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# Theory: Materials and Surfaces

This page collects the theory behind materials and surfaces: surface scattering and diffusion, in-situ road-surface absorption, and acoustic material characterisation from the weighted absorption rating to airflow resistance and the impedance tube. It is part of the [theory reference](theory.md). The ISO 354 reverberation-room measurement that feeds the ISO 11654 rating is covered in [Theory: Rooms and Buildings](theory-rooms-buildings.md).

## Surface scattering and diffusion (ISO 17497-1, ISO 17497-2)

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

A rough surface splits the reflected energy into a specular and a scattered
part; the scattering coefficient $s$ is the non-specular energy fraction.
ISO 17497-1:2004+A1:2014 measures it in a reverberation room with the test
sample on a turntable: four reverberation times, stationary and rotating,
each without and with the sample (Table 2), give the random-incidence
absorption $\alpha_s$ (clause 8.1.1, Formula 1) and the *specular* absorption
$\alpha_{spec}$ (clause 8.1.2, Formula 4). Rotation decorrelates the scattered
reflections between decays, so they average out and register as extra
"absorption", and the scattering coefficient follows (clause 8.1.3,
Formula 5):

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

each $\alpha$ being a two-condition Sabine difference
$55.3 (V/S) [1/(c_b T_b) - 1/(c_a T_a)] - 4 (V/S)(m_b - m_a)$ with
$c = 343.2 \sqrt{(273.15 + t)/293.15}$ (Formula 2) and $m$ from ISO 9613-1
via $m = \alpha_{dB}/(10 \lg e)$ (Formula 3). The base plate itself must
scatter little: Table 1 caps its coefficient (Formula 6) at 0.05–0.25 across
100 Hz – 5 kHz (clause 6.2). Negative $s$ is truncated to zero for
presentation (clause 8.3), but values above 1 near grazing bands are kept
(clause 6.3.2). The Annex A uncertainty chain ($u_\alpha$, Formulae A.3/A.4;
$u_s$, Formula A.5; $U = 2 u_s$) is implemented. Since the standard prints no
worked example, the oracle is a synthetic end-to-end chain
($V = 200$ m³, $S = 10$ m², $T = 8.0/6.0/7.5/5.0$ s → $s = 0.093$) plus the
Formula A.5 hand value $u_s = 0.0297$.

### Directional diffusion coefficient (ISO 17497-2)

ISO 17497-2:2012 measures, in the free field, how uniformly a surface spreads
its reflected polar response over $n$ microphones. The autocorrelation-based
coefficient (clause 8.1, Formula 5) is

$$
d_\theta = \frac{\left( \sum_i p_i \right)^2 - \sum_i p_i^2}{(n - 1) \sum_i p_i^2},
\qquad p_i = 10^{L_i/10},
$$

1 for a perfectly uniform response and tending to 0 for a single specular
lobe; Formula 6 is the area-weighted form with $N_i = A_i / A_{min}$ from the
Formula 8 solid-angle factors ($A_i = (4\pi/\Delta\phi) \sin^2(\Delta\theta/4)$
at the zenith). Normalizing against a flat reference reflector of the same
size removes edge diffraction (clause 8.2, Formula 7):
$d_{\theta,n} = (d_\theta - d_{\theta,r})/(1 - d_{\theta,r})$. The
random-incidence value averages the source angles with weights 1:3:3:3:3 for
0°, ±30°, ±60° (clause 8.4). Anchors: the model-predicted 37-receiver arc of
the published six-period N = 7 QRD (Cox & D'Antonio 3rd ed., Appendix B;
Hargreaves et al. 2000, Table I) at 1000 Hz gives $d_\theta = 0.1099$, its
flat reference $0.0049$ and $d_{\theta,n} = 0.1055$; the band-averaged model
predictions match the published Appendix B BEM normalised diffusion in the
200-400 Hz bands within 0.01 (a low-band anchor: the broadband 100-5000 Hz
mean absolute deviation is about 0.09); zenith area factor 1.5710.

See the [Surface Scattering guide](surface-scattering.md) for usage.

