---
title: "materials.scattering_diffusion"
description: "Public API of phonometry.materials.scattering_diffusion (auto-generated)."
sidebar:
  label: "scattering_diffusion"
---

> Auto-generated from the source docstrings by `scripts/generate_api_docs.py` (`make api-docs`). Do not edit by hand.

Random-incidence scattering and directional diffusion coefficients.

Two complementary free-field / reverberation-room surface descriptors are
implemented, each faithful to its own standard:

* **ISO 17497-1:2004+A1:2014** - random-incidence *scattering* coefficient in
  a reverberation room. Four reverberation times (Table 2) taken with and
  without the test sample, with a static and a rotating turntable, give two
  Sabine-form absorption coefficients: the random-incidence absorption
  coefficient `alpha_s` (Clause 8.1.1, Eq. (1)) and the specular absorption
  coefficient `alpha_spec` (Clause 8.1.2, Eq. (4)). Their ratio yields the
  scattering coefficient `s = (alpha_spec - alpha_s) / (1 - alpha_s)`
  (Clause 8.1.3, Eq. (5)). The turntable base plate is qualified through its
  own scattering coefficient (Clause 8.1.4, Eq. (6)) against the Table 1
  limits (Clause 6.2). Air properties come from the speed-of-sound and
  energy-attenuation relations of Clause 8 (Eqs. (2)/(3), after ISO 9613-1),
  and measurement accuracy from Annex A (Eqs. (A.1)-(A.5)).

* **ISO 17497-2:2012** - directional *diffusion* coefficient in a free field.
  From the set of reflected sound-pressure levels `L_i` on a semicircle or
  hemisphere the autocorrelation diffusion coefficient `d_theta` is formed
  for equal-area receivers (Clause 8.1, Formula (5)) or with per-receiver area
  weights `N_i` (Formula (6)); the area weights follow from the solid-angle
  factors of Clause 8.3 (Formula (8)). Finite-panel effects are removed by
  normalising to the reference flat surface (Clause 8.2, Formula (7)), and the
  random-incidence coefficient is the (weighted) average of the directional
  coefficients over the source positions (Clause 8.4).

Neither part of ISO 17497 contains a numeric worked example; the two methods
are distinct measurements and the helpers are named per part so they are
never mixed.

The Part 2 diffusion coefficient is the design target of subwavelength diffuser
panels such as the metadiffusers of Jiménez, Cox, Romero-García & Groby
(2017, *Scientific Reports* 7, 5389, doi:10.1038/s41598-017-05710-5), whose
slow-sound resonant slots reach the diffusion of a Schroeder or
quadratic-residue diffuser in a fraction of the depth.

## absorption_coefficient_uncertainty

```python
absorption_coefficient_uncertainty(
    volume: float,
    area: float,
    *,
    c: ArrayLike,
    T_a: ArrayLike,
    u_a: ArrayLike,
    T_b: ArrayLike,
    u_b: ArrayLike,
) -> Real
```

Uncertainty of a Sabine absorption coefficient (ISO 17497-1, Eqs. (A.3)/(A.4)).

`u_alpha = (55,3 V) / (c S) * sqrt((u_b / T_b^2)^2 + (u_a / T_a^2)^2)`.

With situations `(T1, T2)` this is `u(alpha_s)` (Eq. (A.3)); with
`(T3, T4)` it is `u(alpha_spec)` (Eq. (A.4)). The unsubscripted `c` of
the standard is taken as a single (mean) speed of sound.

**Parameters**

| Name | Description |
| :--- | :--- |
| `volume` | Reverberation-room volume `V`, in cubic metres. |
| `area` | Test-sample area `S`, in square metres. |
| `c` | Speed of sound `c`, in m/s. |
| `T_a` | Reverberation time of the first situation, in seconds. |
| `u_a` | Standard uncertainty of `T_a` (Eq. (A.1)), in seconds. |
| `T_b` | Reverberation time of the second situation, in seconds. |
| `u_b` | Standard uncertainty of `T_b` (Eq. (A.1)), in seconds. |

**Returns:** Combined standard uncertainty of the absorption coefficient (per band).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for non-positive `V`, `S`, `c` or `T`. |

## air_attenuation_coefficient

```python
air_attenuation_coefficient(
    pressure_attenuation_db_per_m: ArrayLike,
) -> Real
```

Energy attenuation coefficient `m` (ISO 17497-1:2004, Clause 8, Eq. (3)).

