Source: https://jmrplens.github.io/phonometry/guides/materials/ # 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. **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. *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.*

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```python 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() ```

```python 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 $\alpha_p$ 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. To keep the one-third-octave $\alpha_s$ 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 $\alpha_p$, rates it and retains the input $\alpha_s$ 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. ```python 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 ``` ### 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 $\alpha_p$ table beside the practical-versus-shifted-reference plot (the result's own `.plot()`), the boxed $\alpha_w$ 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 $\alpha_s$ table with the octave $\alpha_p$ on the matching rows, exactly as accredited certificates print it. It uses the same `ReportMetadata` container (documented under [Field insulation](https://jmrplens.github.io/phonometry/guides/insulation-field/#report-metadata-reportmetadata)) 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 $\alpha_w$ 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 (`pip install phonometry[report]`); 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")`. ```python 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, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso11654_absorption_example.pdf) *Weighted absorption fiche (`AbsorptionRatingResult.report`), $\alpha_w$ with its class.* ### ISO 354 measurement (`measure_sound_absorption`) 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 Eq. (5)/(7)), $A = 55.3\,V/(c\,T) - 4\,V\,m$, with the speed of sound from Eq. (6), $c = 331 + 0.6\,t$; the sound absorption coefficient is $\alpha_s = (A_2 - A_1)/S$ (Eq. (8)/(9)). The coefficient may exceed 1.0 from edge and diffraction effects (Clause 3.7 NOTE 2) and is never clamped. Air attenuation enters only through the per-band coefficient $m$ (default 0, the zero-attenuation reference). ISO 354 is a characterisation: it produces the $\alpha_s$ spectrum, not a single-number rating. The weighted coefficient $\alpha_w$ is an ISO 11654 quantity; feed the measured $\alpha_s$ to `weighted_absorption_from_third_octave` above to obtain it. ```python 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]) m = materials.measure_sound_absorption( freqs, t_empty, t_specimen, volume=200.0, area=10.8, temperature=20.0 ) print(m.alpha_s[7]) # alpha_s at 500 Hz: 0.328... m.plot() # alpha_s versus one-third-octave frequency ``` *The Sabine inversion of the two decay times: the porous sample barely changes the long low-frequency decays ($\alpha_s \to 0$), 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).*

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```python 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]) m = 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. m.plot() plt.show() # By hand, from the result's fields: fig, ax = plt.subplots() ax.plot(np.arange(freqs.size), m.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() ```

### 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 $\alpha_s$ table beside the $\alpha_s$ 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; 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 (`pip install phonometry[report]`); only `engine="reportlab"` is supported. Pass `language="es"` for a Spanish fiche (translated fixed strings and a comma decimal separator). ```python from phonometry import materials, ReportMetadata m = materials.measure_sound_absorption( freqs, t_empty, t_specimen, volume=200.0, area=10.8, temperature=20.0, humidity=54.0, ) m.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 ``` The example fiche, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso354_absorption_example.pdf) *Reverberation-room absorption fiche (`SoundAbsorptionMeasurement.report`), the $\alpha_s$ spectrum.* ## 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. **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. *The slightly super-linear pressure drop is fitted through the origin; the specific airflow resistance is the fit read at 0.5 mm/s.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials 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 = materials.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() ```

