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. The ISO 354 reverberation-room measurement that feeds the ISO 11654 rating is covered in Rooms and Buildings.

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

The two parts of ISO 17497 answer different questions about the same surface, and their answers do not substitute for one another. Part 1 measures how much of the reflected energy leaves the specular direction: one number per band, which is exactly what a geometrical or ray-based room model consumes as its scattering probability. Part 2 measures how uniformly the reflected energy is spread over angle, which is what tells a designer whether a diffuser works, and it is deliberately blind to how much energy was redirected at all. The classic failure is to read one as the other: a set of battens that merely sends the reflection off in a different direction scores a high scattering coefficient and a low normalised diffusion coefficient, because redirection is not dispersion (Cox & D’Antonio 3rd ed., §5.1 and Fig. 5.2). The measurement conditions differ too — Part 1 is a random-incidence reverberation-room method restricted to shallow samples that absorb little (${\alpha }_s\le 0.50$, clause 6.3.4), Part 2 a free-field method with neither restriction — so a surface can legitimately have only one of the two coefficients published.

Both run from 0 to 1, and neither reaches 1 in practice. Scattering rises with frequency as the surface relief becomes comparable with the wavelength, and a surface meant to be flat stays low across the band: ISO 17497-1 Table 1 caps the base plate of the apparatus itself at 0.05 to 0.25, which is the standard’s own definition of “not scattering”. Real diffusers rarely exceed a diffusion coefficient of 0.7 (ISO 17497-2, definition 3.9, Note 1).

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 $m={\alpha }_{dB}/(10{\log }_{10}e)$ (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; $U=2u_s$) is implemented. Since the standard prints no worked example, the reference value 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 $u_s=0.0297$.

What the four decays require. The mechanism only works if the rotation is sampled properly and the sample is big enough to be a surface rather than an object. At full scale ($N=1$) the sample is circular with a diameter of at least 3.0 m, or square with a 2.65 m edge on a base plate of at least 3.75 m diameter, and its structural depth must stay below $d/16$ of the turntable diameter or the method stops measuring roughness (clauses 6.3.1–6.3.2); no part of the turntable comes closer than 1.0 m to a room wall (6.2), and a sample with rotational symmetry is offset from the rotation axis by at least $d/8$ (6.3.3). Every linear dimension, and the room volume of $V\ge 200$ m³, scales by $1/N$ (by $N^{-3}$ for the volume) in a scale model. Each of the four reverberation times is the arithmetic mean over at least two source and three microphone positions, six combinations in all (clause 7.3). The rotating decays are built from $n$ phase-locked averages at equal steps of $360^{\circ }/n$ with $60\le n\le 120$, $n=72$ (5°) preferred, and the excitation must be identical from step to step (clause 7.3) — it is precisely the angle-to-angle decorrelation that registers as the extra “absorption” the coefficient is built from, so a drifting signal fakes scattering. For the same reason the room must be time-invariant across the whole set: diffusers fixed, no rotating vanes, no ventilation or air circulation running (6.1.1), and temperature and relative humidity measured before and after each of the four situations (7.4).

A random-incidence scattering coefficient rising with frequency as the surface roughness becomes comparable with the wavelength.

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. Those microphones map a semicircle around the sample’s reference point (a two-dimensional goniometer, in an anechoic room or a reflection-free zone) for a single-plane surface, or a hemisphere (three-dimensional goniometer) for one that scatters in both planes (clause 4), at an angular resolution of 5° or finer in both azimuth and elevation — which is where an arc of 37 receivers over 180° comes from. Far-field conditions are taken as met when at least 80 % of the receivers lie outside the specular zone, for which the standard’s working geometry is a 10 m source distance and a 5 m receiver radius (clause 6.2.2); nearer arrangements are legitimate for a near-field application but must be reported with the source and receiver locations and the sample dimensions (6.2.1), because $d$ is a property of that surface at that size and distance, not a material constant. Each source-receiver pair needs the impulse response with the test surface present and the one with it absent, which is subtracted to remove background reflections, plus a source response measured on the reference point whenever more than one source or microphone is used (clause 7.3); the reflection is then time-windowed away from the direct sound and the edge diffraction before the levels $L_i$ are formed. The flat-reference measurement that the normalisation below removes is a further, complete run on a plane surface of the same size.