`m = alpha / (10 * lg(e)) approx. alpha / 4,343` (1/m), where `alpha` is
the sound-*pressure* attenuation coefficient in dB/m obtained from
ISO 9613-1 using the measured temperature and relative humidity.

**Parameters**

| Name | Description |
| :--- | :--- |
| `pressure_attenuation_db_per_m` | Pressure attenuation coefficient `alpha` from ISO 9613-1, in decibels per metre (scalar or per band). |

**Returns:** Energy (power) attenuation coefficient `m`, in reciprocal metres.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | if any value is negative or non-finite. |

## area_factors

```python
area_factors(
    elevations: ArrayLike,
    *,
    delta_theta: float,
    delta_phi: float | None = None,
) -> Real
```

Per-receiver area weights `N_i` (ISO 17497-2, Clause 8.3, Formula (8)).

For a hemispherical measurement the solid-angle area sampled by a receiver
at elevation `theta` (with angular spacings `delta_theta`, `delta_phi`)
is:

```text
A_i = (4 pi / delta_phi) * sin^2(delta_theta / 4)   for theta = 0 deg
A_i = 2 sin(theta) sin(delta_theta / 2)             for theta != 0, 90 deg
A_i = sin(delta_theta / 2)                          for |theta| = 90 deg
```

and `N_i = A_i / A_min` (Formula (8)), with `A_min` the smallest `A_i`.
All angles are handled internally in **radians**; the `theta = 0` form in
particular requires `delta_phi` in radians to be dimensionally consistent
with the `4 pi` factor.

**Parameters**

| Name | Description |
| :--- | :--- |
| `elevations` | Receiver elevation angles `theta` from the reference normal, in **degrees** (1-D), over the measurement domain `0 <= theta <= 90` (Figure 7). Formula (8) assumes a single receiver at `theta = 0` (the zenith); duplicate zenith entries would each take the full zenith area. |
| `delta_theta` | Elevation spacing between adjacent receivers, in degrees (typically 5). |
| `delta_phi` | Azimuth spacing between adjacent receivers, in degrees; defaults to `delta_theta`. Required (implicitly) for the `theta = 0` receiver. |

**Returns:** Per-receiver area weights `N_i` (dimensionless, min value 1).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-1-D input or non-positive spacings. |

## BASE_PLATE_BANDS

*Constant* (`tuple`).

```python
BASE_PLATE_BANDS = (100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000)
```

## BASE_PLATE_MAX_SCATTERING

*Constant* (`dict`).

```python
BASE_PLATE_MAX_SCATTERING = {100: 0.05, 125: 0.05, 160: 0.05, 200: 0.05, 250: 0.05, 315: 0.05, 400: 0.05, 500: 0.05, 630: 0.1, 800: 0.1, 1000: 0.1, 1250: 0.15, 1600: 0.15, 2000: 0.15, 2500: 0.2, 3150: 0.2, 4000: 0.2, 5000: 0.25}
```

## base_plate_scattering

```python
base_plate_scattering(
    volume: float,
    area: float,
    *,
    c1: ArrayLike,
    T1: ArrayLike,
    c3: ArrayLike,
    T3: ArrayLike,
    m1: ArrayLike = 0.0,
    m3: ArrayLike = 0.0,
) -> Real
```

Scattering coefficient of the base plate alone (ISO 17497-1, Eq. (6)).

`s_base = 55,3 * (V / S) * (1 / (c3 T3) - 1 / (c1 T1)) - (4 V / S) * (m3 - m1)`.

Ideally `T1 == T3`; a slightly non-symmetrical base plate shortens `T3`
and this quality metric captures the resulting spurious scattering, which
must not exceed the Table 1 limits (Clause 6.2). See
[`check_base_plate_scattering`](/phonometry/reference/api/materials/scattering-diffusion/#check_base_plate_scattering).

