← [Documentation index](README.md)

# Surface Scattering, Diffusion and In-situ Absorption

How a surface returns incident sound (how much it scatters away from the
specular direction, how uniformly it spreads what it scatters, and how much it
absorbs) is measured by a family of dedicated methods. The **reverberation
room** gives the random-incidence *scattering coefficient* of a surface by
comparing decays with the sample held still and rotating (ISO 17497-1). A
**free-field goniometer** measures the polar response of the reflected sound and
condenses it into a *diffusion coefficient* (ISO 17497-2). And out on a road, a
loudspeaker and a single microphone recover the *in-situ absorption* of the
pavement, either over an extended surface by subtracting the incident wave
(ISO 13472-1) or through a small tube pressed onto the surface (ISO 13472-2).
This page covers all four.

The scattering and diffusion coefficients answer different questions and are not
interchangeable: scattering is *how much* energy leaves the specular direction;
diffusion is *how evenly* the reflected energy is spread over angle.

## 1. Random-incidence scattering coefficient (ISO 17497-1)

The scattering coefficient $s$ is the fraction of reflected energy that does
**not** leave the surface in the specular direction. ISO 17497-1 measures it in a
reverberation room from four reverberation-time situations: with the test sample
mounted on a turntable and held **stationary**, and with the turntable
**rotating** (which averages the phase-coherent specular reflection away), each
with and without a reflecting base plate.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_scattering_reverb_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_scattering_reverb.svg" alt="ISO 17497-1 random-incidence scattering setup: a reverberation room with the test sample on a turntable, a rotating loudspeaker boom and a microphone, measuring reverberation time with the sample stationary (giving the random-incidence absorption) and rotating (giving the specular absorption), from which the scattering coefficient is derived" width="92%"></picture>

**Absorption from reverberation time (Clause 6).** Each situation converts to a
Sabine absorption coefficient with the standard's own air-attenuation term:

$$
\alpha = 55.3\,\frac{V}{S}\left(\frac{1}{c_2 T_2} - \frac{1}{c_1 T_1}\right)
        - 4\,\frac{V}{S}\,(m_2 - m_1),
$$

where $V$ is the room volume, $S$ the sample area, $T$ the reverberation time,
$c$ the speed of sound (Eq. (2): $c = 343.2\sqrt{(273.15+t)/293.15}$) and $m$ the
power attenuation coefficient of air. The **stationary** pair gives the
random-incidence absorption $\alpha_s$ (Eq. (1)); the **rotating** pair gives the
specular absorption $\alpha_{spec}$ (Eq. (4)).

**Scattering coefficient (Eq. (5)).** The two combine into

$$
s = \frac{\alpha_{spec} - \alpha_s}{1 - \alpha_s}.
$$

A fully specular surface reflects all its non-absorbed energy in the specular
direction, so $\alpha_{spec} = \alpha_s$ and $s = 0$; a strong diffuser sends
energy everywhere, raising $\alpha_{spec}$ towards 1 and $s$ towards 1.

```python
from phonometry import materials

# Four reverberation-time situations reduced to two absorption coefficients.
# alpha_s from the stationary pair (Eq. 1); alpha_spec from the rotating pair
# (Eq. 4). V = 200 m^3, S = 10 m^2, c = 343.2 m/s throughout.
alpha_s = materials.random_incidence_absorption(200.0, 10.0, c1=343.2, T1=8.0,
                                         c2=343.2, T2=6.0)
alpha_spec = materials.specular_absorption_coefficient(200.0, 10.0, c3=343.2, T3=7.5,
                                                c4=343.2, T4=5.0)
s = materials.scattering_coefficient(alpha_spec, alpha_s)   # Eq. (5)
print(round(float(alpha_s), 4))     # 0.1343
print(round(float(alpha_spec), 4))  # 0.2148
print(round(float(s), 4))           # 0.0931
```

Over a full one-third-octave measurement, `scattering_coefficient_spectrum`
pairs the per-band $\alpha_{spec}$ and $\alpha_s$ with their band centres and
returns a plottable `ScatteringResult`:

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

# A 13-band measurement (250-4000 Hz): the random-incidence absorption alpha_s
# (stationary sample) and the specular absorption alpha_spec (rotating
# turntable). A diffuser scatters more with frequency, so s(f) rises.
freqs = np.array([250, 315, 400, 500, 630, 800, 1000,
                  1250, 1600, 2000, 2500, 3150, 4000], float)
alpha_s = np.full_like(freqs, 0.10)
alpha_spec = 0.11 + 0.75 * (np.log10(freqs / 250) / np.log10(4000 / 250))

result = materials.scattering_coefficient_spectrum(freqs, alpha_spec, alpha_s)
print(np.round(result.scattering[[0, 6, 12]], 3))   # [0.011 0.428 0.844]
result.plot()   # s(f) on a log-frequency axis, 0 to 1 (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/scattering_coefficient_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/scattering_coefficient.svg" alt="The random-incidence scattering coefficient s of a diffusing surface over the 13 one-third-octave bands from 250 to 4000 Hz, rising smoothly from near zero at low frequency towards 0.84 at 4 kHz" width="88%"></picture>

*The scattering coefficient climbs with frequency: at low frequency the surface
relief is small compared with the wavelength and the reflection stays specular
($s \to 0$); as the wavelength shrinks the relief scatters more energy out of
the specular direction ($s \to 1$).*

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

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

# A 13-band measurement (250-4000 Hz): the random-incidence absorption alpha_s
# (stationary sample) and the specular absorption alpha_spec (rotating
# turntable). A diffuser scatters more with frequency, so s(f) rises.
freqs = np.array([250, 315, 400, 500, 630, 800, 1000,
                  1250, 1600, 2000, 2500, 3150, 4000], float)
alpha_s = np.full_like(freqs, 0.10)
alpha_spec = 0.11 + 0.75 * (np.log10(freqs / 250) / np.log10(4000 / 250))
result = materials.scattering_coefficient_spectrum(freqs, alpha_spec, alpha_s)

# result is the ScatteringResult computed above. One line:
result.plot()
plt.show()

# By hand, from the result's fields, mirroring what ScatteringResult.plot() draws:
fig, ax = plt.subplots()
ax.semilogx(result.frequencies, result.scattering, "o-", color="#1f77b4")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Scattering coefficient s")
ax.set_ylim(0.0, 1.0)
ax.set_title("Random-incidence scattering coefficient (ISO 17497-1)")
plt.show()
```

</details>

**Base-plate check (Clause 6.4, Table 1).** The empty base plate must itself
scatter only negligibly, or it would bias the result. ISO 17497-1 caps the
base-plate scattering coefficient per one-third-octave band; the library exposes
those limits and a checker.

```python
from phonometry import materials

# The normative per-band ceilings (Table 1): 0.05 up to 500 Hz, rising to 0.25.
# materials.BASE_PLATE_BANDS is the band tuple; materials.BASE_PLATE_MAX_SCATTERING maps band -> ceiling.
print(materials.BASE_PLATE_BANDS[0], materials.BASE_PLATE_MAX_SCATTERING[100])   # 100 0.05