The autocorrelation-based coefficient (clause 8.1, Formula 5) is

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

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 ($A_i=(4\pi /\Delta \varphi ){\sin }^2(\Delta \theta /4)$ 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 over the source positions: for a two-dimensional boundary measurement the preferred sources sit at 0°, ±30° and ±60° from the reference normal (clause 6.2.2) and are averaged with weights 1:3:3:3:3, the outer four standing in for the solid angle they represent, while a hemispherical measurement averages the Table 1 source positions with equal weights (clause 8.4).

The object the coefficient is computed from, in the presentation clause 8.5 prescribes: one polar response per one-third-octave band and source position, on a semicircular plot with a decibel radial axis. This is the six-period $N=7$ quadratic-residue diffuser of the anchor below, at 1 kHz. Its energy sits in a fan of discrete grating lobes rather than in one specular lobe, which is why $d_{\theta }$ is 0.11 where the flat reference of the same size scores 0.005 — and why it is still nowhere near 1: the numerator rewards an even response, and a comb of lobes is not even.

Reference values: 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 Diffusers guide for usage.

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

Extended surface method (ISO 13472-1)

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

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

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.

Spot method (ISO 13472-2)

Where Part 1 works at a stand-off of about a metre in the open, ISO 13472-2:2010 presses a small impedance tube against the surface, trading the 250 Hz – 4 kHz range for 250–1600 Hz and a sample the size of the tube mouth. It 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 ($f_u=0.58c_0/d$; microphone spacing bounds $0.45c_0/f_{max}$ and $0.05c_0/f_{min}$, clause 5.4) and the Annex A subtractive correction for internal system losses.

See the In-situ Road-Surface Absorption guide 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)”.

The ISO 11654 rating: practical absorption against the shifted reference, with the unfavourable deviation shaded and the weighted coefficient read at 500 Hz.

See the Sound Absorption Measurement and Rating guide for usage.

Airflow resistance (ISO 9053-1/2)

Airflow resistivity $\sigma =RA/d$ is the key transport parameter of a porous absorber: the Delany-Bazley and Miki regressions predict a material’s complex impedance and wavenumber from $\sigma$ alone, and the Johnson-Champoux-Allard model takes it as the first of its five parameters. What governs absorption, though, is the resistance of the whole layer, $R_s=\sigma d$, against the characteristic impedance of air: a rigidly backed layer works best for ${\rho }_0{c}_0\lesssim \sigma d\lesssim 4{\rho }_0{c}_0$, so a 50 mm blanket wants roughly 8 to 33 kPa·s/m², which is where commercial absorbers cluster. Below that window the layer is acoustically transparent and above it the wave reflects off the front face before the pores can do any work. Since a rigidly backed layer still needs about a quarter wavelength of depth to develop its absorption, no resistivity buys absorption at frequencies whose quarter wavelength exceeds the depth available, which is why a thin, very resistive layer is a worse low-frequency absorber than a thick, moderately resistive one; the Airflow Resistance guide draws the window and gives the material ranges.

ISO 9053-1:2018 (static method) drives a steady flow through the specimen and fits $\Delta p=au+b{u}^2$ through the origin (clause 7.5); since $R_s=\Delta p/u=a+bu$, the linear coefficient is the zero-velocity specific resistance, reported at the reference velocity $u=0.5$ mm/s.

The fit, and the single abscissa it is read at. The curve is slightly super-linear because the $b{u}^2$ term is real, but the reported quantity is the linear coefficient $a$ — the fit extrapolated back to zero velocity — evaluated at the 0.5 mm/s reference velocity, which is why the reference velocity has to be stated with any resistivity that is quoted.