**Parameters**

| Name | Description |
| :--- | :--- |
| `volume` | Reverberation-room volume `V`, in cubic metres. |
| `area` | Test-sample area `S`, in square metres. |
| `c1` | Speed of sound during `T1`, in m/s. |
| `T1` | Reverberation time with the static base plate, in seconds. |
| `c3` | Speed of sound during `T3`, in m/s. |
| `T3` | Reverberation time with the rotating base plate, in seconds. |
| `m1` | Energy attenuation coefficient during `T1`, in 1/m; defaults to 0. |
| `m3` | Energy attenuation coefficient during `T3`, in 1/m; defaults to 0. |

**Returns:** Base-plate scattering coefficient `s_base` (per band).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for non-positive `V`, `S`, `c` or `T`. |

## check_base_plate_scattering

```python
check_base_plate_scattering(
    scattering: Mapping[Any, float] | Sequence[float] | ArrayLike,
) -> tuple[int, ...]
```

Verify base-plate scattering against Table 1 (ISO 17497-1, Clause 6.2).

Every band whose measured base-plate scattering coefficient exceeds the
[`BASE_PLATE_MAX_SCATTERING`](/phonometry/reference/api/materials/scattering-diffusion/#base_plate_max_scattering) limit is collected and a single
[`ScatteringDiffusionWarning`](/phonometry/reference/api/materials/scattering-diffusion/#scatteringdiffusionwarning) is issued when any band is over the
limit.

**Parameters**

| Name | Description |
| :--- | :--- |
| `scattering` | Measured base-plate scattering coefficients, either a mapping keyed by one-third-octave centre frequency (Hz) or a sequence of 18 values ordered as [`BASE_PLATE_BANDS`](/phonometry/reference/api/materials/scattering-diffusion/#base_plate_bands). |

**Returns:** Tuple of the centre frequencies (Hz) that exceed the limit, in ascending order (empty if the base plate is compliant).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a mapping missing a band or a sequence of the wrong length. |

## diffusion_spectrum

```python
diffusion_spectrum(
    frequencies: ArrayLike,
    diffusion: ArrayLike,
    *,
    normalized: ArrayLike | None = None,
) -> DiffusionSpectrum
```

Diffusion-coefficient spectrum `d(f)` (ISO 17497-2, Clause 8.5).

Pairs the per-band diffusion coefficients `d` with their band centres and
returns a plottable, reportable [`DiffusionSpectrum`](/phonometry/reference/api/materials/scattering-diffusion/#diffusionspectrum). The coefficient
is the *directional* coefficient `d_theta` (Formula (5)/(6)) when it comes
from a single source position, or the *random-incidence* coefficient `d`
when it is the per-band average of the directional coefficients over the
source positions (Clause 8.4, e.g. via [`random_incidence_diffusion`](/phonometry/reference/api/materials/scattering-diffusion/#random_incidence_diffusion)
band by band). The optional normalised coefficients `d_n` (Formula (7))
are carried through when supplied.

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequencies` | One-third-octave band centres, in hertz (1-D). |
| `diffusion` | Diffusion coefficient `d` per band. |
| `normalized` | Optional normalised diffusion coefficient `d_n` per band; `None` when the reference flat surface was not measured. |

**Returns:** A [`DiffusionSpectrum`](/phonometry/reference/api/materials/scattering-diffusion/#diffusionspectrum) with `.plot()` and `.report()`.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | if the inputs differ in length, are empty or not 1-D. |

## DiffusionResult

```python
DiffusionResult(angles: Real, levels: Real, coefficient: float)
```

A measured polar response and its diffusion coefficient (ISO 17497-2).

**Attributes**

| Name | Description |
| :--- | :--- |
| `angles` | Receiver angles of the polar response, in degrees. |
| `levels` | Reflected sound-pressure level at each angle, in decibels. |
| `coefficient` | Autocorrelation diffusion coefficient `d` (Formula (5)). |

### DiffusionResult.plot()

```python
DiffusionResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes
```

Plot the polar response with the diffusion coefficient annotated.

Requires matplotlib (`pip install phonometry[plot]`); returns the
polar `Axes` and never calls `plt.show`.

### DiffusionResult.report()

```python
DiffusionResult.report(
    path: str,
    *,
    metadata: ReportMetadata | None = None,
    engine: str = 'reportlab',
    verbose: bool = False,
    language: str = 'en',
) -> str
```

Render an ISO 17497-2 polar-response test-report fiche to a PDF.

Writes a one-page accredited free-field diffusion report for a single
source position (ISO 17497-2:2012, Clause 8.5): the standard-basis line,
an optional metadata header block, a two-panel body with the corrected
polar-response table (receiver angle and reflected sound-pressure level
`L`, rounded to 0,1 dB) beside the semicircular polar plot, a boxed
directional diffusion coefficient `d_theta` (Formula (5)/(6)) and a
footer with the fixed disclaimer. ISO 17497-2 is a characterisation, so
there is no pass/fail verdict.

