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Diffusers and Their Coefficients

A diffuser is judged by where it sends the sound it reflects, and two standardised coefficients grade that job from two different angles. The scattering coefficient $s$ (ISO 17497-1) is measured in a reverberation room and does energy bookkeeping: what fraction of the reflected energy leaves the specular direction. The diffusion coefficient $d$ (ISO 17497-2) is measured on a free-field goniometer and grades spatial quality: how evenly the reflected energy covers a polar arc of receivers. This guide covers both measurements, the prediction of the diffusion coefficient from a Schroeder diffuser design before a sample is built, the published tables to reach for when no room is available, and the recurring confusion between the two coefficients.

The two 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. The closing section makes the distinction precise. The deep-subwavelength relatives of the Schroeder designs treated here, panels a few centimetres thick that scatter like wells tens of centimetres deep, have their own guide: Metadiffusers.

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.

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

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 $\alpha_\mathrm{s}$ (Eq. (1)); the rotating pair gives the specular absorption $\alpha_\mathrm{spec}$ (Eq. (4)).

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

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

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 $\alpha_\mathrm{spec}$ and $\alpha_\mathrm{s}$ with their band centres and returns a plottable ScatteringResult:

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)

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 ($s \to 1$).

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()

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.

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 $\alpha_\mathrm{s}$ 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 $\alpha_\mathrm{spec}$ column and language="es" renders the Spanish fiche; it needs the report extra (pip install phonometry[report]).

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.

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

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 rig those formulas assume is worth drawing to scale. plot_goniometer_geometry lays out the standard arrangement in plan: the 37-microphone semicircle every 5 degrees at 5 m around the sample, the source at 10 m on the normal.

The polar arc behind every $L_i$ of Formula (5), to scale: 37 receivers every 5 degrees at 5 m, the source twice as far out on the normal, and the sample a sliver at the centre.

import matplotlib.pyplot as plt
from phonometry import materials

# The standard arrangement: 37 microphones every 5 degrees at 5 m, the
# source at 10 m on the normal.
materials.plot_goniometer_geometry()
plt.show()

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 ($d = 0.63$).

Watch the high-resolution video (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° spacing, $-90°$ to $+90°$). 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).

The surface itself is worth drawing before predicting anything from it. plot_qrd_geometry turns the depth sequence into the to-scale well profile below, and a predicted DiffuserPolarResponse retains its geometry, so qrd.plot_geometry() draws the surface it was computed for.

Two of the six periods, to scale: the quadratic residues $n^2 \bmod 7$ turned into a buildable surface. The 490 Hz design frequency is what puts the deepest well at exactly 200 mm.

import matplotlib.pyplot as plt
from phonometry import materials

# The published N = 7 QRD: well width 3.6 m / 42, deepest well 0.2 m.
depths = materials.qrd_well_depths(7, 490.0, speed_of_sound=343.0)
pitch = 3.6 / 42                  # 42 wells across the 3.6 m panel
fin = 0.005                       # thin fins, included in the pitch
materials.plot_qrd_geometry(depths, pitch - fin, fin_width=fin, repetitions=2)
plt.show()

# A predicted DiffuserPolarResponse retains its geometry, so
# qrd.plot_geometry() draws the profile it was computed for (all 6 repetitions).

from phonometry import materials

# The published geometry: N = 7 QRD, 6 repetitions, 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, repetitions=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, repetitions=6)

qrd.plot()   # predicted polar response, d in the title (needs matplotlib)

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:

import numpy as np
from phonometry import materials

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

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

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.