# A base plate whose measured scattering stays under the ceiling passes silently;
# an over-limit band raises a ScatteringDiffusionWarning listing the offenders.
materials.check_base_plate_scattering([0.02] * len(materials.BASE_PLATE_BANDS))
```

**Test-report fiche.** `ScatteringResult.report(path)` renders a one-page
accredited scattering test report (ISO 17497-1): a metadata header, the
per-one-third-octave table of the random-incidence absorption $\alpha_s$ and the
scattering coefficient $s$ beside the $s(f)$ curve on a categorical band axis,
and a boxed characterisation headline (no pass/fail). `verbose=True` adds the
specular absorption $\alpha_{spec}$ column and `language="es"` renders the
Spanish fiche; it needs the report extra (`pip install phonometry[report]`).

[![ISO 17497-1 scattering example report: a metadata header, the per-one-third-octave table of the random-incidence absorption and the scattering coefficient beside the s(f) band-axis curve, and the boxed characterisation headline over the tested frequency range](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso17497_scattering_example.webp)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso17497_scattering_example.pdf)

## 2. Diffusion coefficient (ISO 17497-2)

The diffusion coefficient $d$ measures the **spatial uniformity** of the
reflected sound, not how much is scattered. A goniometer sweeps a receiver over a
polar arc and records the reflected level $L_i$ at each angle; the coefficient is
the normalised autocorrelation of the polar energy distribution.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_diffusion_goniometer_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_diffusion_goniometer.svg" alt="ISO 17497-2 free-field diffusion goniometer: a test sample on a turntable, a fixed loudspeaker source, and a semicircular arc of receiver microphones sampling the reflected polar response, from which the autocorrelation diffusion coefficient is computed" width="92%"></picture>

**Autocorrelation (Formula (5)).** For $n$ receivers at equal angular spacing,
with $p_i = 10^{L_i/10}$ the band energy at receiver $i$,

$$
d = \frac{\left(\sum_i p_i\right)^2 - \sum_i p_i^2}
         {(n-1)\,\sum_i p_i^2}.
$$

A perfectly uniform polar response ($L_i$ all equal) gives $d = 1$; a single
sharp specular lobe gives $d = 0$. When receivers subtend unequal solid angles,
Formula (6) area-weights each energy by $N_i$ from Formula (8), and those area
factors are evaluated in **radians**, which is why a 5° spacing at the zenith
produces a weight near 1.57, not 51.9.

The animation below runs that goniometer experiment numerically: the same
plane wavefront hits a flat rigid panel and a Schroeder diffuser (an N = 7
quadratic-residue profile), and the scattered energy on the receiver arc turns
a collimated specular beam (d = 0.32) into a wide fan (d = 0.63).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_diffusion_dark.gif"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_diffusion.gif" alt="Animation: a 2D FDTD simulation of a plane wavefront hitting a flat rigid panel and a Schroeder quadratic-residue diffuser side by side; the flat panel throws a collimated specular beam back while the diffuser's phase-step wells spread the same energy into a wide fan, and the scattered field on a receiver arc yields diffusion coefficients of 0.32 versus 0.63" width="640" height="360" loading="lazy"></picture>

[Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_diffusion.webm)

The single-plane response below is the far-field prediction of a published
diffuser geometry: an N = 7 quadratic-residue diffuser of 6 periods, 3.6 m
total width and 0.2 m maximum well depth (the "N = 7 QRD, 6 periods, 0.2 m
deep" row of Cox & D'Antonio, *Acoustic Absorbers and Diffusers*, 3rd ed.,
Appendix B; the commercial N = 7 QRD measured by Hargreaves, Cox, Lam &
D'Antonio, *J. Acoust. Soc. Am.* 108(4), 1710-1720, 2000, Table I) and its
equal-footprint flat reference panel, at 1000 Hz, normal incidence, on the
standard 37-point semicircle (5 deg spacing, -90 to +90 deg). The same model
levels drive the phonometry conformance suite as an arithmetic oracle for
Formulas (5) and (7); the external third-party anchor checks the model's
band-averaged normalised diffusion against the published Appendix B BEM table
in the 200-400 Hz bands (agreement within 0.01; over the full published
100-5000 Hz range the model-vs-BEM mean absolute deviation is about 0.09,
because edge diffraction is outside the Fraunhofer model).

```python
from phonometry import materials

# The published geometry: N = 7 QRD, 6 periods, 3.6 m wide, 0.2 m deep
# (Cox & D'Antonio 3rd ed., Appendix B; Hargreaves et al. 2000, Table I).
# Design frequency 490 Hz puts the deepest well at exactly 0.2 m.
depths = materials.qrd_well_depths(7, 490.0)   # [0, 0.05, 0.2, 0.1, ...] m
qrd = materials.predict_diffuser_polar_response(
    3.6 / 42, 1000.0, depths=depths, periods=6)
# The flat reference panel: the same 3.6 m footprint with zero-depth wells.
flat = materials.predict_diffuser_polar_response(
    3.6 / 42, 1000.0, depths=[0.0] * 7, periods=6)

d = materials.directional_diffusion_coefficient(qrd.levels)    # Formula (5)
print(round(float(d), 4))            # 0.1099
d_ref = materials.directional_diffusion_coefficient(flat.levels)
print(round(float(d_ref), 4))        # 0.0049

# Normalise against the flat reference to isolate the diffuser's own effect
# (Formula (7)): d_n = (d - d_ref) / (1 - d_ref).
d_n = materials.normalized_diffusion_coefficient(d, d_ref)
print(round(float(d_n), 4))          # 0.1055