**Parameters**

| Name | Description |
| :--- | :--- |
| `path` | Destination path of the PDF file. |
| `metadata` | Optional [`ReportMetadata`](/phonometry/reference/api/building/insulation/#reportmetadata); `None` produces a body-and-disclaimer fiche. The applicable descriptive fields are `client`, `manufacturer`, `specimen`, `mounting`, `test_room`, `test_date`, `temperature`, `relative_humidity`, `pressure`, `measurement_standard`, `laboratory`, `operator`, `report_id` and `notes`. The `requirement` field is ignored (ISO 17497-2 has no verdict). |
| `engine` | Rendering back end; only `"reportlab"` is supported. |
| `verbose` | Accepted for signature parity; the polar-response fiche has no extended table, so it renders the same body. |
| `language` | Fiche language: `"en"` (default, English, decimal point) or `"es"` (Spanish, decimal comma). |

**Returns:** The written `path` as a `str`.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | If `engine` is not `"reportlab"`. |
| ImportError | If reportlab is not installed (`pip install phonometry[report]`). |

## DiffusionSpectrum

```python
DiffusionSpectrum(
    frequencies: Real,
    diffusion: Real,
    normalized: Real | None = None,
)
```

A diffusion-coefficient spectrum `d(f)` (ISO 17497-2, Clause 8.5).

Where [`DiffusionResult`](/phonometry/reference/api/materials/scattering-diffusion/#diffusionresult) holds the polar response of a single
one-third-octave band, this holds the diffusion coefficient across the
measured bands, so it can be tabulated and plotted against frequency as
Clause 8.5 requires. The per-band coefficient is a *directional* diffusion
coefficient `d_theta` (Formula (5)/(6)) when it comes from one source
position, or a *random-incidence* diffusion coefficient `d` when it is the
per-band average of the directional coefficients over the source positions
(Clause 8.4); the standard defines both as frequency-dependent quantities,
so this carries a spectrum rather than a single number.

**Attributes**

| Name | Description |
| :--- | :--- |
| `frequencies` | One-third-octave band centre frequencies, in hertz. |
| `diffusion` | Diffusion coefficient `d` per band (directional per source, or random-incidence when averaged over source positions). |
| `normalized` | Optional normalised diffusion coefficient `d_n` per band (Formula (7)), or `None` when the reference flat surface was not measured. |

### DiffusionSpectrum.plot()

```python
DiffusionSpectrum.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes
```

Plot the diffusion coefficient `d` versus frequency.

Requires matplotlib (`pip install phonometry[plot]`); returns the
`Axes` and never calls `plt.show`.

### DiffusionSpectrum.report()

```python
DiffusionSpectrum.report(
    path: str,
    *,
    metadata: ReportMetadata | None = None,
    engine: str = 'reportlab',
    verbose: bool = False,
    language: str = 'en',
) -> str
```

Render an ISO 17497-2 diffusion-coefficient test-report fiche to a PDF.

Writes a one-page accredited free-field diffusion report
(ISO 17497-2:2012, Clause 8.5): the standard-basis line, an optional
metadata header block, a two-panel body with the per-band table
(frequency, the diffusion coefficient `d` and, when present, the
normalised `d_n`) beside the `d(f)` curve on a categorical band
axis, a boxed characterisation headline over the tested frequency range,
and a footer with the fixed disclaimer. ISO 17497-2 is a
characterisation, so there is no pass/fail verdict.

**Parameters**

| Name | Description |
| :--- | :--- |
| `path` | Destination path of the PDF file. |
| `metadata` | Optional [`ReportMetadata`](/phonometry/reference/api/building/insulation/#reportmetadata); `None` produces a body-and-disclaimer fiche whose header shows only the measured frequency range. The applicable descriptive fields are `client`, `manufacturer`, `specimen`, `mounting`, `test_room`, `test_date`, `temperature`, `relative_humidity`, `pressure`, `measurement_standard`, `laboratory`, `operator`, `report_id` and `notes`. The `requirement` field is ignored (ISO 17497-2 has no verdict). |
| `engine` | Rendering back end; only `"reportlab"` is supported. |
| `verbose` | When `True` and a normalised spectrum is present, the value table adds the normalised `d_n` column. |
| `language` | Fiche language: `"en"` (default, English, decimal point) or `"es"` (Spanish, decimal comma). |

