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# Acoustic Materials

Characterising an absorber or a partition is a laboratory exercise with three
distinct instruments behind it. The **reverberation room** delivers a spectrum of
sound-absorption coefficients that ISO 11654 collapses into a single rated number
$\alpha_w$ with a letter class. The **flow rig** — steady or oscillating —
measures how hard air is to push through a porous sample, the airflow resistance
that governs its low-frequency behaviour (ISO 9053-1/-2). And the **impedance
tube** recovers, at normal incidence, the full complex surface impedance,
reflection factor and absorption of a small sample, and — with four microphones —
its transmission loss (ISO 10534-1/-2, ASTM E2611). This page covers all three.

## 1. Sound-absorption rating (ISO 11654)

ISO 354 measures the sound-absorption coefficient $\alpha_s$ of a material in
one-third-octave bands in a reverberation room. ISO 11654:1997 turns that
spectrum into a single-number rating comparable across products.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_iso11654_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_iso11654.svg" alt="ISO 11654 rating flow: measured alpha_s becomes practical alpha_p per octave band, the reference curve is shifted to best fit, alpha_w is read at 500 Hz with shape indicators, giving the absorption class A to E" width="82%"></picture>

**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:

$$
\alpha_{p,i} = \tfrac{1}{3}\big(\alpha_{i1} + \alpha_{i2} + \alpha_{i3}\big),
$$

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 $\alpha_p$, in steps of $0.05$ until
the sum of the **unfavourable** deviations — taken only where the measurement
lies below the shifted curve, with magnitude $(\text{curve} - \text{measured})$ —
is no more than $0.10$. The weighted coefficient $\alpha_w$ 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 $\alpha_w$ 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 $\alpha_w$ is always a multiple of $0.05$ these
ranges partition the grid exactly.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/absorption_rating_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/absorption_rating.svg" alt="ISO 11654 weighted sound absorption rating: the practical absorption spectrum plotted against the shifted reference curve over 250 Hz to 4000 Hz, with the unfavourable deviation at 250 Hz shaded and the weighted coefficient alpha_w read at 500 Hz" width="80%"></picture>

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

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

```python
import matplotlib.pyplot as plt
from phonometry import weighted_absorption

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

</details>

```python
from phonometry import weighted_absorption, absorption_class

# 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 = 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

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

`weighted_absorption` accepts the five octave-band $\alpha_p$ values (as a
sequence or a `{frequency: value}` mapping); pass the fifteen one-third-octave
$\alpha_s$ values to `practical_absorption_coefficient` first if you are starting
from raw ISO 354 data. The result carries the shifted reference curve and the
per-band deviations, and its `.plot()` renders the figure above.

## 2. Airflow resistance (ISO 9053-1/-2)

The airflow resistance quantifies how strongly a porous material opposes a steady
or slowly-oscillating flow. Both parts share the same three quantities and units
(ISO 9053-1:2018, Clause 3):

$$
R = \frac{\Delta p}{q_v}\ \left[\text{Pa·s/m}^3\right], \qquad
R_s = R\,A\ \left[\text{Pa·s/m}\right], \qquad
\sigma = \frac{R_s}{d}\ \left[\text{Pa·s/m}^2\right],
$$

with $\Delta p$ the pressure difference across the specimen, $q_v$ the volumetric
flow, $A$ the cross-section and $d$ the thickness. Note the specific airflow
resistance $R_s$ is in **Pa·s/m** (not Pa·s/m²); the airflow resistivity $\sigma$
is $R_s$ per metre of thickness.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_airflow_resistance_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_airflow_resistance.svg" alt="Airflow resistance measurement rigs: the ISO 9053-1 static method with a specimen in a holder, a steady laminar flow q_v and a differential manometer reading the pressure drop; and the ISO 9053-2 alternating method with an oscillating piston driving a cavity terminated by the specimen or an airtight plug, and a microphone reading the cavity level" width="92%"></picture>

**Static method (ISO 9053-1:2018).** A steady laminar flow is stepped up and the
pressure difference plotted against the linear velocity $u = q_v/A$. A regression
of at least second order **constrained through the origin**, $\Delta p = a\,u +
b\,u^2$, is fitted, and $\Delta p$ and $R_s$ are read at the reference velocity
$u = 0.5\ \text{mm/s}$ (Clause 7.5); the highest velocity must not exceed
15 mm/s. Because $R_s = \Delta p/u = a + b\,u$, the linear term $a$ is the
zero-velocity specific airflow resistance.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/airflow_resistance_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/airflow_resistance.svg" alt="ISO 9053-1 static-method airflow resistance: the measured pressure drop against linear airflow velocity, fitted with a through-origin quadratic, with the specific airflow resistance evaluated at the 0.5 mm/s reference velocity" width="80%"></picture>