# Random-incidence value: average the band coefficients over source positions,
# with the standard's 2-D weighting (0 deg -> 1, +/-30/+/-60 deg -> 3).
d_random = materials.random_incidence_diffusion(
    [0.5, 0.2, 0.2, 0.2, 0.2], weights=materials.TWO_DIMENSIONAL_SOURCE_WEIGHTS)
print(round(float(d_random), 4))     # 0.2231
```

`directional_diffusion` keeps the receiver angles beside the levels of a full
goniometer sweep and returns a plottable `DiffusionResult`. Reusing the QRD
polar levels from above:

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

# qrd.levels: the 37 predicted QRD levels from the example above.
angles = np.arange(-90.0, 90.5, 5.0)

result = materials.directional_diffusion(angles, qrd.levels)
print(round(result.coefficient, 2))   # 0.11
result.plot()   # polar reflected response, d in the title (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diffusion_polar_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diffusion_polar.svg" alt="A polar plot of the predicted reflected sound-pressure level of a six-period N=7 quadratic-residue diffuser over 37 receivers from -90 to 90 degrees, the energy concentrated in a fan of discrete grating lobes, giving an autocorrelation diffusion coefficient d of about 0.11" width="72%"></picture>

*The periodic QRD array splits the reflected energy into a fan of discrete
grating lobes rather than one specular spike, but six repetitions of the same
period concentrate the energy in those few directions, so the autocorrelation
diffusion coefficient stays modest ($d \approx 0.11$). The flat reference
panel collapses into the specular direction alone and drops to
$d \approx 0.005$; this lobing penalty of periodic arrays is exactly why
Cox & D'Antonio recommend modulated arrangements.*

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

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

# qrd.levels: the 37 predicted levels L_i of the six-period N = 7 QRD from
# the example above (Fraunhofer far-field model, published Cox & D'Antonio
# Appendix B geometry, 1000 Hz, normal incidence) on the -90..90 deg, 5 deg
# semicircle. The periodic array concentrates the energy into grating lobes,
# so the Formula (5) coefficient d is modest.
angles = np.arange(-90.0, 90.5, 5.0)
result = materials.directional_diffusion(angles, qrd.levels)

# result is the DiffusionResult computed above. One line:
result.plot()
plt.show()

# By hand: a polar plot of the levels, with d annotated in the title.
fig, ax = plt.subplots(subplot_kw={"projection": "polar"})
theta = np.radians(result.angles)
ax.plot(theta, result.levels, "o-", color="#1f77b4")
ax.fill(theta, result.levels, color="#1f77b4", alpha=0.15)
ax.set_theta_zero_location("N")
ax.set_theta_direction(-1)
ax.set_thetamin(-90)
ax.set_thetamax(90)
ax.set_title(f"Directional diffusion  d = {result.coefficient:.2f}  (ISO 17497-2)")
plt.show()
```

</details>

**Test-report fiche.** Collected across the one-third-octave bands, the
diffusion coefficient $d(f)$ forms a `DiffusionSpectrum`
(`materials.diffusion_spectrum(freqs, d, ...)`), whose `report(path)` renders a
one-page diffusion test report (ISO 17497-2, Clause 8.5): the per-band table of
$d$ beside the $d(f)$ band-axis curve, with a boxed characterisation headline.
Per Clause 8.4 the random-incidence coefficient is itself a per-band quantity,
the average of the directional coefficients over the source positions band by
band (not a mean across frequency). `verbose=True` adds the normalised $d_n$
column to the table (the curve always draws $d_n$ as a companion when present).
A single band's polar response is itself reportable through
`DiffusionResult.report(path)`.

[![ISO 17497-2 diffusion example report: a metadata header, the per-one-third-octave table of the diffusion coefficient d beside the d(f) band-axis curve (with the normalised d_n drawn as a companion curve), and the boxed characterisation headline over the tested frequency range](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso17497_diffusion_example.webp)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso17497_diffusion_example.pdf)

## Scattering or diffusion? Two coefficients, two jobs

The two coefficients above are routinely treated as interchangeable, in
product data sheets and occasionally in simulation manuals. They are not, and
neither can be computed from the other. The scattering coefficient $s$
(ISO 17497-1) does **energy bookkeeping**: what fraction of the reflected
energy leaves the specular direction. It says nothing about where that energy
goes. The diffusion coefficient $d$ (ISO 17497-2) grades **spatial quality**:
how evenly the reflected energy covers the receiver arc, one source direction
at a time. It says nothing about how the energy divides between the specular
lobe and the rest.