**Returns:** The written `path` as a `str`.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | If `engine` is not `"reportlab"`. |
| ImportError | If reportlab is not installed (`pip install phonometry[report]`). |

## directional_diffusion

```python
directional_diffusion(
    angles: ArrayLike,
    levels: ArrayLike,
    *,
    weights: ArrayLike | None = None,
) -> DiffusionResult
```

Diffusion coefficient of a polar response (ISO 17497-2, Formula (5)/(6)).

Convenience wrapper over [`directional_diffusion_coefficient`](/phonometry/reference/api/materials/scattering-diffusion/#directional_diffusion_coefficient) that
keeps the receiver angles alongside the levels and returns a plottable
[`DiffusionResult`](/phonometry/reference/api/materials/scattering-diffusion/#diffusionresult).

**Parameters**

| Name | Description |
| :--- | :--- |
| `angles` | Receiver angles of the polar response, in degrees (1-D). |
| `levels` | Reflected sound-pressure level at each angle, in decibels. |
| `weights` | Optional area weights `N_i` (Formula (8)); `None` uses the equal-area Formula (5). |

**Returns:** A [`DiffusionResult`](/phonometry/reference/api/materials/scattering-diffusion/#diffusionresult) with `.plot()`.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | if `angles` and `levels` differ in length or are shorter than two receivers. |

## directional_diffusion_coefficient

```python
directional_diffusion_coefficient(
    levels: ArrayLike,
    *,
    area_weights: ArrayLike | None = None,
) -> float
```

Directional diffusion coefficient `d_theta` (ISO 17497-2, Formulas (5)/(6)).

For a fixed source position and one-third-octave band, from the `n`
reflected sound-pressure levels `L_i` (dB). With equal-area receivers
(`area_weights is None`, Formula (5)):

```text
d_theta = ((sum p_i)^2 - sum p_i^2) / ((n - 1) * sum p_i^2)
```

where `p_i = 10^(L_i / 10)`. When each receiver samples a different area
(Formula (6)) the per-receiver weights `N_i` (from [`area_factors`](/phonometry/reference/api/materials/scattering-diffusion/#area_factors))
enter:

```text
d_theta = ((sum p_i N_i)^2 - sum N_i p_i^2)
          / ((sum N_i - 1) * sum N_i p_i^2)
```

which reduces to Formula (5) for uniform weights. The coefficient is 0 when
only one receiver has non-zero scattered energy and 1 when all receivers
are equal.

**Parameters**

| Name | Description |
| :--- | :--- |
| `levels` | The `n >= 2` reflected sound-pressure levels `L_i`, in decibels (a level of `-inf` denotes a receiver with zero energy). |
| `area_weights` | Optional per-receiver area weights `N_i` (Formula (8)); `None` selects the equal-area Formula (5). |

**Returns:** Directional diffusion coefficient `d_theta` (a scalar).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for fewer than two receivers, a non-1-D input, a length mismatch, or non-positive total weight. |

## normalized_diffusion_coefficient

```python
normalized_diffusion_coefficient(
    d_theta: ArrayLike,
    d_theta_reference: ArrayLike,
) -> Real
```

Normalised directional diffusion coefficient (ISO 17497-2, Formula (7)).

`d_theta_n = (d_theta - d_theta_r) / (1 - d_theta_r)`, removing the
finite-panel diffusion of the reference flat surface `d_theta_r` (same
projected footprint as the test surface). It maps `d_theta = d_theta_r`
to 0 and `d_theta = 1` to 1.

**Parameters**

| Name | Description |
| :--- | :--- |
| `d_theta` | Directional diffusion coefficient of the test surface. |
| `d_theta_reference` | Directional diffusion coefficient of the reference flat surface `d_theta_r`. |

**Returns:** Normalised directional diffusion coefficient `d_theta_n`.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | if any reference coefficient equals 1 (undefined ratio). |

## random_incidence_absorption

```python
random_incidence_absorption(
    volume: float,
    area: float,
    *,
    c1: ArrayLike,
    T1: ArrayLike,
    c2: ArrayLike,
    T2: ArrayLike,
    m1: ArrayLike = 0.0,
    m2: ArrayLike = 0.0,
) -> Real
```

Random-incidence absorption coefficient `alpha_s` (ISO 17497-1, Eq. (1)).

