Sound Absorption Measurement and Rating

Standards: ISO 354 ISO 11654 ISO 12999Key references: Cox & D'Antonio 2017

Of all the numbers a materials laboratory produces, the sound-absorption coefficient is the one that travels furthest: it leaves the reverberation room on a datasheet, enters a Sabine estimate, an EN 12354-6 absorption budget or a public-tender requirement, and is rarely questioned again. This guide covers that number’s whole laboratory life. The measurement is ISO 354: a sample of 10 m² to 12 m² on the floor of a reverberation room, two decay times, and Sabine’s formula run backwards to get the random-incidence coefficient $α_{s}$ per one-third-octave band. The rating is ISO 11654: the spectrum collapsed into the weighted coefficient $α_{w}$ with its letter class A to E, the single number absorber datasheets quote. And because a coefficient without an uncertainty is only half a result, ISO 12999-2 supplies the measurement uncertainty of both. The normal-incidence counterpart measured on small samples lives in the impedance tube guide; the closing section here, after the uncertainty, explains when each of the two is the right tool and why the two numbers do not match.

1. Reverberation-room measurement (ISO 354)

Plan of an ISO 354 reverberation room with non-parallel walls: the 10.8 square metre rectangular specimen on the floor with its edges deliberately not parallel to the room edges and a 0.75 metre clearance dimensioned to the nearest boundary, three microphone positions and two loudspeaker positions with their separation rules, suspended diffuser panels near the ceiling, an inset showing the empty room giving T1 and A1 above the room with the specimen giving T2 and A2, and a strip comparing the Type A and Type E-400 mountings in section

The reverberation-room measurement itself is measure_sound_absorption. It takes the one-third-octave reverberation time of the empty room ( $T_{1}$ ) and of the room with the specimen installed ( $T_{2}$ ), the room volume $V$ and the specimen area $S$ , and returns a frozen SoundAbsorptionMeasurement. The equivalent sound absorption areas follow from Sabine’s equation (ISO 354:2003 Eqs. (5) and (7)), and the coefficient from their difference (Eqs. (8)/(9)):

A = \frac{55.3 V}{c T} - 4 V m, α_{s} = \frac{A _{2} - A _{1}}{S}, c = 331 + 0.6 t,

with $c$ the speed of sound built from the room air temperature $t$ in °C (Eq. (6), valid 15–30 °C) and $m$ the power attenuation coefficient of air in 1/m. The coefficient may exceed 1.0 from edge and diffraction effects (Clause 3.7 NOTE 2) and is never clamped; the closing section explains why.

The air-attenuation term $m$ , and why the default 0 is not a shortcut. $α_{s}$ is a difference of two absorption areas, so a single $m$ enters $A_{1}$ and $A_{2}$ identically and the $- 4 V m$ term cancels exactly: measure_sound_absorption broadcasts one $m$ into both, and the default $m = 0$ therefore changes nothing as long as both decays were measured under the same climate. Only the difference $m_{2} - m_{1}$ survives, which is why Clause 6.3.2 asks for $T_{1}$ and $T_{2}$ to be measured at almost the same temperature and relative humidity, inside $30 % \leq RH \leq 90 %$ and $t \geq 15$ °C throughout the test. The term itself is anything but small: for the 200 m³ room of the example below at 21.4 °C and 54 % RH, the ISO 9613-1 attenuation makes $4 V m$ about 0.9 m² at 1 kHz, 4.9 m² at 4 kHz and 7.2 m² at 5 kHz — against an empty-room absorption area of about 7.7 m² in that top band. A few percent of humidity drift between the two runs therefore moves $α_{s}$ by more than most of the effect being measured. When the climate did drift, drop to absorption_coefficient and supply a separate value per measurement as m1/m2 (with temperature1/temperature2); measure_sound_absorption assumes one climate. The per-band $m$ comes from an ISO 9613-1 attenuation $α$ in dB/m — environment.air_attenuation(freqs, temperature=21.4, relative_humidity=54.0, pressure=101.325) — converted with materials.attenuation_from_alpha(alpha), which is ISO 354’s own $m = α / (10 l g e)$ .

ISO 354 is a characterisation: it produces the $α_{s}$ spectrum, not a single-number rating. The weighted coefficient $α_{w}$ is an ISO 11654 quantity; feed the measured $α_{s}$ to weighted_absorption_from_third_octave in section 3 to obtain it.

import numpy as np
from phonometry import materials

freqs = np.array([100, 125, 160, 200, 250, 315, 400, 500, 630, 800,
                  1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000], float)
t_empty = np.array([9.0, 9.0, 8.8, 8.6, 8.4, 8.2, 8.0, 7.8, 7.5, 7.2,
                    6.9, 6.6, 6.2, 5.8, 5.4, 5.0, 4.6, 4.2])
t_specimen = np.array([8.4, 8.2, 7.7, 7.2, 6.5, 5.7, 4.9, 4.2, 3.6, 3.15,
                       2.85, 2.65, 2.55, 2.5, 2.55, 2.6, 2.7, 2.85])

meas = materials.measure_sound_absorption(
    freqs, t_empty, t_specimen, volume=200.0, area=10.8, temperature=20.0
)
print(meas.alpha_s[7])   # alpha_s at 500 Hz: 0.328...
meas.plot()              # alpha_s versus one-third-octave frequency

ISO 354 reverberation-room sound absorption: the alpha_s spectrum of a porous absorber sample over the one-third-octave bands from 100 Hz to 5000 Hz, rising from near zero at low frequency to a broad maximum around 0.69 near 1600 Hz and easing off above

The Sabine inversion of the two decay times: the porous sample barely changes the long low-frequency decays ( $α_{s} \to 0$ ), bites hardest where its thickness approaches a quarter wavelength, and eases off at high frequency, a typical thin-porous-absorber signature (Cox & D’Antonio 3e, Ch. 5).

