  from the equipment supplier. It is the workhorse of industrial-plant and
  environmental impact predictions assessed with ISO 1996-2.
* **CNOSSOS-EU** (Common Noise Assessment Methods in Europe) is the mandatory
  common framework for the strategic noise maps of the Environmental Noise
  Directive 2002/49/EC, adopted as its Annex II by
  [Commission Directive (EU) 2015/996](https://eur-lex.europa.eu/eli/dir/2015/996/oj/eng).
  It bundles *emission* models for road, rail and industrial sources (and
  delegates aircraft to ECAC Doc 29) with its own propagation part derived
  from the French NMPB 2008 method, producing the $L_{den}$/$L_{night}$
  indicators the Directive reports.

The propagation parts disagree by design, not by accident. CNOSSOS-EU
evaluates every path twice, once under a homogeneous atmosphere and once
under favourable (downward-refracting) conditions, and combines the two
long-term with the local occurrence probability of favourable conditions per
path direction; ISO 9613-2 computes only the favourable case and subtracts a
scalar $C_{met}$. The ground effect in CNOSSOS-EU is built from a
path-averaged ground factor over a fitted mean plane, with expressions that
change between the two atmospheres, where ISO 9613-2 uses the fixed
three-region Table 3. Diffraction, too, follows a different formulation that
couples the ground effect on each side of the edge. Run both on the same
geometry and the octave-band results can differ by several decibels, each
internally consistent. The practical rule: END strategic maps and anything
that must be comparable across EU member states use CNOSSOS-EU; a plant
permit, a compliance prediction against a measured sound power, or work
under regulations that cite ISO 9613 use this module. (ISO published a
revised ISO 9613-2 in 2024; this library implements the 1996 edition, the one
most national regulations and the validation literature still reference.)

See the [Theory](/phonometry/reference/theory/environment-transport/) page for the full derivation, the
[Room Acoustics guide](/phonometry/guides/room-acoustics/) for how $\alpha$ feeds
ISO 354, and the [Occupational Noise Exposure guide](/phonometry/guides/occupational-exposure/) for the ISO 9612 occupational
exposure that consumes A-weighted levels.

## 4. Prediction reports (`.report()`)

Both outdoor-propagation results render a one-page PDF **prediction** fiche.
They are clearly labelled predictions, not measurements: the sheet states the
meteorological and ground assumptions and, for the barrier, names the actual
diffraction model behind the number.

`OutdoorAttenuation.report(path)` boxes the octave-band range of the total
attenuation on its own; pass a `SourceEmission` (the source sound power and
directivity) to add the source power and downwind level to the table and box the
A-weighted downwind level $L_{AT}(\text{DW})$ at the receiver instead. A limit
level supplied through the metadata `requirement` then adds a PASS/FAIL verdict
(a lower level is better). Pass `language="es"` for the Spanish fiche.

```python
import numpy as np
from phonometry import (
    Barrier,
    ReportMetadata,
    SourceEmission,
    outdoor_propagation_attenuation,
)

freqs = np.array([63, 125, 250, 500, 1000, 2000, 4000, 8000], dtype=float)
lw = np.array([95, 100, 103, 105, 104, 101, 95, 88], dtype=float)
result = outdoor_propagation_attenuation(
    200.0, 4.0, 2.0, freqs, 1.0, 1.0, 1.0,
    barrier=Barrier(source_to_edge=105.0, edge_to_receiver=105.0),
    temperature=10.0, relative_humidity=70.0,
)
result.report(
    "outdoor_attenuation.pdf",
    metadata=ReportMetadata(
        specimen="Industrial fan plant (point source)",
        test_room="Nearest dwelling facade",
        requirement=50.0,  # maximum acceptable A-weighted downwind level
    ),
    source_emission=SourceEmission(sound_power_level=lw),
)
```

<ReportPreview
	name="iso9613_outdoor_attenuation_example"
	title="ISO 9613-2 outdoor propagation example report (PDF)"
	description="One-page ISO 9613-2 prediction fiche: a metadata header with the propagation distance, a per-band table of the source power level Lw and the divergence, atmospheric, ground and barrier attenuation terms with the total A and the downwind level LfT(DW), the attenuation-breakdown plot and the boxed A-weighted downwind level LAT(DW) = 29.9 dB with a PASS verdict against a 50 dB limit."
	caption="Outdoor propagation fiche (OutdoorAttenuation.report), the A-weighted downwind level at the receiver."
/>

`BarrierInsertionLoss.report(path)` boxes the mean insertion loss over the
octave bands; a minimum required insertion loss supplied through `requirement`
adds a PASS/FAIL verdict (a higher insertion loss is better).

```python
import numpy as np
from phonometry import ReportMetadata, barrier_insertion_loss

freqs = np.array([63, 125, 250, 500, 1000, 2000, 4000, 8000], dtype=float)
result = barrier_insertion_loss(freqs, 1.0, 50.0, 4.0, 100.0, 1.5)
result.report(
    "barrier_insertion_loss.pdf",
    metadata=ReportMetadata(
        specimen="Roadside noise barrier, 4 m high",
        requirement=8.0,  # minimum required mean insertion loss
    ),
)
```

<ReportPreview
	name="iso9613_barrier_insertion_loss_example"
	title="Barrier insertion loss example report (PDF)"
	description="One-page barrier insertion-loss prediction fiche: a metadata header naming the ground model, a per-band table of the insertion loss, the insertion-loss spectrum plot and the boxed mean insertion loss IL = 12.9 dB with a PASS verdict against an 8 dB minimum. The basis line names the wave-theoretic rigid-screen diffraction model, a complement to the ISO 9613-2 screening term."
	caption="Barrier insertion loss fiche (BarrierInsertionLoss.report), the mean insertion loss over the octave bands."
/>

## See also

- API reference: [`environmental.outdoor_propagation`](/phonometry/reference/api/environment/outdoor-propagation/) and [`environmental.air_absorption`](/phonometry/reference/api/environment/air-absorption/).