`alpha_s = 55,3 * (V / S) * (1 / (c2 T2) - 1 / (c1 T1)) - (4 V / S) * (m2 - m1)`.

Situation 1 is the empty room with the (static) base plate present;
situation 2 adds the test sample, still without turntable rotation
(Table 2, rows T1 and T2).

**Parameters**

| Name | Description |
| :--- | :--- |
| `volume` | Reverberation-room volume `V`, in cubic metres. |
| `area` | Test-sample area `S`, in square metres. |
| `c1` | Speed of sound during `T1`, in m/s (see [`speed_of_sound`](/phonometry/reference/api/materials/scattering-diffusion/#speed_of_sound)). |
| `T1` | Reverberation time without sample (base plate only), in seconds. |
| `c2` | Speed of sound during `T2`, in m/s. |
| `T2` | Reverberation time with the test sample, in seconds. |
| `m1` | Energy attenuation coefficient during `T1`, in 1/m (see [`air_attenuation_coefficient`](/phonometry/reference/api/materials/scattering-diffusion/#air_attenuation_coefficient)); defaults to 0. |
| `m2` | Energy attenuation coefficient during `T2`, in 1/m; defaults to 0. |

**Returns:** Random-incidence absorption coefficient `alpha_s` (per band).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for non-positive `V`, `S`, `c` or `T`. |

## random_incidence_diffusion

```python
random_incidence_diffusion(
    directional_coefficients: ArrayLike,
    *,
    weights: ArrayLike | None = None,
) -> float
```

Random-incidence diffusion coefficient `d` (ISO 17497-2, Clause 8.4).

The (normalised or non-normalised) directional coefficients are averaged
over the source positions. Hemispherical measurements use **equal**
weightings (`weights is None`); two-dimensional (single-plane)
measurements use the source weighting of Clause 8.4 - weight 1 for the
0 deg source and weight 3 for each of the four +/-30 deg, +/-60 deg sources
(see [`TWO_DIMENSIONAL_SOURCE_WEIGHTS`](/phonometry/reference/api/materials/scattering-diffusion/#two_dimensional_source_weights)).

**Parameters**

| Name | Description |
| :--- | :--- |
| `directional_coefficients` | Directional diffusion coefficients `d_theta` (or `d_theta_n`), one per source position (1-D). |
| `weights` | Optional source-position weights; `None` averages with equal weight. |

**Returns:** Random-incidence diffusion coefficient `d` (a scalar).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for an empty or non-1-D input, a length mismatch, or non-positive total weight. |

## reverberation_time_uncertainty

```python
reverberation_time_uncertainty(times: ArrayLike) -> Real
```

Standard uncertainty of a reverberation time (ISO 17497-1, Eq. (A.1)).

`u = sqrt( sum_i (T_i - Tbar)^2 / (N (N - 1)) )` with `Tbar` the mean of
the `N` spatially-averaged measurements (Eq. (A.2)); this is the standard
error of the mean.

**Parameters**

| Name | Description |
| :--- | :--- |
| `times` | The `N >= 2` reverberation-time measurements, in seconds. |

**Returns:** Standard uncertainty `u` of the mean reverberation time (0-d).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | if fewer than two measurements are supplied. |

## scattering_coefficient

```python
scattering_coefficient(
    alpha_spec: ArrayLike,
    alpha_s: ArrayLike,
    *,
    truncate_negative: bool = True,
) -> Real
```

Random-incidence scattering coefficient `s` (ISO 17497-1, Eq. (5)).

`s = 1 - (1 - alpha_spec) / (1 - alpha_s)
= (alpha_spec - alpha_s) / (1 - alpha_s)`.

Following the presentation rule of Clause 8.3, negative results are
truncated to 0 while values greater than 1 (which can occur through edge
effects, Clause 6.3.2) are **kept** and reported. Rounding to 0,01 for a
results table is left to the caller.