Show the code for this figure

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

freqs = np.array([100, 125, 160, 200, 250, 315, 400, 500, 630, 800,
                  1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000], float)
t_empty = np.array([9.0, 9.0, 8.8, 8.6, 8.4, 8.2, 8.0, 7.8, 7.5, 7.2,
                    6.9, 6.6, 6.2, 5.8, 5.4, 5.0, 4.6, 4.2])
t_specimen = np.array([8.4, 8.2, 7.7, 7.2, 6.5, 5.7, 4.9, 4.2, 3.6, 3.15,
                       2.85, 2.65, 2.55, 2.5, 2.55, 2.6, 2.7, 2.85])
meas = materials.measure_sound_absorption(
    freqs, t_empty, t_specimen, volume=200.0, area=10.8, temperature=20.0
)

# One line: the alpha_s spectrum over the one-third-octave band axis.
meas.plot()
plt.show()

# By hand, from the result's fields:
fig, ax = plt.subplots()
ax.plot(np.arange(freqs.size), meas.alpha_s, "o-")
ax.set_xticks(np.arange(freqs.size))
ax.set_xticklabels([f"{f:g}" if f < 1000 else f"{f/1000:g}k" for f in freqs],
                   rotation=45)
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound absorption coefficient alpha_s")
plt.show()

1.1 How $T_{1}$ and $T_{2}$ are actually measured (Clause 7)

The two reverberation times are ordinary decay measurements — the machinery is the one described under room acoustics — but ISO 354 constrains the geometry and the averaging much harder than a general room survey does, because the whole result is the difference of two numbers that are almost equal at low frequency.

Where the transducers go (Clause 7.1). Both the microphones and the sound source are omnidirectional. Microphone positions are at least 1.5 m apart, 2 m from any sound source and 1 m from any room surface and from the test specimen; source positions are at least 3 m apart. Decay curves measured at different microphone positions shall not be combined in any way — they are separate decays, averaged only as reverberation times.

How many decays (Clause 7.1.4). At least twelve spatially independent decay curves: the number of microphone positions times the number of source positions must reach 12, with a minimum of three microphone positions and two source positions. Several sources may be driven simultaneously if their radiated powers agree within 3 dB in every one-third-octave band, and that buys a reduction to six curves.

How the room is excited (Clause 7.2 or 7.3). Two methods are allowed. The interrupted-noise method drives a loudspeaker with broadband or band-limited noise of continuous spectrum: with broadband noise and a real-time analyser the resulting sound pressure levels must differ by less than 6 dB between adjacent one-third-octave bands, and band-limited noise must be at least one one-third octave wide. The excitation must run long enough to reach the steady state — at least half the expected reverberation time — and start high enough that the bottom of the evaluation range still clears the background by 10 dB. Because a single interrupted decay is a statistical sample, averaging at one microphone/source position is mandatory: at least three decays, and at least ten if the repeatability is to match the other method. The integrated impulse response method (a sweep or a maximum-length sequence deconvolved into the impulse response) is deterministic and needs no averaging at all, which is why it is the method to prefer when the instrumentation allows it.

How each decay is read (Clause 7.4.1). Per band: start 5 dB below the initial level, evaluate over a 20 dB range, and keep the bottom of that range at least 10 dB above the overall background noise — a $T_{20}$ , extrapolated to 60 dB. $T_{1}$ and $T_{2}$ are then the arithmetic mean over all the measurements in that band, carried to at least two decimal places (Clause 8.1.1).

Then do it again. The whole grid is repeated with the specimen installed and nothing else changed: same positions, same excitation, same climate. Any difference between the two runs other than the specimen ends up inside $A_{2} - A_{1}$ and is reported as absorption.

1.2 The specimen and its mounting (Clause 6.2 and Annex B)

Geometry (Clause 6.2.1). The test area is between 10 m² and 12 m²; above a room volume of 200 m³ the upper limit is multiplied by $(V /200)^{2/3}$ , and a weakly absorbing specimen should use the upper limit. The specimen is rectangular with a width-to-length ratio between 0.7 and 1, placed so that no part is closer than 1 m to a room boundary and never closer than 0.75 m, with its edges preferably not parallel to the nearest room edge — both rules exist to keep the specimen out of the pressure maxima at the room boundaries. A heavy specimen may instead stand vertically against a wall resting on the floor, and then the 0.75 m clearance does not apply.

The mounting is part of the result. Annex B is normative and defines the mounting catalogue a datasheet quotes:

Type	What it is
A	Specimen laid or fixed directly against a room surface, normally the reverberation-room floor, with no air space
B	Glued to a gypsum board with adhesive dabs and 3 mm shims at the four corners, so a thin air space is retained
E-xxx	Mounted over a sealed air space in a fixture of at least 20 kg/m²; the suffix is the distance in millimetres from the exposed face of the specimen to the room surface behind it, tested at E-200, E-300 or E-400
G-xxx	Hung parallel to a room surface (curtains, drapes, blinds); the suffix is again the face-to-surface distance, tested at G-100, other spacings in multiples of 50 mm
I	Sprayed or trowelled materials, applied to a substrate and then tested as a Type A with a frame
J	Unit absorbers — pads and baffles — in two or three parallel rows over 10 m² to 15 m² of floor, surrounded by a non-absorptive barrier

The suffix is a cavity depth, and cavity depth moves the answer more than most material choices do. Predicting the same 50 mm blanket ( $σ = 20$ kPa·s/m²) with the multilayer models gives $α_{w} = 0.70 (H)$ , class C, laid flat as a Type A, and $α_{w} = 0.90$ , class A, on a Type E-400 mounting — two classes from one product, because the 350 mm of air behind it moves the layer into the particle-velocity maximum at 250 Hz (where the predicted coefficient goes from 0.37 to 0.86). A quoted $α_{s}$ or $α_{w}$ without its mounting is therefore not a material property but an installation result, which is why the mounting field of the fiche below is not decoration.

The perimeter edge (Annex B.2). The specimen edges shall be sealed or covered so that they do not absorb — unless the product is normally installed with exposed edges, in which case they stay open and the area of the edges is added into $S$ . Either way the treatment goes in the report. Where a reflective frame is used it must be solid rather than hollow, with no air space between frame, specimen and room surface, at least 1.0 mm steel or 12.5 mm gypsum board or wood, tightly butted and sealed, its exposed face flush with the face of the specimen. One consequence is easy to miss: Clause 6.2.1.3 requires the empty-room measurement to be made without the frame or the specimen’s side walls (the barrier of a Type J mounting being the exception), or the frame’s own absorption never cancels out of $A_{2} - A_{1}$ .