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_m 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_deg = np.arange(-90.0, 90.5, 5.0)
result = materials.directional_diffusion(angles_deg, 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_deg)
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()

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).

import numpy as np
from phonometry import materials

# One diffusion coefficient per band (here a closed-form example). In practice
# each band's random-incidence d is the source-position average of the
# directional coefficients: for band k, average directional_diffusion_coefficient
# over the source positions with random_incidence_diffusion (Clause 8.4).
freqs = np.array([250, 500, 1000, 2000, 4000], float)
d = np.array([0.30, 0.45, 0.60, 0.75, 0.88])
spectrum = materials.diffusion_spectrum(freqs, d)
spectrum.plot()    # d(f) on the band axis (needs matplotlib)
spectrum.report("diffusion.pdf")   # one-page fiche (needs phonometry[report])

A single band's polar response is itself reportable: DiffusionResult.report(path) renders the corrected receiver-angle / reflected-level table beside the semicircular polar plot, boxing that band's directional diffusion coefficient.

Single-source polar-response fiche (DiffusionResult.report), the reflected level over the receiver arc. The spectrum fiche above grades a surface across frequency; this one is the raw evidence behind a single band of it, and it is what a goniometer session actually produces.

3. Designing a Schroeder diffuser

The measurements above grade a surface that already exists. The classic way to design one is Schroeder's phase grating: divide the surface into wells of equal width and different depths, so that each well returns the incident wave with a different phase. A rigid-bottomed well of depth $d_n$ reflects with the pressure coefficient

the round trip down and back up the well, so the depth sequence is a phase sequence in disguise. Schroeder's insight is number-theoretic: if the depths follow a quadratic residue sequence,

for a prime $N$ and a chosen design frequency $f_0$ (Cox & D'Antonio Eqs. (10.2)/(10.3)), then at $f_0$ the reflection phases $-4\pi s_n / (2N)$ sample a sequence whose discrete Fourier transform has constant magnitude, and the energy radiated into the grating lobes of the periodic surface is the same in every lobe. The surface spreads the reflection as evenly as a periodic surface can.

A handful of design rules follow directly from the geometry:

• The design frequency sets the depth. The deepest well is $d_\mathrm{max} = \max(s_n)\,\lambda_0/(2N)$, close to half a wavelength for large $N$; halve $f_0$ and the diffuser gets twice as deep. This is the cost that the metadiffuser attacks: at 500 Hz an $N = 5$ design already calls for wells 27.4 cm deep.
• The sequence repeats what the prime provides. $s_n$ is symmetric about $n = 0$ and periodic in $N$, so one period has $N$ wells and the profile is built by repeating it. The flat-phase property also holds at the harmonics $2f_0, 3f_0, \ldots$, except at multiples of$N f_0$, where every well reflects in phase again and the panel momentarily behaves like a flat plate.
• The well width sets the ceiling. Each well is a small waveguide; the single-plane-wave picture inside it holds only while the width $w$ stays below half a wavelength, giving an upper working limit near $f_\mathrm{max} = c/(2w)$. Narrower wells raise the ceiling but add viscous losses and fins.
• Periodicity concentrates, modulation spreads. Repeating one period locks the reflected energy into grating lobes at $\sin\theta_m = \sin\psi + m\lambda/L$ for period length $L$: even a perfect phase sequence then feeds a finite set of directions, which is the lobing penalty measured on the polar response above. Modulating two different sequences (or a sequence and its inverse) breaks the periodicity and recovers diffusion; primitive-root sequences ($s_n = r^n \bmod N$) additionally suppress the specular lobe at the design frequency.

qrd_well_depths evaluates the depth sequence and predict_diffuser_polar_response grades the design before anything is built, which is the subject of the next section. Only the quadratic-residue sequence has a dedicated helper: a primitive-root, modulated or otherwise custom profile enters through the explicit depths or reflection arguments.

Predicting diffusion from a diffuser design

The autocorrelation coefficient above reduces a measured polar response. The same coefficient can be predicted from the physical design of a Schroeder phase-grating diffuser, so a well-depth sequence can be graded before a sample is built. predict_diffuser_polar_response evaluates the single-plane Fraunhofer (far-field) model of Cox and D'Antonio: each rigid-bottom well of depth $d_n$ contributes a pressure reflection coefficient $R_n = e^{-2 j k d_n}$, and the scattered pressure at reflection angle $\theta$ for a source at incidence $\psi$ is the sum over the wells of the periodic surface (Eq. (5.8)),

with $x_n$ the well centre and $k = 2\pi f / c$. The predicted polar levels $L_i = 20\log_{10}|p(\theta_i)|$ feed the same directional_diffusion_coefficient as a measurement.