**Parameters**

| Name | Description |
| :--- | :--- |
| `alpha_spec` | Specular absorption coefficient `alpha_spec` (Eq. (4)). |
| `alpha_s` | Random-incidence absorption coefficient `alpha_s` (Eq. (1)). |
| `truncate_negative` | If `True` (default), clip negative `s` to 0 per Clause 8.3; values above 1 are never clipped. |

**Returns:** Scattering coefficient `s` (per band).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | if any `alpha_s` equals 1 (undefined ratio). |

## scattering_coefficient_spectrum

```python
scattering_coefficient_spectrum(
    frequencies: ArrayLike,
    specular_absorption: ArrayLike,
    random_absorption: ArrayLike,
    *,
    truncate_negative: bool = True,
) -> ScatteringResult
```

Scattering-coefficient spectrum `s(f)` (ISO 17497-1, Eq. (5)).

Convenience wrapper over [`scattering_coefficient`](/phonometry/reference/api/materials/scattering-diffusion/#scattering_coefficient) that pairs the
per-band specular `alpha_spec` (Eq. (4)) and random-incidence `alpha_s`
(Eq. (1)) absorptions with their band centres and returns a plottable
[`ScatteringResult`](/phonometry/reference/api/materials/scattering-diffusion/#scatteringresult).

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequencies` | One-third-octave band centres, in hertz (1-D). |
| `specular_absorption` | Specular absorption `alpha_spec` per band. |
| `random_absorption` | Random-incidence absorption `alpha_s` per band. |
| `truncate_negative` | Clip negative `s` to 0 (Clause 8.3 default). |

**Returns:** A [`ScatteringResult`](/phonometry/reference/api/materials/scattering-diffusion/#scatteringresult) with `.plot()`.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | if the three inputs differ in length, are empty, or any `alpha_s` equals 1. |

## scattering_coefficient_uncertainty

```python
scattering_coefficient_uncertainty(
    alpha_spec: ArrayLike,
    alpha_s: ArrayLike,
    u_alpha_spec: ArrayLike,
    u_alpha_s: ArrayLike,
) -> ScatteringUncertainty
```

Uncertainty of the scattering coefficient (ISO 17497-1, Eq. (A.5)).

`u_s = |(alpha_spec - 1) / (1 - alpha_s)|
* sqrt((u_alpha_spec / (alpha_spec - 1))^2 + (u_alpha_s / (1 - alpha_s))^2)`,

with the expanded uncertainty `U = 2 u_s` (95 % confidence).

**Parameters**

| Name | Description |
| :--- | :--- |
| `alpha_spec` | Specular absorption coefficient `alpha_spec` (Eq. (4)). |
| `alpha_s` | Random-incidence absorption coefficient `alpha_s` (Eq. (1)). |
| `u_alpha_spec` | Standard uncertainty of `alpha_spec` (Eq. (A.4)). |
| `u_alpha_s` | Standard uncertainty of `alpha_s` (Eq. (A.3)). |

**Returns:** A [`ScatteringUncertainty`](/phonometry/reference/api/materials/scattering-diffusion/#scatteringuncertainty) with `u_s` and `U = 2 u_s`.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | if any `alpha_s` equals 1 or any `alpha_spec` equals 1. |

## ScatteringDiffusionWarning

Advisory for out-of-range scattering/diffusion measurement conditions.

## ScatteringResult

```python
ScatteringResult(
    frequencies: Real,
    scattering: Real,
    random_incidence: Real,
    specular: Real,
)
```

A random-incidence scattering-coefficient spectrum (ISO 17497-1).

**Attributes**

| Name | Description |
| :--- | :--- |
| `frequencies` | One-third-octave band centre frequencies, in hertz. |
| `scattering` | Scattering coefficient `s` per band (Eq. (5)). |
| `random_incidence` | Random-incidence absorption `alpha_s` (Eq. (1)). |
| `specular` | Specular absorption `alpha_spec` (Eq. (4)). |

### ScatteringResult.plot()

```python
ScatteringResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes
```

Plot the scattering coefficient `s` versus frequency.

Requires matplotlib (`pip install phonometry[plot]`); returns the
`Axes` and never calls `plt.show`.

### ScatteringResult.report()

```python
ScatteringResult.report(
    path: str,
    *,
    metadata: ReportMetadata | None = None,
    engine: str = 'reportlab',
    verbose: bool = False,
    language: str = 'en',
) -> str
```

Render an ISO 17497-1 scattering-coefficient test-report fiche to a PDF.