1.3 Is the room fit to measure in? (Clause 6.1 and Annex A)

Three checks decide whether a room may be used at all, and none of them is performed by this library — measure_sound_absorption only raises an advisory AbsorptionWarning on the volume and the specimen area.

Volume and shape (Clauses 6.1.1, 6.1.2). $V \geq 150$ m³, and at least 200 m³ is strongly recommended for a new room; above roughly 500 m³ the high-frequency bands become inaccurate because air absorption dominates the decay. The shape must satisfy $l_{max} < 1.9 V^{1/3}$ , with $l_{max}$ the longest straight line inside the room (the major diagonal of a rectangular room), and no two room dimensions may stand in the ratio of small whole numbers — both conditions spread the low-frequency modes.
Empty-room absorption (Clause 6.1.4). $A_{1}$ shall not exceed the Table 1 ceiling, which is 6.5 m² from 100 Hz to 800 Hz for a 200 m³ room and rises through 7.0 m² at 1 kHz to 14.0 m² at 5 kHz; for another volume the whole table is scaled by $(V /200)^{2/3}$ . The $A_{1} (f)$ curve must also be smooth, with no dip or peak differing by more than 15 % from the mean of the two adjacent one-third-octave bands. This is the single check that says the room itself is quiet enough acoustically to see a specimen at all: a leaky, lined or cluttered room spends its absorption budget on itself.
Diffusivity (Annex A). Diffusers are sheets of low absorption, about 5 kg/m², individually 0.8 m² to 3 m² on one side, slightly curved, randomly oriented and spread through the room; rotating vanes are the alternative, and their rotation frequency must not stand in a small-whole-number ratio to the decay repetition frequency. The qualification procedure of A.2 is empirical: mount a homogeneous porous specimen 5 cm to 10 cm thick whose absorption exceeds 0.9 over 500 Hz to 4000 Hz, measure it with no diffusers, then with about 5 m² of diffusers, then in further 5 m² steps, and plot the mean coefficient over 500 Hz to 5000 Hz against diffuser area. The curve rises and then flattens; the optimum is where it stops rising. In a rectangular room that typically lands at 15 % to 25 % of the room’s total surface area (counting both sides of each diffuser).

What the report must carry (Clause 9). The fiche below holds the specimen description, $S$ , the mounting, $V$ , $c$ , the temperature and humidity and the $T_{1}$ / $T_{2}$ and $A_{1}$ / $A_{2}$ tables (with verbose=True) — that is items (c), (e) in part, (f) and (g). Four Clause 9 items have no ReportMetadata field and must be appended by the laboratory: the shape of the room and its total surface area $S_{t}$ , the number and size of the diffusers, the number of microphone and source positions, and the treatment given to the specimen edges. Results are reported with $A_{T}$ rounded to 0.1 m² and $α_{s}$ to 0.01 (Clause 8.3).

ISO 354 report (`.report()`)

SoundAbsorptionMeasurement.report(path) renders a one-page PDF fiche laid out like an accredited reverberation-room absorption test report (ISO 354:2003): a standard-basis line, a metadata header block, the one-third-octave $α_{s}$ table beside the $α_{s}$ curve (the result’s own .plot()), a boxed characterisation headline and a footer with the fixed disclaimer. ISO 354 has no pass/fail verdict and no single-number rating, so the fiche carries neither. Setting verbose=True adds the reverberation times $T_{1}$ / $T_{2}$ and the equivalent absorption areas $A_{1}$ / $A_{2}$ to the table.

It uses the same ReportMetadata container and rendering engine as the other fiches. The specimen area $S$ , room volume $V$ , speed of sound $c$ , temperature and humidity are taken from the measurement result (they drove the Sabine inversion); the descriptive ReportMetadata fields that apply here are client, manufacturer, specimen, mounting, test_room, test_date, pressure, measurement_standard, laboratory, operator, report_id and notes. The requirement field is ignored (ISO 354 has no verdict). Rendering needs reportlab and, for the figure the fiche embeds, matplotlib (pip install "phonometry[report,plot]"); only engine="reportlab" is supported. The fiche renders in English by default; pass language="es" for a Spanish fiche (translated fixed strings and a comma decimal separator).

from phonometry import materials, ReportMetadata

meas = materials.measure_sound_absorption(
    freqs, t_empty, t_specimen, volume=200.0, area=10.8,
    temperature=20.0, humidity=54.0,
)
meas.report(
    "alpha_s_fiche.pdf",
    metadata=ReportMetadata(
        specimen="50 mm porous absorber over a 100 mm air gap",
        mounting="Type A (against a rigid wall)",
        measurement_standard="ISO 354",
        test_room="Reverberation room R1",
        laboratory="Phonometry Reference Laboratory",
    ),
)                                  # one-third-octave alpha_s, 100 Hz to 5000 Hz

ISO 354 absorption example report (PDF) — Reverberation-room absorption fiche (SoundAbsorptionMeasurement.report), the alpha_s spectrum.

2. The same inversion on bare arrays: `absorption_area` and `absorption_coefficient`

Section 1 returns a whole SoundAbsorptionMeasurement: it carries the room and specimen metadata, plots itself and renders the ISO 354 fiche. When all that is wanted is the arithmetic — inside a loop, a fit, a spreadsheet import, or when the quantity of interest is the room’s own equivalent absorption area rather than a specimen coefficient — the same Eqs. (5) to (9) are exposed as two plain array functions. The equivalent absorption area $A$ is also what drives $R^{'}$ , $L_{n}^{'}$ , the ISO 3744 $K_{2}$ environmental correction and the ISO 3741 absorption term, and absorption_area is the entry point those calculations use.

import numpy as np
from phonometry import materials

# Third-octave reverberation times of a 200 m^3 room, empty (T1) and with a
# 10.8 m^2 absorber sample installed (T2).
t1 = np.array([5.0, 4.0, 3.0])
t2 = np.array([3.0, 2.5, 2.0])

a_empty = materials.absorption_area(t1, volume=200.0, temperature=20.0)
print(np.round(a_empty, 2))                    # [ 6.45  8.06 10.75] m^2

alpha = materials.absorption_coefficient(t1, t2, volume=200.0, sample_area=10.8,
                               temperature1=20.0)
print(np.round(alpha, 3))                      # [0.398 0.448 0.498]

$T_{1}$ and $T_{2}$ are exactly the reverberation times room_parameters returns, so an ISO 3382-2 decay measurement of the empty and treated room flows straight into absorption_coefficient.