import numpy as np
from phonometry import materials

# An N = 7 quadratic residue diffuser, design frequency 500 Hz.
depths = materials.qrd_well_depths(7, 500.0)   # Eqs. (10.2)/(10.3)
print(np.round(depths * 100, 1))               # well depths in cm:
                                               # [ 0.   4.9 19.6  9.8  9.8 19.6  4.9]

# Predicted far-field polar response at one frequency (5 repeated repetitions,
# 10 cm wells), reduced to the ISO 17497-2 directional diffusion coefficient.
surface = materials.predict_diffuser_polar_response(0.10, 2000.0, depths=depths,
                                                    repetitions=5)
print(round(surface.coefficient, 3))           # 0.210
surface.plot()   # predicted polar response, d in the title (needs matplotlib)

# The spectrum variant normalises band by band against the same-footprint flat
# reference (Formula (7)): the flat panel maps to exactly zero, the QRD well
# above it.
freqs = np.array([500, 1000, 2000, 4000], float)
qrd = materials.predicted_diffusion_spectrum(0.10, freqs, depths=depths, repetitions=5)
print(np.round(qrd.normalized, 3))             # [0.352 0.275 0.208 0.073]

flat = materials.predicted_diffusion_spectrum(0.10, freqs,
                                        depths=np.zeros_like(depths), repetitions=5)
print(np.round(flat.normalized, 3))            # [0. 0. 0. 0.]

Predicted from the design alone: the $N = 7$ QRD spreads the reflected energy far more evenly than the flat panel of the same footprint, so its predicted diffusion coefficient sits well above the near-specular flat reference across the band. This is a far-field design estimate, not a substitute for an ISO 17497-2 measurement; like every Fourier diffuser model it loses accuracy at low frequency, at grazing angles and for strongly absorbing surfaces. An arbitrary complex per-well reflection sequence can be supplied through the reflection argument, so an admittance or resonator-loaded surface computed elsewhere can be graded the same way.

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

# N = 7 QRD (design frequency 500 Hz, 10 cm wells, five repetitions) versus the
# flat panel of the same footprint, over the one-third-octave bands.
freqs = np.array([250, 315, 400, 500, 630, 800, 1000, 1250, 1600,
                  2000, 2500, 3150, 4000, 5000], float)
depths = materials.qrd_well_depths(7, 500.0)
qrd = materials.predicted_diffusion_spectrum(0.10, freqs, depths=depths, repetitions=5)

# The DiffusionSpectrum plots the predicted d(f) directly. One line:
qrd.plot()
plt.show()

# By hand: predicted d(f) for the QRD against the flat reference.
flat = materials.predicted_diffusion_spectrum(0.10, freqs,
                                depths=np.zeros_like(depths), repetitions=5,
                                normalize=False)
fig, ax = plt.subplots()
ax.semilogx(freqs, qrd.diffusion, "o-", color="#1f77b4", label="N = 7 QRD design")
ax.semilogx(freqs, flat.diffusion, "s--", color="#d62728", label="Flat panel")
ax.set_ylim(0.0, 1.0)
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Predicted diffusion coefficient d")
ax.legend()
plt.show()

4. Published values, when the room is not yours

Sections 1 and 2 assume you can measure, and section 3 assumes the surface is a Schroeder diffuser you are designing yourself. Neither holds in the usual case: the reverberation room with a turntable in it belongs to somebody else, the goniometer arc even more so, and the number is needed this afternoon for a surface that already exists and is not a Schroeder diffuser. Cox & D'Antonio print three appendices of diffuser values for exactly that, B, C and D, and this library holds them row by row, with the page each one was read from.