A pair of counterexamples keeps them apart. A curved or tilted element that
*redirects* the reflection concentrates nearly all the reflected energy into
one strong lobe away from the specular direction: almost everything is
non-specular, so $s$ is high, yet the beam is as collimated as a mirror's, so
$d$ stays low. Conversely, a small flat panel measured at low frequency sends
nearly all the reflected energy toward the specular direction, so $s$ stays
near zero, yet edge diffraction spreads that reflection so broadly over the
receiver arc that the measured $d$ comes out surprisingly high. High $s$ does
not mean uniform; decent $d$ does not mean much energy was scattered at all.

**Which one for design.** The two numbers serve different consumers:

- **The scattering coefficient feeds room-acoustics simulation.**
  Geometrical-acoustics engines decide at every wall reflection how much
  energy continues specularly and how much is redistributed; the per-band
  random-incidence $s$ of each surface is precisely that split, which is what
  ISO 17497-1 was written to supply. Getting it wrong shows up as the wrong
  reverberation-time and clarity predictions in non-mixing rooms.
- **The diffusion coefficient qualifies diffusers.** When the task is to break
  up an echo, a flutter or a focusing reflection, what matters is that the
  reflected energy is spread over angle, and $d$ measures exactly that. The
  normalised $d_n$ (Formula (7)) additionally subtracts the edge diffraction
  that every finite panel exhibits, so products of different sizes can be
  compared fairly.

Swapping them fails in both directions: a diffusion coefficient dropped into a
simulator's scattering slot biases the energy split, and a scattering
coefficient quoted as proof of "diffusion" may describe a surface that merely
redirects the problem reflection somewhere else. Both coefficients are
one-third-octave-band functions that generally rise once the surface relief is
no longer small against the wavelength; a frequency-blind single number
("scatters 90 % of the sound") is neither of them.

## 3. In-situ road absorption: subtraction technique (ISO 13472-1)

Out in the field there is no reverberation room. ISO 13472-1 measures the sound
absorption of a road surface (or any extended flat surface) *in situ* by firing
an impulse from a loudspeaker at height $d_s$ down onto the surface and recording
the impulse response at a microphone at height $d_m$. The **incident** and
**reflected** components are separated in time with an Adrienne window; their
transfer function gives the reflection factor and hence the absorption.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_insitu_subtraction_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_insitu_subtraction.svg" alt="ISO 13472-1 in-situ road absorption by the subtraction technique: a loudspeaker at 1.25 m and a microphone at 0.25 m above the road surface, with the direct and road-reflected ray paths and a free-field reference measurement, the reflected component isolated by an Adrienne time window" width="92%"></picture>

**Geometrical spreading (Clause 4.1).** The reflected wave travels farther than
the direct wave, so it is attenuated by the geometrical-spreading factor

$$
K_r = \frac{d_s - d_m}{d_s + d_m},
$$

which equals $2/3$ for the mandatory geometry $d_s = 1.25$ m, $d_m = 0.25$ m.
The absorption follows from the windowed incident and reflected spectra
$H_i$, $H_r$:

$$
\alpha(f) = 1 - \frac{1}{K_r^2}\left|\frac{H_r(f)}{H_i(f)}\right|^2.
$$

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

# A band-limited incident impulse response and a synthetic road reflection
# hr = Kr * r0 * delayed(hi): a reflection of magnitude r0 = 0.4, delayed by the
# extra path, and scaled by the geometrical-spreading factor Kr.
fs, n = 48000.0, 4096
t = np.arange(n) / fs
hi = np.zeros(n)
hi[:64] = np.hanning(64) * np.cos(2.0 * np.pi * 1500.0 * t[:64])

kr = materials.geometric_spreading_factor()          # (ds - dm)/(ds + dm) = 2/3
hr = kr * 0.4 * np.roll(hi, 96)

# Narrow-band absorption, then reduced to one-third octaves over 250-4000 Hz.
alpha = materials.insitu_absorption_coefficient(hi, hr)   # 1 - (1/Kr^2)|Hr/Hi|^2
freq = np.fft.rfftfreq(n, 1.0 / fs)
centres, band = materials.one_third_octave_absorption(freq, alpha)
print(round(kr, 4))                # 0.6667
print(round(float(band[2]), 3))    # 0.84  (alpha = 1 - 0.4^2 = 0.84)
```

**Adrienne window (Clause 6.4).** The time window that isolates the reflection
mandates only a sharp leading edge, a 5 ms flat portion and a cosine-squared or
Blackman-Harris trailing edge; the exact durations are reported per measurement,
not fixed, so they are configurable here.