*The slightly super-linear pressure drop is fitted through the origin; the
specific airflow resistance is the fit read at 0.5 mm/s.*

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

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

area = np.pi * 0.05**2                      # 100 mm diameter cell [m^2]
u = np.array([0.5, 1, 2, 4, 8, 12]) * 1e-3  # linear velocity [m/s]
dp = 1.6e4 * u + 4.0e5 * u**2               # measured pressure drop [Pa]
r = static_airflow_resistance(u, dp, area=area, thickness=0.05)

u_fit = np.linspace(0.0, 13e-3, 200)
dp_fit = r.linear_coefficient * u_fit + r.quadratic_coefficient * u_fit**2
fig, ax = plt.subplots()
ax.plot(u_fit * 1e3, dp_fit, label="Through-origin fit  dp = a u + b u^2")
ax.plot(u * 1e3, dp, "o", label="Measured pressure drop")
ax.plot(r.evaluation_velocity * 1e3, r.pressure_drop, "D",
        label="Evaluation at 0.5 mm/s")
ax.set_xlabel("Linear airflow velocity u [mm/s]")
ax.set_ylabel("Pressure drop dp [Pa]")
ax.set_title(f"R_s = {r.specific_resistance:.0f} Pa·s/m")
ax.legend()
plt.show()
```

</details>

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

area = np.pi * 0.05**2            # 100 mm diameter cell [m^2]
u = np.array([0.5, 1, 2, 4, 8, 12]) * 1e-3      # linear velocity [m/s]
dp = 1.6e4 * u + 4.0e5 * u**2                    # measured pressure drop [Pa]

r = static_airflow_resistance(u, dp, area=area, thickness=0.05)
print(round(r.specific_resistance))   # 16200   R_s [Pa*s/m]
print(round(r.resistivity))           # 324000  sigma [Pa*s/m^2]
print(round(r.linear_coefficient))    # 16000   a = R_s at u -> 0
```

**Alternating method (ISO 9053-2:2020).** A piston oscillating at 1–4 Hz drives
an alternating flow into a cavity terminated either by the specimen or by an
airtight plug; the resistance follows from the sound-pressure-level difference
between the two terminations (Formula (2)):

$$
R = \frac{\kappa'\,P_S}{2\pi f\,V}\cdot\frac{h_t}{h_s}\cdot
    10^{(L_{p,s} - L_{p,t})/20},
$$

where $\kappa'$ is the **effective** ratio of specific heats. Heat conduction
between the oscillating air and the cavity walls makes the compression not fully
adiabatic; the normative Annex A corrects $\kappa$ down to

$$
\kappa' = \frac{\kappa}{\sqrt{1 + (\kappa-1)\tfrac{S}{V}b
          + \tfrac{1}{2}\big((\kappa-1)\tfrac{S}{V}b\big)^2}},
\qquad b = \sqrt{\frac{2 c_0 l_h}{\omega}},\quad
l_h = \frac{k_a}{\rho_0 c_0 C_P},
$$

with $S$ and $V$ the cavity surface and volume and $b$ the thermal
boundary-layer thickness. For the Annex A.3 example cavity this gives
$\kappa' = 1.370$, about 2 % below the adiabatic 1.4008.

```python
from phonometry import effective_kappa, alternating_airflow_resistance

# Annex A.3 cavity: closed cylinder 100 mm x 100 mm, piston at 2 Hz
kp = effective_kappa(cavity_surface=0.0471, cavity_volume=7.854e-4, frequency=2.0)
print(round(kp, 3))               # 1.37   effective ratio of specific heats

R = alternating_airflow_resistance(
    level_specimen=74.0, level_termination=90.0,
    piston_stroke_specimen=14e-3, piston_stroke_termination=1.4e-3,
    frequency=2.0, cavity_volume=7.854e-4, kappa_prime=kp,
)
print(round(R))                   # 222956  airflow resistance R [Pa*s/m^3]
```

Pass the `effective_kappa` result to `alternating_airflow_resistance` for an
Annex-A-conforming figure; its `kappa_prime` argument otherwise defaults to the
uncorrected adiabatic 1.4. The call warns (via `AirflowResistanceWarning`) when
the piston frequency leaves 1–4 Hz or the Formula (3)/(4) validity criteria fail.