The result object of section 1 keeps every intermediate of that inversion — t_empty, t_specimen, absorption_area_empty and absorption_area_with_specimen — and they are worth looking at, because they show where the method is strong and where it is running out of signal:

Two stacked panels over the one-third-octave axis from 100 Hz to 5 kHz. The upper panel plots the empty-room reverberation time T1 and the with-specimen time T2, which nearly coincide at 100 Hz and separate widely above 500 Hz. The lower panel plots the equivalent absorption areas A1 and A2 with the difference A2 minus A1 filled between them, and a right-hand axis reading the resulting alpha_s

The whole ISO 354 inversion of the section 1 example, drawn: the two decay times (top) and the two Sabine areas they invert to (bottom), with the filled difference $A_{2} - A_{1}$ divided by $S = 10.8$ m² giving the $α_{s}$ on the right-hand axis. At 100 Hz the two decays differ by 0.6 s out of 9 s and the filled band is barely visible — which is exactly why the uncertainty of section 4 is worst there. By 1.6 kHz the specimen has more than halved the decay and the difference is unmistakable.

Show the code for this figure

import matplotlib.pyplot as plt

# `meas` is the SoundAbsorptionMeasurement of section 1; every quantity the
# figure draws is a field of it.
fig, (ax_t, ax_a) = plt.subplots(2, 1, sharex=True, figsize=(10, 7))
band = np.arange(freqs.size)
ax_t.plot(band, meas.t_empty, "o-", label="T1 (empty room)")
ax_t.plot(band, meas.t_specimen, "s-", label="T2 (specimen installed)")
ax_t.set_ylabel("Reverberation time [s]")
ax_t.legend()

a1 = meas.absorption_area_empty
a2 = meas.absorption_area_with_specimen
ax_a.plot(band, a1, "o-", label="A1 (empty room)")
ax_a.plot(band, a2, "s-", label="A2 (with specimen)")
ax_a.fill_between(band, a1, a2, alpha=0.25, label="A2 - A1")
ax_a.set_ylabel("Equivalent absorption area [m$^2$]")
ax_a.set_xticks(band)
ax_a.set_xticklabels([f"{f:g}" if f < 1000 else f"{f/1000:g}k" for f in freqs],
                     rotation=45)
ax_a.set_xlabel("Frequency [Hz]")
ax_a.legend()
plt.show()

print(np.round((a2 - a1) / 10.8 - meas.alpha_s, 12).max())   # 0.0: same alpha_s

Each of those two numbers is read off a decay, and the clip below shows how one is read: the squared impulse response is integrated backwards from the tail, the Schroeder curve emerges, and the T20 and T30 regressions are fitted to a straight portion of it. That is the operation behind $T_{1}$ , and again behind $T_{2}$ — the clip shows a single room, not the pair, so it answers “where does one $T$ come from” and not “what does subtracting two of them cost”. The second question is the one that governs this measurement, and section 4 puts a number on it: because $α_{s}$ is a difference of two reciprocal decay times, its uncertainty is worst exactly where the two decays are most alike, at the low-frequency end.

Download the animation (WebM)

A room volume below the 150 m³ minimum or a sample area outside 10–12 m² raises an advisory AbsorptionWarning; the result still returns.

`absorption_area()` / `absorption_coefficient()` parameters

Parameter	Type	Units	Range / default	Notes
`t60` / `t1`, `t2`	1D array	s	> 0	Reverberation time(s); `t1` empty, `t2` with specimen
`volume`	float	m³	> 0	Room volume $V$ (advisory below 150 m³)
`sample_area`	float	m²	> 0	Area $S$ the specimen covers (coefficient only)
`temperature` / `temperature1`, `temperature2`	float	°C	default `20.0`, 15–30	Sets $c$ via Eq. (6); `temperature2` defaults to `temperature1`
`speed_of_sound` (`…1`, `…2`)	float, optional	m/s	> 0	Overrides the temperature-derived $c$
`m` (`m1`, `m2`)	float or 1D array	1/m	≥ 0, default `0`	Air power attenuation coefficient

absorption_area() returns the equivalent absorption area $A$ (m²) with the shape of t60; absorption_coefficient() returns $α_{s}$ ; attenuation_from_alpha(alpha) converts an ISO 9613-1 $α$ (dB/m) to $m$ .

3. Weighted rating and absorption class (ISO 11654)

The measurement of section 1 delivers the $α_{s}$ spectrum in one-third-octave bands. ISO 11654:1997 turns that spectrum into a single-number rating comparable across products.

Practical absorption coefficient (Clause 4.1). The one-third-octave data are first grouped into octave bands, each the arithmetic mean of its three thirds:

α_{p, i} = \frac{1}{3} (α_{i 1} + α_{i 2} + α_{i 3}),

evaluated to the second decimal and then rounded in steps of $0.05$ (the Clause 4.1 NOTE fixes the rounding, e.g. $0.92 \to 0.90$ ); rounded means above $1.00$ are set to $1.00$ . The five rating bands are 250, 500, 1000, 2000 and 4000 Hz.

Weighted absorption (Clause 4.2). A fixed reference curve ${250 : 0.80, 500 : 1.00, 1000 : 1.00, 2000 : 1.00, 4000 : 0.90}$ is shifted downwards, towards the measured $α_{p}$ , in steps of $0.05$ until the sum of the unfavourable deviations (taken only where the measurement lies below the shifted curve, with magnitude $(curve - measured)$ ) is no more than $0.10$ . The weighted coefficient $α_{w}$ is the shifted-curve value read at 500 Hz.