```python
from phonometry import materials

# Default: 0.5 ms leading edge, 5 ms flat top, 5 ms Blackman-Harris trailing.
w = materials.adrienne_window(48000.0)
print(w.shape[0])          # 504 samples at 48 kHz
print(round(float(w.max()), 3))   # 1.0  (flat top and edges meet at unity)
```

**End-to-end spectrum.** `insitu_absorption_spectrum` runs the whole chain (the
windowed incident and reflected impulse responses to the narrow-band absorption
and on to one-third-octave bands) and returns a plottable
`InsituAbsorptionResult`:

```python
import numpy as np
from phonometry import materials
from scipy.signal import firwin, lfilter

# A synthetic-but-realistic measurement. hi is a unit incident impulse; the road
# reflection hr = Kr * r0 * roll(hi, shift) uses the geometrical-spreading
# factor Kr, a mildly frequency-dependent r0 (a gentle low-pass, so a porous
# surface reflects less as frequency rises) and the reflected-path delay
# shift = round(2 dm / c * fs).
fs, n = 48000.0, 8192
kr = materials.geometric_spreading_factor()           # (ds - dm)/(ds + dm) = 2/3
hi = np.zeros(n)
hi[0] = 1.0
taps = firwin(41, 1200.0, fs=fs)
taps = taps / taps.sum()
shift = int(round(2.0 * 0.25 / 340.0 * fs))     # reflected-path delay 2 dm / c
hr = kr * 0.85 * np.roll(lfilter(taps, 1.0, hi), shift)

result = materials.insitu_absorption_spectrum(hi, hr, fs)
print(result.frequencies[[0, -1]].astype(int))     # [ 250 4000]
print(np.round(result.absorption[[0, 6, 12]], 2))  # [0.31 0.65 1.  ]
result.plot()   # alpha(f) bar chart over 250-4000 Hz (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insitu_absorption_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insitu_absorption.svg" alt="An in-situ one-third-octave road-surface absorption spectrum computed by the reflection-factor route from a synthetic road reflection, rising from about 0.3 at 250 Hz to near 1.0 above 2 kHz" width="88%"></picture>

*The absorption rises with frequency because the surface reflects less of the
high-frequency energy, exactly as the low-pass reflection factor $r_0(f)$
dictates through $\alpha = 1 - (1/K_r^2)\,|H_r/H_i|^2$.*

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

```python
import matplotlib.pyplot as plt
import numpy as np
from scipy.signal import firwin, lfilter
from phonometry import materials

# A synthetic-but-realistic measurement. hi is a unit incident impulse; the road
# reflection hr = Kr * r0 * roll(hi, shift) uses the geometrical-spreading
# factor Kr, a mildly frequency-dependent r0 (a gentle low-pass, so a porous
# surface reflects less as frequency rises) and the reflected-path delay
# shift = round(2 dm / c * fs).
fs, n = 48000.0, 8192
kr = materials.geometric_spreading_factor()           # (ds - dm)/(ds + dm) = 2/3
hi = np.zeros(n)
hi[0] = 1.0
taps = firwin(41, 1200.0, fs=fs)
taps = taps / taps.sum()
shift = int(round(2.0 * 0.25 / 340.0 * fs))     # reflected-path delay 2 dm / c
hr = kr * 0.85 * np.roll(lfilter(taps, 1.0, hi), shift)
result = materials.insitu_absorption_spectrum(hi, hr, fs)

# result is the InsituAbsorptionResult computed above. One line:
result.plot()
plt.show()

# By hand: a bar chart of alpha over the one-third-octave bands.
freqs = result.frequencies
positions = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(positions, np.nan_to_num(result.absorption), width=0.7, color="#1f77b4")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Absorption coefficient alpha")
ax.set_ylim(0.0, 1.0)
ax.set_title("In-situ road-surface absorption (ISO 13472-1)")
plt.show()
```

</details>

**Maximum sampled area (Annex A).** The finite time window limits how much of the
surface contributes to the reflection. The maximum sampled area is a circle whose
radius the library computes from the geometry and window width; the Annex A worked
example ($d_s = 1.25$ m, $d_m = 0.25$ m, $c = 340$ m/s, 5 ms flat window) gives
about 1.34 m.

```python
from phonometry import materials
print(round(materials.max_sampled_area_radius(5.0e-3), 3))   # 1.343  (metres)
```