## 3. Impedance tube (ISO 10534-1/-2, ASTM E2611)

The impedance tube measures a small sample at **normal incidence**. Two standards
share the geometry but differ in method and sign convention, so the library keeps
their helpers separate and never mixes them.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_impedance_tube_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_impedance_tube.svg" alt="ISO 10534-2 two-microphone impedance tube: a loudspeaker radiating a plane wave down the tube, two microphones flush in the wall at spacing s and distance x1 from the specimen face, the test specimen against a rigid backing, and the incident and reflected waves" width="92%"></picture>

**Standing-wave-ratio method (ISO 10534-1).** A probe traverses the standing wave
and reads the level difference $\Delta L = L_\text{max} - L_\text{min}$ between a
pressure maximum and the adjacent minimum. The standing-wave ratio, reflection
magnitude and absorption follow in closed form:

$$
s = 10^{\Delta L/20}, \qquad |r| = \frac{s-1}{s+1}, \qquad
\alpha = 1 - |r|^2.
$$

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/impedance_tube_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/impedance_tube.svg" alt="ISO 10534-1 standing-wave-ratio method: the absorption coefficient and the reflection factor magnitude as functions of the standing-wave level difference, showing that a 9.54 dB difference corresponds to a standing-wave ratio of 3, a reflection magnitude of 0.5 and an absorption of 0.75" width="80%"></picture>

*A small level difference means a near-perfect absorber; a level difference of
9.54 dB gives $s = 3$, $|r| = 0.5$ and $\alpha = 0.75$.*

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import (
    standing_wave_absorption, standing_wave_ratio_from_level,
    standing_wave_reflection_magnitude,
)

level_diff = np.linspace(0.5, 40.0, 300)    # L_max - L_min [dB]
swr = standing_wave_ratio_from_level(level_diff)
fig, ax = plt.subplots()
ax.plot(level_diff, standing_wave_absorption(swr),
        label="Absorption coefficient alpha")
ax.plot(level_diff, standing_wave_reflection_magnitude(swr), "--",
        label="Reflection factor magnitude |r|")
ax.set_xlabel("Standing-wave level difference L_max - L_min [dB]")
ax.set_ylabel("alpha, |r|")
ax.legend()
plt.show()
```

</details>

```python
from phonometry import standing_wave_absorption, standing_wave_ratio_from_level

s = float(standing_wave_ratio_from_level(9.542))   # level difference [dB] -> SWR
print(round(s, 2))                                  # 3.0
print(round(float(standing_wave_absorption(s)), 2)) # 0.75
```

**Transfer-function method (ISO 10534-2).** Two fixed microphones measure the
complex transfer function $H_{12}$; from it the reflection factor at the sample
face, the absorption and the normalised surface impedance follow (Eqs. (17)–(19)):

$$
r = \frac{H_{12} - H_I}{H_R - H_{12}}\,e^{\,2 j k_0 x_1}, \qquad
\alpha = 1 - |r|^2, \qquad \frac{Z}{\rho c_0} = \frac{1+r}{1-r},
$$

with $H_I = e^{-j k_0 s}$, $H_R = e^{+j k_0 s}$, microphone spacing $s$ and $x_1$
the distance from the sample to the farther microphone.

```python
import numpy as np
from phonometry import (
    tube_wavenumber, reflection_factor,
    absorption_from_reflection, normalized_surface_impedance,
)

f = np.array([500.0, 1000.0, 1800.0])
x1, spacing, c0 = 0.12, 0.03, 343.2
k0 = tube_wavenumber(f, c0)

# A measured transfer function H12 (here synthesised from r = 0.3 - 0.4j)
target = 0.3 - 0.4j
x2 = x1 - spacing
h12 = (np.exp(1j*k0*x2) + target*np.exp(-1j*k0*x2)) / \
      (np.exp(1j*k0*x1) + target*np.exp(-1j*k0*x1))