Shape indicators (Clause 4.3). When a practical coefficient exceeds the shifted curve by $0.25$ or more, a shape indicator is appended: L at 250 Hz, M at 500 or 1000 Hz, H at 2000 or 4000 Hz (e.g. 0.60(M)).

Absorption class (Table B.1). Finally $α_{w}$ maps to a class: A (0.90–1.00), B (0.80–0.85), C (0.60–0.75), D (0.30–0.55), E (0.15–0.25), or “not classified” (0.00–0.10). Because $α_{w}$ is always a multiple of $0.05$ these ranges partition the grid exactly.

What the rating cannot say. ISO 11654 scopes itself in Clause 1.2, and the four limits it states there are exactly the cases a practitioner meets. The reference curve starts at the 250 Hz octave, so everything the specimen does at 100 Hz to 200 Hz is outside the rating by construction and the standard says the rating “is not appropriate below this frequency”; a bass treatment and a mid-frequency panel can share an $α_{w}$ while differing by half a coefficient at 125 Hz. The rating is likewise “not applicable unless the applications cover the whole frequency range of the reference curve” — if only part of the range matters, the standard sends the reader back to the spectrum, and points at the shape indicators as the hint that a modest $α_{w}$ with an (L) may be the better product for a low-frequency problem. It is scoped to routine applications (normal offices, corridors, classrooms, hospitals) and declares itself inappropriate for “qualified environments requiring careful acoustical design by expertise”, where only the full spectrum will do. And it is “often not suitable for application to single items, such as chairs, baffles, etc.” — whose absorption is an equivalent area per object, not a coefficient — “nor is it applicable to road barriers and road surfaces”, which have EN 1793-1 and ISO 13472 of their own. This is why the fiche always prints the whole spectrum beside the single number.

The Annex A.2 worked example: the reference curve is shifted down by 0.40 until the unfavourable deviations sum to 0.05 ( $\leq 0.10$ ), giving $α_{w} = 0.60$ ; the 500 Hz peak overshoots the shifted curve by $\geq 0.25$ , adding the M indicator, so the rating is $0.60 (M)$ , class C.

Show the code for this figure

import matplotlib.pyplot as plt
from phonometry import materials

# ISO 11654 Annex A.2 practical coefficients at 250/500/1000/2000/4000 Hz
result = materials.weighted_absorption([0.35, 1.00, 0.65, 0.60, 0.55])
result.plot()   # practical curve vs shifted reference, deviations shaded
plt.show()

from phonometry import materials

# ISO 11654 Annex A.2 practical coefficients at 250/500/1000/2000/4000 Hz
alpha_p = [0.35, 1.00, 0.65, 0.60, 0.55]

result = materials.weighted_absorption(alpha_p)
print(result.rating_label)          # 0.60(M)
print(result.alpha_w)               # 0.6
print(result.absorption_class)      # C
print(round(result.unfavourable_sum, 2))  # 0.05
result.plot()   # the figure above: practical curve vs shifted reference

# A bare alpha_w also maps straight to its class (Table B.1)
print(materials.absorption_class(0.85))       # B

weighted_absorption accepts the five octave-band $α_{p}$ values (as a sequence or a {frequency: value} mapping); pass the fifteen one-third-octave $α_{s}$ values to practical_absorption_coefficient first if you are starting from raw ISO 354 data. To keep the one-third-octave $α_{s}$ on the result (so the fiche can print the full table every accredited ISO 354 certificate carries), rate it in one step with weighted_absorption_from_third_octave(alpha_s), which forms $α_{p}$ , rates it and retains the input $α_{s}$ and its band centres (third_octave_alpha_s, third_octave_bands). The result carries the shifted reference curve and the per-band deviations, and its .plot() renders the figure above.

from phonometry import materials

# Fifteen one-third-octave alpha_s (200 Hz to 5000 Hz), as an ISO 354 report gives
alpha_s = [0.30, 0.35, 0.40, 1.00, 1.00, 1.00, 0.62, 0.66, 0.67,
           0.58, 0.60, 0.62, 0.53, 0.55, 0.57]
result = materials.weighted_absorption_from_third_octave(alpha_s)
print(result.rating_label)          # 0.60(M)
print(result.third_octave_alpha_s)  # the input alpha_s, retained for the fiche

Is that a good number? Absorption tracks thickness against wavelength, so every halving of the useful frequency costs a doubling of thickness or of cavity depth, and the ladder is short. Predicting a mineral-wool-class material ( $σ = 20$ kPa·s/m²) hard against a wall with the multilayer models gives a random-incidence coefficient at 500 Hz of about 0.32 for a 25 mm blanket, 0.68 for 50 mm and 0.89 for 100 mm; at 250 Hz the same three give 0.14, 0.37 and 0.74. Spacing the 50 mm layer off the wall on a 100 mm cavity lifts its 250 Hz value from 0.37 to 0.68 — the cavity buys roughly the octave that doubling the material would have bought. A painted or sealed hard surface sits at a few hundredths across the whole range. The practical reading: class A is nearly always a thick or a spaced construction rather than a clever surface, and a thin sample that reports a mid-frequency $α_{s}$ above about 1.1, or a rising low-frequency tail, is showing an edge effect or a mounting leak rather than a good material. Measured ISO 354 values also run above these predicted ones, by roughly 0.1 to 0.2 in the mid bands, for the finite-sample reason set out in the closing section.