## 4. In-situ road absorption: spot method (ISO 13472-2)

For smaller patches, ISO 13472-2 seals a short circular tube onto the surface and
measures the absorption with the two-microphone transfer-function method of
ISO 10534-2. The library provides the spot-method geometry and validity helpers;
the transfer-function DSP itself is the impedance-tube routine
`two_microphone_impedance` (see [Acoustic Materials](materials.md)).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_spot_tube_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_spot_tube.svg" alt="ISO 13472-2 spot method: a short circular tube sealed onto the road surface with a loudspeaker at the top and two microphones flush in the tube wall at spacing s, measuring absorption over 250 to 1600 Hz via the ISO 10534-2 two-microphone transfer-function method" width="92%"></picture>

**Plane-wave limits (Clause 5.4).** The tube supports only plane waves below

$$
f_u = 0.58\,\frac{c_0}{d},
$$

with $d$ the tube diameter, and the microphone spacing $s$ must sit between
$0.05\,c_0/f_{min}$ and $0.45\,c_0/f_{max}$. The reported range is the
one-third-octave bands 250–1600 Hz.

```python
from phonometry import materials

# Upper usable frequency of a 100 mm tube and the valid spacing window.
print(round(materials.spot_tube_upper_frequency(0.100, 343.0), 1))      # 1989.4 Hz
s_min, s_max = materials.spot_microphone_spacing_bounds(
    343.0, f_min=220.0, f_max=1800.0)
print(round(s_min, 3), round(s_max, 3))    # 0.078 0.086  (metres)
```

## References

- Cox, T. J., & D'Antonio, P. (2017). *Acoustic absorbers and diffusers:
  Theory, design and application* (3rd ed.). CRC Press.
  ISBN 978-1-4987-4099-9.
  [doi:10.1201/9781315369211](https://doi.org/10.1201/9781315369211).
  The reference monograph on diffuser theory and design, by the authors
  behind the ISO 17497-2 diffusion-coefficient method: the
  scattering-versus-diffusion distinction, the
  measurement rigs and the design guidance this page condenses. Appendix B
  (Normalized diffusion coefficient table, pp. 481-485) is the published BEM
  anchor for the diffuser-prediction model on this page.
- Hargreaves, T. J., Cox, T. J., Lam, Y. W., & D'Antonio, P. (2000). Surface
  diffusion coefficients for room acoustics: Free-field measures of
  single-plane diffusion. *The Journal of the Acoustical Society of America*,
  108(4), 1710-1720.
  [doi:10.1121/1.1310192](https://doi.org/10.1121/1.1310192).
  The free-field diffusion-coefficient method behind ISO 17497-2; its Table I
  documents the commercial N = 7 QRD geometry (0.2 m maximum well depth) used
  in the worked example above.
- International Organization for Standardization. (2004). *Acoustics —
  Sound-scattering properties of surfaces — Part 1: Measurement of the
  random-incidence scattering coefficient in a reverberation room*
  (ISO 17497-1:2004+A1:2014, the edition implemented here).
  [iso.org catalogue](https://www.iso.org/standard/31397.html).
  The turntable method implemented in section 1 of this page and the
  base-plate limits of the standard's Table 1.
- International Organization for Standardization. (2012). *Acoustics —
  Sound-scattering properties of surfaces — Part 2: Measurement of the
  directional diffusion coefficient in a free field* (ISO 17497-2:2012).
  [iso.org catalogue](https://www.iso.org/standard/55293.html).
  The goniometer autocorrelation coefficient behind section 2 of this page,
  its area weighting and the normalised $d_n$.

## Standards

ISO 17497-1:2004+A1:2014 (scattering
coefficient), ISO 17497-2:2012 (diffusion coefficient), ISO 13472-1:2002 (in-situ
absorption, extended surface), ISO 13472-2:2010 (in-situ absorption, spot
method), and ISO 9613-1:1993, from which only the pure-tone attenuation coefficient
α consumed by the ISO 17497-1 Clause 8 air-attenuation relations (Eqs. (2)/(3)); the
full atmospheric-absorption model is covered in
[Outdoor Sound Propagation](outdoor-propagation.md). Numerical conformance against the standards' worked examples and closed
forms is tracked in [CONFORMANCE.md](CONFORMANCE.md).