r = reflection_factor(h12, spacing=spacing, x1=x1, wavenumber=k0)
print(np.round(absorption_from_reflection(r), 3))     # [0.75 0.75 0.75]
print(np.round(normalized_surface_impedance(r), 2))   # Z / rho c0
# [1.15-1.23j 1.15-1.23j 1.15-1.23j]
```

The high-level `two_microphone_impedance` wraps this chain and returns an
`ImpedanceTubeResult` with absorption, reflection factor, surface impedance and
normalised impedance, applying the plane-wave frequency-range check and optional
tube attenuation; correct any microphone mismatch beforehand with
`apply_mic_calibration`. Its `.plot()` draws the absorption spectrum
$\alpha(f)$ with the reflection-factor magnitude $|r|$ overlaid.

**Transmission loss (ASTM E2611).** With four microphones — two upstream, two
downstream of the sample — a two-load (or one-load) measurement recovers the
sample's transfer matrix, whose entries give the normal-incidence transmission
loss, reflection and wavenumber:

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_astm_tube_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_astm_tube.svg" alt="ASTM E2611 four-microphone transmission-loss tube: a sound source, two microphones upstream and two downstream of the test specimen at spacings s1 and s2 and offsets l1 and l2, an adjustable termination for the two-load method, the upstream A and B and downstream C and D travelling waves, and the transfer matrix and transmission-loss relations" width="92%"></picture>

$$
\mathrm{TL} = 20\log_{10}\left|\frac{T_{11} + T_{12}/\rho c
             + \rho c\,T_{21} + T_{22}}{2}\right|.
$$

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

# An air layer is a known transfer matrix; TL = 0 dB (nothing is lost)
f = np.array([500.0, 1000.0, 2000.0])
k0 = 2*np.pi*f / 343.2
rho_c = 1.186 * 343.2
tm = air_layer_transfer_matrix(thickness=0.05, wavenumber=k0,
                               characteristic_impedance=rho_c)
print(np.round(tm.transmission_loss(rho_c), 6))   # [0. 0. 0.]
```

`transfer_matrix_two_load` / `transfer_matrix_one_load` build the
`TransferMatrix` from the four measured microphone transfer functions
`(H1, H2, H3, H4)` of each load; its methods
(`transmission_loss`, `reflection_hard_backed`, `absorption_hard_backed`,
`characteristic_impedance_material`, `material_wavenumber`) then read off the
ASTM E2611 quantities.

## 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
([Building acoustics](building-acoustics.md)).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/absorption_uncertainty_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/absorption_uncertainty.svg" alt="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" width="80%"></picture>

**One-third-octave bands (Clause 5).** For the sound-absorption coefficient the
reproducibility standard deviation is `σR = m·αs + n` (Formula (1)), and for the
equivalent absorption area `σR = m·AT + n·S` with `S = 10 m²` (Formula (2)), where
`m` and `n` are the frequency-dependent constants of Table 1 (63–5000 Hz). The
repeatability value is `σr = 0.6·σR` (Formula (3)).

**Practical coefficient (Clause 6).** For the ISO 11654 practical coefficient
`σR = m·αp + n` in octave bands with the constants of Table 2 (250–4000 Hz);
again `σr = 0.6·σR`.

**Single numbers (Clause 7).** The weighted coefficient `αw` has a constant
standard uncertainty (`σR = 0.035`, `σr = 0.020`); the EN 1793-1 single-number
rating `DLα,NRD` scales with the value (`σR = 0.10·DLα`, `σr = 0.02·DLα`).

**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). The reported
`U` is rounded to two decimals for absorption coefficients and one decimal for
the equivalent area and `DLα,NRD`.

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

```python
import matplotlib.pyplot as plt
from phonometry import sound_absorption_coefficient_uncertainty

# 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 = sound_absorption_coefficient_uncertainty(alpha_s, freqs, confidence=0.95)
result.plot()   # alpha_s with the +/-U (k = 2) reproducibility ribbon
plt.show()
```
</details>

```python
from phonometry import (
    sound_absorption_coefficient_uncertainty,
    weighted_coefficient_uncertainty,
    single_number_rating_uncertainty,
)

# Reproducibility uncertainty of alpha_s at 1000 Hz (Table 1: m=0.040, n=0.015).
r = 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)

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

---

**Standards.** ISO 11654:1997 (weighted sound-absorption rating); ISO 9053-1:2018
(static airflow resistance); ISO 9053-2:2020 (alternating airflow resistance);
BS EN ISO 10534-1:2001 and BS EN ISO 10534-2:2001 (impedance tube); ASTM E2611-19
(four-microphone transmission loss); ISO 12999-2:2020 (sound-absorption
measurement uncertainty). Every equation is derived from the standard
text; the [conformance report](CONFORMANCE.md) validates the library against the
standards' own worked examples (ISO 11654 Annex A, ISO 9053-2 Annex A.3,
ISO 12999-2 Tables 4/5) and closed-form identities.