ISO 11654 report (`.report()`)

AbsorptionRatingResult.report(path) renders a one-page PDF fiche laid out like an accredited absorption test report (an ISO 354 reverberation-room measurement rated per ISO 11654): a standard-basis line, an optional metadata header block, the octave-band $α_{p}$ table beside the practical-versus-shifted-reference plot (the result’s own .plot()), the boxed $α_{w}$ single number with its absorption class and applied shift, an optional verdict row and a footer with the fixed disclaimer. When the rating was built with weighted_absorption_from_third_octave, the left table becomes the full ISO 354 one-third-octave $α_{s}$ table with the octave $α_{p}$ on the matching rows, exactly as accredited certificates print it. It uses the same ReportMetadata container and rendering engine as the ISO 717 insulation fiche; passing metadata=None produces a lightweight prediction fiche, and a supplied requirement is read as the minimum $α_{w}$ for the PASS/FAIL verdict. Setting verbose=True swaps the two-column table for the ISO 11654 evaluation columns (practical coefficient, shifted reference, unfavourable deviation). Rendering needs reportlab and, for the figure the fiche embeds, matplotlib (pip install "phonometry[report,plot]"); only engine="reportlab" is supported. The fiche renders in English by default; pass language="es" for a Spanish fiche (translated fixed strings and a comma decimal separator), e.g. result.report("alpha_w_fiche_es.pdf", language="es").

from phonometry import materials, ReportMetadata

# Rate from the fifteen one-third-octave alpha_s so the fiche prints the full
# ISO 354 table; materials.weighted_absorption([...]) also works from alpha_p.
alpha_s = [0.30, 0.35, 0.40, 1.00, 1.00, 1.00, 0.62, 0.66, 0.67,
           0.58, 0.60, 0.62, 0.53, 0.55, 0.57]
result = materials.weighted_absorption_from_third_octave(alpha_s)
result.report(
    "alpha_w_fiche.pdf",
    metadata=ReportMetadata(
        specimen="50 mm porous absorber over a 100 mm air gap",
        area=10.8, mounting="Type A (against a rigid wall)",
        measurement_standard="ISO 354",
        temperature=21.4, relative_humidity=54.0,
        laboratory="Phonometry Reference Laboratory",
        requirement=0.55,          # adds the PASS/FAIL verdict row
    ),
)                                  # alpha_w (shape) + absorption class

The example fiche is regenerated with make reports and kept rendered in the repository; click the preview to open the PDF.

ISO 11654 absorption example report (PDF) — Weighted absorption fiche (AbsorptionRatingResult.report), alpha_w with its class.

4. Sound-absorption measurement uncertainty (ISO 12999-2)

A rated absorption coefficient means little without its uncertainty. ISO 12999-2:2020 gives the standard uncertainty $u$ of the quantities produced by a reverberation-room measurement (ISO 354) and its ratings (ISO 11654, EN 1793-1), estimated from inter-laboratory tests to ISO 5725. It is the sound-absorption companion of the sound-insulation uncertainty of ISO 12999-1 (Field Insulation Measurement (ISO 16283)).

One-third-octave bands (Clause 5). For the sound-absorption coefficient the reproducibility standard deviation is $σ_{R} = m_{R} α_{s} + n_{R}$ (Formula (1)), and for the equivalent absorption area $σ_{R} = m_{R} A_{T} + n_{R} S$ with $S = 10 m^{2}$ (Formula (2)), where $m_{R}$ and $n_{R}$ are the frequency-dependent regression constants of Table 1 (63–5000 Hz) — nothing to do with the air power attenuation coefficient $m$ of section 1, which shares the standards’ letter and not its meaning. The repeatability value is $σ_{r} = 0.6 σ_{R}$ (Formula (3)). Note that the $n_{R}$ term of Formula (2) is multiplied by a fixed reference area of 10 m² rather than by the specimen’s own area, so a small specimen’s $A_{T}$ is proportionally far less certain than a large one’s — which is one more reason ISO 354 pins the test area between 10 m² and 12 m².

Practical coefficient (Clause 6). For the ISO 11654 practical coefficient $σ_{R} = m_{R} α_{p} + n_{R}$ in octave bands with the constants of Table 2 (250–4000 Hz); again $σ_{r} = 0.6 σ_{R}$ . Table 2 covers those five octaves only, so the 125 Hz octave has no tabulated uncertainty and practical_coefficient_uncertainty rejects it rather than returning a blank.

Single numbers (Clause 7). The weighted coefficient $α_{w}$ has a constant standard uncertainty ( $σ_{R} = 0.035$ , $σ_{r} = 0.020$ ); the EN 1793-1 single-number rating $D L_{α, NRD}$ scales with the value ( $σ_{R} = 0.10 D L_{α}$ , $σ_{r} = 0.02 D L_{α}$ ).

Reporting (Clause 8). The expanded uncertainty is $U = k u$ (Formula (10)) with the Table 3 coverage factor $k$ ( $k = 2.0$ at 95 %, Gaussian; read another confidence off the same table with absorption_coverage_factor, which returns 2.6 at 99 %). The reported $U$ is rounded to two decimals for absorption coefficients and one decimal for the equivalent area and $D L_{α, NRD}$ .

ISO 12999-2 sound absorption coefficient uncertainty: the measured alpha_s spectrum over one-third-octave bands from 63 Hz to 5000 Hz with a shaded plus-or-minus U band at coverage factor k = 2, reproducing the standard's worked Table 4 example

The Table 4 worked example: the measured $α_{s}$ spectrum with its $\pm U$ ribbon at $k = 2$ (95 %), where $σ_{R} = m_{R} α_{s} + n_{R}$ takes $m_{R}$ and $n_{R}$ from Table 1. The ribbon is a shallow U: $\pm 0.33$ in the 63 Hz band, tightening to $\pm 0.08$ from 250 Hz to 2 kHz and opening again to $\pm 0.16$ at 5 kHz. The width is set almost entirely by the band, not by the absorber — the low end is the modal field of the room, the high end is the air absorption the two decays must be corrected for. The plotted $U$ is the reported one, rounded to two decimals as Clause 8 requires.