## In-situ road surface absorption (ISO 13472-1, ISO 13472-2)

ISO 13472-1:2002 (extended surface method) recovers the normal-incidence
absorption of a road surface in place, from one microphone above it: the
direct and reflected components of an impulse response are separated by the
**subtraction technique** and the **Adrienne window** (clause 6.4: a sharp
leading edge, a mandated 5 ms flat top and a Blackman-Harris trailing edge),
and

$$
\alpha(f) = 1 - \frac{1}{K_r^2} \left| \frac{H_r(f)}{H_i(f)} \right|^2,
\qquad K_r = \frac{d_s - d_m}{d_s + d_m} = \frac{2}{3}
$$

for the mandatory geometry $d_s = 1.25$ m, $d_m = 0.25$ m (clause 4.2,
Annex C); $K_r$ is the spherical-spreading ratio between the direct and the
image path. Ratioing the road measurement against one on a highly reflective
reference surface cancels the entire electro-acoustic chain along with $K_r$
(Annex B). The 5 ms window bounds the sampled area (Annex A closed form:
radius ≈ 1.34 m for the standard geometry) and the valid range is
250 Hz – 4 kHz in one-third octaves. ISO 13472-2:2010 (spot method,
250–1600 Hz) instead couples a small impedance tube to the surface and defers
the mathematics to the ISO 10534-2 transfer-function method below (its
clauses 4/5.7/6.6); the implementation reuses that module, adding the Part 2
geometry and validity limits ($f_u = 0.58\ c_0/d$; microphone spacing bounds
$0.45\ c_0/f_{max}$ and $0.05\ c_0/f_{min}$, clause 5.4) and the Annex A
subtractive correction for internal system losses.

See the [Surface Scattering guide](surface-scattering.md) for usage.

## Acoustic material characterisation (ISO 11654, ISO 9053-1/2, ISO 10534-1/2, ASTM E2611)

### Weighted sound absorption (ISO 11654)

ISO 11654:1997 condenses an ISO 354 third-octave absorption curve into a
single number. The practical coefficient $\alpha_p$ averages the three thirds
of each octave 250 Hz – 4 kHz and rounds to steps of 0.05 (clause 4.1). The
reference curve (0.80, 1.00, 1.00, 1.00, 0.90 at 250–4000 Hz) is then shifted
downward in 0.05 steps until the sum of unfavourable deviations, counted only
where the measurement falls *below* the shifted curve, is $\le 0.10$;
$\alpha_w$ is the shifted curve at 500 Hz (clause 4.2). A shape indicator
flags excess absorption $\ge 0.25$ above the shifted curve: L at 250 Hz, M at
500/1000 Hz, H at 2000/4000 Hz (clause 4.3), and the informative Annex B maps
$\alpha_w$ to the absorption classes A–E. Because every quantity is a multiple
of 0.05, the implementation does the whole grid arithmetic in integer
twentieths, making the shift search and class boundaries exact and
float-safe. The two Annex A worked examples are reproduced:
$\alpha_p = (0.35, 0.70, 0.65, 0.60, 0.55)$ → $\alpha_w = 0.60$, class C; and
raising 500 Hz to 1.00 keeps $\alpha_w = 0.60$ but adds the indicator, "0.60(M)".

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

Airflow resistivity $\sigma = R\,A/d$ is the key transport parameter of a
porous absorber. ISO 9053-1:2018 (static method) drives a steady flow through
the specimen and fits $\Delta p = a\,u + b\,u^2$ through the origin
(clause 7.5); since $R_s = \Delta p / u = a + b\,u$, the linear coefficient is
the zero-velocity specific resistance, reported at the reference velocity
$u = 0.5$ mm/s. ISO 9053-2:2020 (alternating method) replaces the flowmeter
with a ~2 Hz piston and a microphone in a closed cavity (clause 8.7,
Formula 2):

$$
R = \kappa'\ \frac{p_s}{2 \pi f V}\ \frac{h_t}{h_s}\ 10^{(L_{ps} - L_{pt})/20}
$$

Only a level *difference* enters, so the sound-level device needs no
absolute calibration. The effective exponent $\kappa'$ (Annex A,
Formula A.7) corrects the adiabatic $\kappa$ for wall heat conduction through
the thermal boundary layer $b = \sqrt{2 c_0 l_h / \omega}$ (Formulae A.4/A.5).
The Annex A.3 worked example (100 mm closed cylinder at 2 Hz: $b = 1.83$ mm,
$\kappa' = 1.370 = 0.978\,\kappa$) is reproduced, and the validity guards of
Formula 3 (transfer ratio < 0.3) and Formula 4 (10 dB background margin) are
enforced.