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):

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

Only a level difference enters, so the sound-level device needs no absolute calibration. The effective exponent ${\kappa }^{\prime }$ (Annex A, Formula A.7) corrects the adiabatic $\kappa$ for wall heat conduction through the thermal boundary layer ${\delta }_h=\sqrt{2c_0{l}_h/\omega }$ (Formulae A.4/A.5; ISO 9053-2 writes it $b$, renamed here to keep it apart from the quadratic fit coefficient above). The Annex A.3 worked example (100 mm closed cylinder at 2 Hz: ${\delta }_h=1.83$ mm, ${\kappa }^{\prime }=1.370=0.978\kappa$) is reproduced, and the validity guards of Formula 3 (transfer ratio < 0.3) and Formula 4 (10 dB background margin) are enforced.

See the Airflow Resistance guide for usage.

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

A tube below its cut-on frequency ($fd<0.58c_0$ circular, $<0.50c_0$ rectangular; microphone-spacing limits $fs<0.45c_0$ and $f>c_0/(20s)$, where $s$ is the microphone spacing throughout this subsection and not the scattering coefficient of the first section; clauses 4.2–4.5) carries only plane waves, so the surface reflection factor of a sample is fully observable. ISO 10534-2:1998 (transfer-function method — the clause and equation numbers here are that edition’s, and ISO 10534-2:2023 renumbers them) 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):

$$r=\frac{H_{12}-H_I}{H_R-H_{12}}{e}^{2j{k}_0{x}_1},\alpha =1-|r{|}^2,\frac{Z}{\rho c_0}=\frac{1+r}{1-r},$$

with the complex wavenumber’s attenuation lower bound $k_0^{\prime \prime }=1.94\times 10^{-2}\sqrt{f}/(c_{0}d)$ (Eq. A.18).

$H_{12}$ as measured is not the $H_{12}$ this formula wants. It carries the amplitude and, far more damagingly, the phase mismatch of the two channels — microphone, preamplifier and analyser channel together — and clause 7.5 requires one of two corrections before anything above is evaluated: interchange the two microphones for every specimen measurement and combine the two transfer functions (7.5.1, Eq. 8), or measure a calibration factor $H_c$ once, as the geometric mean of the interchanged pair on an absorptive specimen, and divide it out of every subsequent measurement (7.5.2, Eqs. 10 and 13) — which is the route mic_calibration_factor and apply_mic_calibration implement. It matters more than it looks: the method infers a complex reflection factor from the small phase difference between two positions a few centimetres apart, so at the low-frequency end that difference is itself small and a residual channel phase error of a fraction of a degree is indistinguishable from real specimen reactance. The symptom is a low-frequency absorption curve that drifts below zero or above unity and does not repeat between mountings.

The specimen mounting is part of the same contract (clause 6). The sample fits snugly in the holder but is neither compressed nor squeezed until it bulges, its edge is sealed against leaks, and its front face sits normal to the tube axis at a position known to ±0.5 mm — the same tolerance the standard puts on the reference plane’s distance to the nearest microphone (7.1), because that distance is $x_1$ in the formula above. An uneven back face is levelled with a putty layer against the backing plate, since an unintended air gap turns a porous layer into a resonant absorber and moves the whole curve; at least two specimens are measured under identical mounting conditions, and more when the material is laterally inhomogeneous. The two spacing bounds quoted above are what force a wide-band measurement into two spacer positions.

What the formula returns: $\alpha$ and $|r|$ for a 50 mm porous absorber over the working band of a 100 mm tube. The two curves are the same information — $\alpha =1-|r{|}^2$ — so the figure is really one measurement drawn twice, and the rise with frequency is the layer thickness growing against the wavelength.