Show the code for this figure

import matplotlib.pyplot as plt
from phonometry import materials

# ISO 12999-2 Table 4 worked example: alpha_s per one-third-octave band.
freqs = [63, 80, 100, 125, 160, 200, 250, 315, 400, 500,
         630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000]
alpha_s = [0.33, 0.35, 0.39, 0.38, 0.37, 0.36, 0.36, 0.36, 0.43, 0.49,
           0.58, 0.63, 0.68, 0.71, 0.73, 0.75, 0.77, 0.79, 0.81, 0.81]
result = materials.sound_absorption_coefficient_uncertainty(alpha_s, freqs, confidence=0.95)
result.plot()   # alpha_s with the +/-U (k = 2) reproducibility ribbon
plt.show()

from phonometry import materials

# Reproducibility uncertainty of alpha_s at 1000 Hz (Table 1: m=0.040, n=0.015).
r = materials.sound_absorption_coefficient_uncertainty([0.68], [1000], confidence=0.95)
print(round(float(r.standard_uncertainty[0]), 4))            # 0.0422 (sigma_R)
print(float(r.reported_expanded_uncertainty[0]))             # 0.08  (U, k=2)
r.plot()   # the figure above: alpha_s with its +/-U (k = 2) ribbon

# Same band, same value, but a repeat on my own bench rather than another
# laboratory: sigma_r = 0.6 sigma_R (Formula (3)).
rr = materials.sound_absorption_coefficient_uncertainty(
    [0.68], [1000], condition="repeatability"
)
print(round(float(rr.standard_uncertainty[0]), 5))           # 0.02532 = 0.6 x 0.0422

# The other two quantities this page produces, Formula (2) and Clause 6.
at = materials.equivalent_area_uncertainty([12.4], [1000], confidence=0.95)
print(float(at.reported_expanded_uncertainty[0]))            # 1.3 m^2 on A_T = 12.4 m^2
ap = materials.practical_coefficient_uncertainty(
    [0.35, 1.00, 0.65, 0.60, 0.55], [250, 500, 1000, 2000, 4000]
)
print(list(ap.reported_expanded_uncertainty))                # [0.07, 0.08, 0.08, 0.08, 0.1]

# Single-number ratings (Clause 7 worked examples).
print(float(materials.weighted_coefficient_uncertainty(0.70).reported_expanded_uncertainty[0]))  # 0.07
print(float(materials.single_number_rating_uncertainty(8.1).reported_expanded_uncertainty[0]))   # 1.6

What those numbers mean. An expanded uncertainty of 0.08 on a one-third-octave $α_{s}$ is a statement about two laboratories: measuring the same product, both correctly, they may legitimately report 0.64 and 0.72, so a specification written to two decimals is not testable and a datasheet value that misses your own measurement by less than the combined $U$ is agreement, not a discrepancy. Within one laboratory the repeatability figure applies instead — 0.6 of the reproducibility value, a spread about 40 % narrower — so a repeat on your own bench that falls outside the narrower band points at the specimen, the mounting or the room, not at the method. The single numbers are worse than they look: $U (α_{w}) = 0.07$ exceeds the 0.05 step $α_{w}$ is quoted on, so a product measured at 0.85 (class B) cannot be told apart from one at 0.90 (class A) by a single test. A tender that hinges on a class letter is a tender decided by measurement noise; ask for the spectrum and the uncertainty instead, and quote $U$ beside $α_{w}$ in the fiche.

Tube or reverberation room?

The tube and the reverberation room both deliver a number called the “absorption coefficient”, and the two are routinely confused. They are different physical quantities, measured under different sound fields, and they do not match, sometimes not even closely.

The tube measures a normal-incidence coefficient: one plane wave, one angle, a specimen a few centimetres across, and a complex reflection factor that keeps magnitude and phase. ISO 354 measures the random-incidence coefficient $α_{s}$ : a diffuse field striking a sample of 10 m² to 12 m² from every direction at once, recovered from the change in the room’s decay time through Sabine’s formula, an energy average with no phase left in it. Because the diffuse field finds more ways into an absorber than the single normal-incidence wave (oblique waves travel a longer path inside the layer), $α_{s}$ usually comes out higher. For a locally reacting surface the two are linked by Paris’ angular average, which defines the statistical absorption coefficient

α_{s t} = \int_{0}^{π /2} α (θ) sin 2 θ d θ,

an integral that weights the oblique angles most heavily. It needs the angle-dependent $α (θ)$ , not the normal-incidence coefficient itself — but for a locally reacting surface $α (θ)$ follows from the normalised surface impedance alone, so the integral has a closed form and a tube result converts in one line: materials.statistical_absorption(result.normalized_impedance), documented with the multilayer models. What that cannot do is cover a bulk-reacting layer, where the oblique behaviour depends on refraction inside the material and the integral has to be evaluated angle by angle from a model — diffuse_field_absorption, on the same page.

The size of the gap is worth seeing, because “usually higher” is not a useful design statement:

Absorption coefficient of a 50 mm hard-backed porous layer against frequency from 125 Hz to 4 kHz: the random-incidence Paris average lies above the normal-incidence curve, with the two converging as the layer becomes thick against the wavelength

What the tube reads and what the room reads, for the same 50 mm hard-backed porous layer ( $σ = 20$ kPa·s/m², Miki): the tube measures $α (0°)$ , the dashed curve, while the room measures the Paris average over all angles, the solid one. The gap is widest where the layer is thin against the wavelength — precisely the low-frequency region where a normal-incidence number substituted into a Sabine or EN 12354-6 budget would underpredict the installed absorption — and closes as the layer becomes electrically thick. It is a model comparison on an infinite layer, so it is a lower bound on the real difference: a finite ISO 354 specimen adds the edge effect on top.