Writes a one-page accredited random-incidence scattering report
(ISO 17497-1:2004+A1:2014): the standard-basis line, an optional
metadata header block (client, specimen, test room, sample area `S`,
temperature, humidity ...), a two-panel body with the per-band table
(frequency, the random-incidence absorption `alpha_s` and the
scattering coefficient `s`) beside the `s(f)` curve on a categorical
band axis, and a footer with the fixed disclaimer. ISO 17497-1 is a
characterisation, so there is no pass/fail verdict and no single-number
rating.

**Parameters**

| Name | Description |
| :--- | :--- |
| `path` | Destination path of the PDF file. |
| `metadata` | Optional [`ReportMetadata`](/phonometry/reference/api/building/insulation/#reportmetadata); `None` produces a body-and-disclaimer fiche whose header shows only the measured frequency range. The applicable descriptive fields are `client`, `manufacturer`, `specimen`, `area`, `room_volume`, `mounting`, `test_room`, `test_date`, `temperature`, `relative_humidity`, `pressure`, `measurement_standard`, `laboratory`, `operator`, `report_id` and `notes`. The `requirement` field is ignored (ISO 17497-1 has no verdict). |
| `engine` | Rendering back end; only `"reportlab"` is supported. |
| `verbose` | When `True`, the value table inserts the specular absorption `alpha_spec` column beside `alpha_s` and `s`. |
| `language` | Fiche language: `"en"` (default, English, decimal point) or `"es"` (Spanish, decimal comma). |

**Returns:** The written `path` as a `str`.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | If `engine` is not `"reportlab"`. |
| ImportError | If reportlab is not installed (`pip install phonometry[report]`). |

## ScatteringUncertainty

```python
ScatteringUncertainty(u_scattering: Real, expanded: Real)
```

Uncertainty of the scattering coefficient (ISO 17497-1, Annex A).

**Attributes**

| Name | Description |
| :--- | :--- |
| `u_scattering` | Combined standard uncertainty `u_s` of the scattering coefficient (Eq. (A.5)). |
| `expanded` | Expanded uncertainty `U = 2 u_s` at 95 % confidence (Annex A). |

## specular_absorption_coefficient

```python
specular_absorption_coefficient(
    volume: float,
    area: float,
    *,
    c3: ArrayLike,
    T3: ArrayLike,
    c4: ArrayLike,
    T4: ArrayLike,
    m3: ArrayLike = 0.0,
    m4: ArrayLike = 0.0,
) -> Real
```

Specular absorption coefficient `alpha_spec` (ISO 17497-1, Eq. (4)).

`alpha_spec = 55,3 * (V / S) * (1 / (c4 T4) - 1 / (c3 T3)) - (4 V / S) * (m4 - m3)`.

Situation 3 is the rotating base plate without the sample; situation 4 is
the sample on the rotating turntable (Table 2, rows T3 and T4). The
apparent (specular) absorption includes the energy lost to scattering.

**Parameters**

| Name | Description |
| :--- | :--- |
| `volume` | Reverberation-room volume `V`, in cubic metres. |
| `area` | Test-sample area `S`, in square metres. |
| `c3` | Speed of sound during `T3`, in m/s. |
| `T3` | Reverberation time, rotating base plate without sample, in seconds. |
| `c4` | Speed of sound during `T4`, in m/s. |
| `T4` | Reverberation time, sample on the rotating turntable, in seconds. |
| `m3` | Energy attenuation coefficient during `T3`, in 1/m; defaults to 0. |
| `m4` | Energy attenuation coefficient during `T4`, in 1/m; defaults to 0. |

**Returns:** Specular absorption coefficient `alpha_spec` (per band).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for non-positive `V`, `S`, `c` or `T`. |

## speed_of_sound

```python
speed_of_sound(temperature: ArrayLike) -> Real
```

Speed of sound in air (ISO 17497-1:2004, Clause 8, Eq. (2)).

`c = 343,2 * sqrt((273,15 + t) / 293,15)` (m/s).

**Parameters**

| Name | Description |
| :--- | :--- |
| `temperature` | Air temperature `t`, in **degrees Celsius** (scalar or per band). |

**Returns:** Speed of sound `c`, in metres per second.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | if any temperature is at or below -273,15 degC. |

## TWO_DIMENSIONAL_SOURCE_WEIGHTS

*Constant* (`tuple`).

```python
TWO_DIMENSIONAL_SOURCE_WEIGHTS = (1, 3, 3, 3, 3)
```