The upper panel is the argument the first section of Appendix B exists to make, and no formula in the book makes it: one semicylinder scores 0.77 at 1 kHz at random incidence, and twelve of the same semicylinder score 0.22. A lone object scatters into every direction because it has nothing to interfere with; a periodic array of it concentrates the energy into grating lobes, and the diffusion coefficient falls accordingly. Reading a single device's polar response and scaling it up gets this wrong by a factor of three. The lower panel is the other axis of the same appendix: six semiellipses of one width, deepened from 1 cm to 30 cm, where the shallowest is acoustically flat as far as 5 kHz is concerned and the deepest is diffusing over most of the range.

What a published value is worth

One geometry, three published numbers, and nothing chosen to make a point: Appendix D prints battens 10 cm high and 10 cm wide on a 20 cm period twice, measured to ISO 17497-1 by two different teams, and Table C.2 computes the same geometry with a boundary element model. Between 400 Hz and 1 kHz the two measurements differ from each other by up to 0.2, which is the width of the method and not anybody's mistake, and the computed curve runs below both. Above 2 kHz all three converge.

That is what a row from these tables is: a value of the right order with a spread of its own, useful to start a simulation, to sanity-check a measurement, or to choose between two surface families. It is not a specification, and a project that turns on the difference between 0.25 and 0.45 at 1 kHz needs its own measurement.

Measured and computed are two catalogues

The library keeps them apart, with a class each and no key in common, because a coefficient a solver produced and one a reverberation room produced are the same number and not the same evidence:

from phonometry import materials

# Measured under ISO 17497-1: forty-six surfaces, each credited to its paper.
battens = materials.scattering_named("Periodic 1D battens")
for row in battens:
    print(row.name, row.spectrum()[1000], row.attributed_to["row"][:24])

# Computed: a hundred and nineteen rows, each naming its solver.
predicted = materials.PUBLISHED_PREDICTED_SCATTERING[
    "cox-2017-table-c2/rectangular_cross_section_batt_h_10_cm_l_20_cm_w_10_cm"
]
print(predicted.model)
print(predicted.scattering_coefficient(1000))

PUBLISHED_DIFFUSION holds the diffusion coefficients of Appendix B the same way, three rows per surface because the page prints one line per angle of incidence. Its random incidence row carries no angle at all: it is an arithmetic mean over ten angles, without the weighting a measurement to ISO 17497-2 would apply, and asking it for the angle says so rather than answering zero, which is what the normal incidence row of the same surface means.

Every row refuses rather than inventing. A band the page left empty is not a zero, and for a scattering coefficient a zero is a surface that sends every ray back along the specular direction:

row = materials.PUBLISHED_SCATTERING[
    "cox-2017-appendix-d/periodic_1d_battens_h_w_10_cm_l_2h_sakuma"
]
print(row.why_missing("scattering_coefficient_5000"))

The whole of all three tables is on the published catalogues page, filterable and with the source of every row.

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.

See also

• Metadiffusers: the deep-subwavelength version of the Schroeder designs of section 3, slits loaded by Helmholtz resonators that reach the same reflection phases from a panel a few centimetres thick.
• In-situ Road-Surface Absorption: the other surface family measured in the field rather than the laboratory, with the ISO 13472 subtraction and spot methods.
• Sound Absorption Measurement and Rating: the same reverberation room used by ISO 17497-1, measuring absorption instead of scattering.
• Outdoor Sound Propagation: the full ISO 9613-1 atmospheric-absorption model whose pure-tone coefficient the ISO 17497-1 air-attenuation relations consume.
• API reference: materials.diffusers.scattering_diffusion and materials.diffusers.design.

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. 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. 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.
• Audio Engineering Society. (2001). AES information document for room acoustics and sound reinforcement systems — Characterization and measurement of surface scattering uniformity (AES-4id-2001). Journal of the Audio Engineering Society, 49(3), 149-165. AES standards in print. The single-plane free-field diffusion-coefficient procedure that ISO 17497-2 later standardised.
• 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. 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. 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), and ISO 9613-1:1993, from which only the pure-tone attenuation coefficient $\alpha$ 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. Numerical conformance against the standards' worked examples and closed forms is tracked in CONFORMANCE.md.