Show the code for this figure

import matplotlib.pyplot as plt

# A 50 mm porous layer on a rigid backing, predicted both ways.
f = np.geomspace(125.0, 4000.0, 200)
layers = [materials.PorousLayer(0.05, materials.miki(f, 20000.0))]
normal = materials.layered_absorber(f, layers)
diffuse = materials.diffuse_field_absorption(f, layers)
ax = diffuse.plot()
ax.plot(f, normal.absorption, ls="--", label="Normal incidence")
ax.legend()
plt.show()

Why ISO 354 values exceed 1. A ratio of absorbed to incident energy cannot exceed one, yet reverberation-room reports of $α_{s} = 1.05$ to $1.20$ for thick porous absorbers are routine and correct by the method. Sabine’s formula converts the decay-time change into an equivalent absorption area $A$ , and $α_{s} = A / S$ divides by the geometric sample area $S$ . Diffraction at the sample edges lets the specimen drain energy from a sound field wider than its footprint (the edge effect), so the equivalent area can exceed the geometric one. Such values are not errors, but they are not portable either: they depend on the sample size and perimeter, which is precisely why ISO 354 fixes both. The ISO 11654 rating simply truncates: practical coefficients above $1.00$ are set to $1.00$ (section 3). Prediction inputs get no such silent clipping in this library: each reverberation-time estimator enforces its own mathematical domain, so Sabine and Eyring accept ISO 354 values at or above one as supplied (Eyring as long as the mean absorption stays below one), while Millington-Sette rejects them (its per-surface logarithm diverges at one) and any adjustment below one is left to the caller. The equivalent-absorption-area budget of EN 12354-6 likewise accepts the coefficients as supplied.

Which to use. They answer different questions. The reverberation-room value is the one that feeds diffuse-field prediction: Sabine reverberation estimates, the equivalent absorption areas of EN 12354-6 and the $α_{w}$ rating and class, all of which expect random incidence over a mounted, finite sample (ISO 354’s Annex B mounting types exist because the mounting is part of the result). The tube value is the laboratory and development tool: it needs only a few square centimetres of material, resolves magnitude and phase, and its surface impedance pins down the parameters of porous-material models in the Allard and Atalla tradition, with the airflow resistivity as the first input; the fitted model then predicts the layer at any angle, thickness or backing. What the tube number is not is a drop-in substitute for $α_{s}$ : feeding normal-incidence coefficients into a Sabine or EN 12354-6 budget systematically underpredicts the installed absorption.

What this guide covers

Covered
The ISO 354:2003 Sabine inversion (Eqs. (5)-(9)) in measure_sound_absorption, and the Clause 4/8.1 conversion of an already-measured $T_{1}$ / $T_{2}$ pair in absorption_area and absorption_coefficient (with attenuation_from_alpha for the air term). ISO 11654:1997’s weighted absorption rating: the practical coefficient of Clause 4.1, the reference-curve shift of Clause 4.2, the shape indicators of Clause 4.3, and the Table B.1 absorption class, built by weighted_absorption and weighted_absorption_from_third_octave. ISO 12999-2:2020’s measurement-uncertainty formulae (Clauses 5-8), one function per quantity: sound_absorption_coefficient_uncertainty (Formula (1)), equivalent_area_uncertainty (Formula (2)), practical_coefficient_uncertainty (Clause 6), weighted_coefficient_uncertainty and single_number_rating_uncertainty (Clause 7), each taking condition="reproducibility" or "repeatability" for Formula (3), with the Table 3 coverage factor in absorption_coverage_factor.
Not covered
Nothing on this page consumes a measured surface impedance: the conversion from a tube result to the statistical coefficient $α_{s t}$ lives with the prediction models (statistical_absorption for a locally reacting surface, diffuse_field_absorption for a bulk-reacting one), and neither is a substitute for an ISO 354 measurement, because neither reproduces the edge effect. The room and specimen requirements of Clause 6 and Annex A are described in sections 1.2 and 1.3 but not checked: nothing here verifies the shape rule, the Table 1 ceiling on $A_{1}$ , the diffusivity qualification or the microphone and source counts, and absorption_area / absorption_coefficient only convert an already-measured $T_{1}$ / $T_{2}$ pair and raise an advisory warning when the room volume or sample area falls outside the Clause 6 limits. Neither is the decay measurement itself in scope — the interrupted-noise and integrated-impulse-response methods of Clause 7 belong to room acoustics. ISO 12999-2’s uncertainty formula for the EN 1793-1 single-number rating $D L_{α, NRD}$ is implemented, but EN 1793-1’s own in-situ measurement method (road-traffic noise-reducing devices) is not.

References

Cox, T. J., & D'Antonio, P. (2017). Acoustic absorbers and diffusers: Theory, design and application (3rd ed.). CRC Press. https://doi.org/10.1201/9781315369211Absorber measurement in practice: the room method, mountings, and the edge-effect discussion behind the tube-or-reverberation-room section. ISBN 978-1-4987-4099-9.
International Organization for Standardization. (1997). Acoustics — Sound absorbers for use in buildings — Rating of sound absorption (ISO 11654:1997). The weighted sound-absorption rating of section 3: the practical coefficient, the shifted reference curve, the shape indicators and the absorption class, validated against the standard's own Annex A worked example.
International Organization for Standardization. (2003). Acoustics — Measurement of sound absorption in a reverberation room (ISO 354:2003). The reverberation-room method of section 1 that produces the alpha_s spectra rated by ISO 11654, including the Annex B specimen mountings.
International Organization for Standardization. (2020). Acoustics — Determination and application of measurement uncertainties in building acoustics — Part 2: Sound absorption (ISO 12999-2:2020). The sound-absorption measurement uncertainty of section 4, validated against the standard's own worked examples (Tables 4/5).

Created and maintained by José Manuel Requena Plens ORCID All projects

Sound Absorption Measurement and Rating

1. Reverberation-room measurement (ISO 354)

1.1 How T1​ and T2​ are actually measured (Clause 7)

1.2 The specimen and its mounting (Clause 6.2 and Annex B)

1.3 Is the room fit to measure in? (Clause 6.1 and Annex A)

ISO 354 report (.report())

2. The same inversion on bare arrays: absorption_area and absorption_coefficient

absorption_area() / absorption_coefficient() parameters

3. Weighted rating and absorption class (ISO 11654)

ISO 11654 report (.report())

4. Sound-absorption measurement uncertainty (ISO 12999-2)

Tube or reverberation room?

What this guide covers

See also

References

1.1 How $T_{1}$ and $T_{2}$ are actually measured (Clause 7)

ISO 354 report (`.report()`)

2. The same inversion on bare arrays: `absorption_area` and `absorption_coefficient`

`absorption_area()` / `absorption_coefficient()` parameters

ISO 11654 report (`.report()`)