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

A tube below its cut-on frequency ($f d < 0.58\ c_0$ circular,
$< 0.50\ c_0$ rectangular; microphone-spacing limits $f s < 0.45\ c_0$ and
$f > c_0/(20 s)$; clauses 4.2–4.5) carries only plane waves, so the surface
reflection factor of a sample is fully observable. ISO 10534-2
(transfer-function method) compares the measured two-microphone transfer
function $H_{12}$ with the analytic incident and reflected ones
$H_I = e^{-j k_0 s}$, $H_R = e^{+j k_0 s}$ (Annex D) to give (clause 7,
Eq. 17):

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

with the complex wavenumber's attenuation lower bound
$k_0'' = 1.94 \times 10^{-2} \sqrt{f}/(c_0 d)$ (Eq. A.18). ISO 10534-1
(standing-wave-ratio method) is the closed-form classic:
$|r| = (s - 1)/(s + 1)$ from the max/min ratio $s = 10^{\Delta L/20}$ and the
phase from the first-minimum position (Eqs. 12–26); an SWR of 3 gives exactly
$|r| = 0.5$ and $\alpha = 0.75$. ASTM E2611-19 adds transmission: four
microphones decompose the up- and downstream fields into the $A, B, C, D$
waves (Eqs. 17–20) and a two-load (or symmetric one-load) solve yields the
specimen's 2×2 **transfer matrix** $[p; u]_0 = T\,[p; u]_d$ (Eqs. 16/22–24),
from which the anechoic-backing normal-incidence transmission loss is
(Eqs. 25/26)

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

plus the hard-backed reflection
$R = (T_{11} - \rho c\,T_{21})/(T_{11} + \rho c\,T_{21})$ (Eq. 27), the
material wavenumber $\arccos(T_{11})/d$ (Eq. 29) and the characteristic
impedance $\sqrt{T_{12}/T_{21}}$ (Eq. 30). The three standards deliberately
keep their own sign ansatz and temperature units (ISO in kelvin, ASTM in
Celsius), and near-singular load solves raise a warning. Since neither
standard prints a numeric example, the oracles are physics identities: the
analytic air-layer matrix ($\det T = 1$, $T_{11} = T_{22}$, TL = 0 dB,
hard-backed $|R| = 1$), synthetic round-trips that recover a known $r$, and
two-load recovery of an asymmetric reciprocal specimen.

See the [Materials guide](materials.md) for usage.

## References

- Cox, T. J., & D'Antonio, P. (2017). *Acoustic absorbers and diffusers:
  Theory, design and application* (3rd ed.). CRC Press.
  ISBN 978-1-4987-4099-9.
  [doi:10.1201/9781315369211](https://doi.org/10.1201/9781315369211).
  Absorber and diffuser measurement and design, by the authors behind the
  ISO 17497-2 diffusion-coefficient method.
- Allard, J. F., & Atalla, N. (2009). *Propagation of sound in porous media:
  Modelling sound absorbing materials* (2nd ed.). Wiley.
  ISBN 978-0-470-74661-5.
  [doi:10.1002/9780470747339](https://doi.org/10.1002/9780470747339).
  The porous-material theory linking the airflow-resistance and
  impedance-tube quantities of the characterisation section.
- International Organization for Standardization. (2004). *Acoustics —
  Sound-scattering properties of surfaces — Part 1: Measurement of the
  random-incidence scattering coefficient in a reverberation room*
  (ISO 17497-1:2004+A1:2014, the edition implemented here).
  [iso.org catalogue](https://www.iso.org/standard/31397.html).
  The turntable scattering-coefficient method and its Annex A uncertainty
  chain.
- International Organization for Standardization. (2012). *Acoustics —
  Sound-scattering properties of surfaces — Part 2: Measurement of the
  directional diffusion coefficient in a free field* (ISO 17497-2:2012).
  [iso.org catalogue](https://www.iso.org/standard/55293.html).
  The free-field directional diffusion coefficient and its solid-angle area
  weighting.
- International Organization for Standardization. (1998). *Acoustics —
  Determination of sound absorption coefficient and impedance in impedance
  tubes — Part 2: Transfer-function method* (ISO 10534-2:1998; adopted in
  Europe as EN ISO 10534-2:2001; since revised as
  [ISO 10534-2:2023](https://www.iso.org/standard/81294.html)).
  [iso.org catalogue](https://www.iso.org/standard/22851.html).
  The two-microphone transfer-function method of the impedance-tube
  section.
- ASTM International. (2019). *Standard test method for normal incidence
  determination of porous material acoustical properties based on the
  transfer matrix method* (ASTM E2611-19, the edition implemented here;
  since revised as [ASTM E2611-24](https://store.astm.org/e2611-24.html)).
  [ASTM store](https://store.astm.org/e2611-19.html).
  The four-microphone transfer-matrix decomposition and its transmission
  loss.
- International Organization for Standardization. (2018). *Acoustics —
  Determination of airflow resistance — Part 1: Static airflow method*
  (ISO 9053-1:2018).
  [iso.org catalogue](https://www.iso.org/standard/69869.html).
  The static airflow-resistance method and its reference velocity.