ISO 10534-1:1996 (standing-wave-ratio method) is the closed-form classic: $|r|=(\mathrm{SWR}-1)/(\mathrm{SWR}+1)$ from the max/min ratio $\mathrm{SWR}=10^{\Delta L/20}$ and the phase from the first-minimum position (Eqs. 12–26); an SWR of 3 gives exactly $|r|=0.5$ and $\alpha =0.75$. (The standard writes the ratio $s$; it is spelled out here to keep it apart from the microphone spacing above.) 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)

$$TL=20{\log }_{10}\frac{\left|T_{11}+T_{12}/\rho c+\rho c{T}_{21}+T_{22}\right|}{2},$$

plus the hard-backed reflection coefficient $r_{hb}=(T_{11}-\rho c{T}_{21})/(T_{11}+\rho c{T}_{21})$ (Eq. 27, written $R$ there; renamed here because $R$ is the airflow resistance two subsections above), 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 reference values are physics identities that must hold exactly: the analytic air-layer matrix ($\det T=1$, $T_{11}=T_{22}$, $TL=0\text{dB}$, hard-backed $|r_{hb}|=1$), synthetic round-trips that recover a known $r$, and two-load recovery of an asymmetric reciprocal specimen.

The same four-pole entries answering two different questions: how much sound the free-standing layer lets through (the transmission loss above) and how much the same layer absorbs once it is backed rigidly. A material can be a good absorber and a poor barrier at once, which this pair makes plain.

See the Impedance Tube guide for usage.

References

• Allard, J. F., & Atalla, N. (2009). Propagation of sound in porous media: Modelling sound absorbing materials (2nd ed.). Wiley. https://doi.org/10.1002/9780470747339ISBN 978-0-470-74661-5. The porous-material theory linking the airflow-resistance and impedance-tube quantities of the characterisation 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 four-microphone transfer-matrix decomposition and its transmission loss. The edition implemented here; since revised as ASTM E2611-24 (https://store.astm.org/e2611-24.html).
• Cox, T. J., & D'Antonio, P. (2017). Acoustic absorbers and diffusers: Theory, design and application (3rd ed.). CRC Press. https://doi.org/10.1201/9781315369211ISBN 978-1-4987-4099-9. Absorber and diffuser measurement and design, by the authors behind the ISO 17497-2 diffusion-coefficient method.
• 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). The two-microphone transfer-function method of the impedance-tube section. Adopted in Europe as EN ISO 10534-2:2001; since revised as ISO 10534-2:2023 (https://www.iso.org/standard/81294.html).
• 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: 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). The free-field directional diffusion coefficient and its solid-angle area weighting.
• International Organization for Standardization. (2018). Acoustics — Determination of airflow resistance — Part 1: Static airflow method (ISO 9053-1:2018). The static airflow-resistance method and its reference velocity.
• Jiménez, N., Cox, T. J., Romero-García, V., & Groby, J.-P. (2017). Metadiffusers: Deep-subwavelength sound diffusers. Scientific Reports, 7, 5389. https://doi.org/10.1038/s41598-017-05710-5Open access. Deep-subwavelength panel diffusers whose slow-sound resonant slots reach the diffusion of a Schroeder or quadratic-residue diffuser in a fraction of the depth; co-authored with Cox, behind the ISO 17497-2 diffusion-coefficient method.
• Jiménez, N., Groby, J.-P., Pagneux, V., & Romero-García, V. (2017). Iridescent perfect absorption in critically-coupled acoustic metamaterials using the transfer matrix method. Applied Sciences, 7(6), 618. https://doi.org/10.3390/app7060618Open access. A transfer-matrix-method tutorial for rigidly-backed resonant absorbers with the critical-coupling condition for perfect absorption, using the same layer machinery as the multilayer impedance-tube prediction here.
• Jiménez, N., Romero-García, V., & Groby, J.-P. (2018). Perfect absorption of sound by rigidly-backed high-porous materials. Acta Acustica united with Acustica, 104(3), 396-409. https://doi.org/10.3813/AAA.919183The critical-coupling condition applied to a rigidly-backed layer of ordinary high-porosity absorber, connecting the porous-material models of the characterisation section to perfect single-frequency absorption.