```python import numpy as np from phonometry import materials 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 = materials.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 r.plot() # the figure above: the fitted dp(u) with the evaluation point ``` ### ISO 9053-1 report (`.report()`) `StaticAirflowResult.report(path)` renders a one-page PDF fiche laid out like an accredited airflow-resistance test report (ISO 9053-1:2018, static method): a standard-basis line, an optional metadata header block, a two-panel body with a metrics table (the evaluation velocity, the fitted pressure difference, the airflow resistance $R$, the specific airflow resistance $R_s$, the airflow resistivity $\sigma$ when a thickness is available, and the through-origin fit coefficients $a$ and $b$) beside the fitted $\Delta p(u)$ curve (the result's own `.plot()`), the boxed specific airflow resistance $R_s$ with $R$ and $\sigma$ alongside, and a footer with the fixed disclaimer. ISO 9053-1 is a material characterisation, so the fiche carries no pass/fail verdict. It uses the same `ReportMetadata` container and rendering engine as the other fiches. The descriptive fields that apply here are `client`, `manufacturer`, `specimen`, `thickness` (the specimen thickness $d$, in metres, shown in millimetres), `test_room`, `test_date`, `temperature`, `relative_humidity`, `measurement_standard`, `laboratory`, `operator`, `report_id` and `notes`. The `requirement` field is ignored (ISO 9053-1 has no verdict). The fiche embeds the fitted curve, so rendering needs both reportlab and matplotlib (`pip install "phonometry[report,plot]"`); only `engine="reportlab"` is supported. Pass `language="es"` for a Spanish fiche (translated fixed strings and a comma decimal separator). ```python from phonometry import materials, ReportMetadata r = materials.static_airflow_resistance(u, dp, area=area, thickness=0.05) r.report( "airflow_fiche.pdf", metadata=ReportMetadata( specimen="50 mm porous absorber (open-cell)", thickness=0.050, measurement_standard="ISO 9053-1", test_room="Static airflow rig, 100 mm cell", laboratory="Phonometry Reference Laboratory", ), ) # R_s, R and sigma at u = 0.5 mm/s ``` The example fiche, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso9053_airflow_resistance_example.pdf) *Static airflow-resistance fiche (`StaticAirflowResult.report`), $R_s$, $R$ and $\sigma$ at 0.5 mm/s.* **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 materials # Annex A.3 cavity: closed cylinder 100 mm x 100 mm, piston at 2 Hz kp = materials.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 = materials.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. **Working frequency range (ISO 10534-2, Clause 4).** Everything the tube reports assumes the field inside is a single plane wave, and the geometry sets both ends of the usable band. Above the cut-on of the first cross-sectional mode the field stops being planar: a circular tube of diameter $d$ needs $f\,d < 0.58\,c_0$ (Eq. (2); $f\,d < 0.50\,c_0$ for a rectangular tube, Eq. (3)). The microphone spacing $s$ must also stay clear of the half-wavelength singularity of the transfer-function method, $f\,s < 0.45\,c_0$ (Eq. (4)). At the low end the opposite problem appears: a spacing much shorter than the wavelength leaves almost no phase difference between the microphones to measure, so the Clause 4.2 guideline keeps the spacing above 5 % of the wavelength: $$ \frac{c_0}{20\,s} \;<\; f \;<\; \min\!\left(0.58\,\frac{c_0}{d},\ 0.45\,\frac{c_0}{s}\right). $$ No single tube covers the building-acoustics range. A 100 mm tube with a 100 mm spacing works from roughly 170 Hz to 1.5 kHz; reaching the 5 kHz bands takes a small tube (29 mm) with a close spacing, which in turn cannot see the low bands. Laboratories therefore pair a large and a small tube (or one tube with two spacings) and splice the spectra; the two must agree in the overlap bands, and a mismatch there points at the sample cut or mounting, not at the physics. ```python from phonometry import materials # A 100 mm tube with 100 mm spacing, and a 29 mm tube with 20 mm spacing. f_l, f_u = materials.plane_wave_frequency_range(0.100, 343.2, diameter=0.100) print(round(f_l, 1), round(f_u, 1)) # 171.6 1544.4 f_l, f_u = materials.plane_wave_frequency_range(0.020, 343.2, diameter=0.029) print(round(f_l, 1), round(f_u, 1)) # 858.0 6864.0 ``` Those limits are easier to see on the hardware itself. `plot_impedance_tube_geometry` draws a tube to scale with its plane-wave band worked out, and a measured `two_microphone_impedance` result that was given its geometry (`spacing`, `x1`, `diameter`) redraws its own setup with `result.plot_geometry()`. *Everything the working-range inequalities talk about, in one to-scale side view: the 100 mm bore sets the 1991 Hz top end, the 50 mm spacing sets the 343 Hz bottom end, and the microphones sit at $s$ = 50 mm with the farther one $x_1$ = 150 mm from the specimen face.*

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```python import matplotlib.pyplot as plt from phonometry import materials materials.plot_impedance_tube_geometry(spacing=0.05, x1=0.15, diameter=0.10) plt.show() # A measured result that retains its geometry draws its own tube: # res = materials.two_microphone_impedance( # h12, frequency=f, spacing=0.05, x1=0.15, diameter=0.10, ...) # res.plot_geometry() ```

The same helper covers square and rectangular tubes: pass `shape="square"` or `shape="rectangular"` and the Eq. (3) factor replaces the circular one, with $d$ the maximum side length. The four-microphone branch keeps its own limits in `plane_wave_frequency_range_astm`: ASTM E2611 retains the unrounded circular constant ($f\,d < 0.586\,c$, 6.2.4.1), the same rectangular $f\,d < 0.500\,c$ (6.2.5), a slightly stricter spacing bound $f\,s < 0.40\,c$ (6.5.4) and a laxer low end at 1 % of the wavelength (6.2.3). Passing `diameter=` (and `shape=`) to `two_microphone_impedance`, `wave_decomposition` or the `transfer_matrix_*` solvers turns the matching check into an advisory warning, and for the Annex A attenuation estimate of a rectangular tube `hydraulic_diameter(width, height)` supplies the $4A/P$ diameter that `tube_attenuation_constant` expects. **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: $$ s = 10^{\Delta L/20}, \qquad |r| = \frac{s-1}{s+1}, \qquad \alpha = 1 - |r|^2. $$ *A small level difference means a near-perfect absorber; a level difference of 9.54 dB gives $s = 3$, $|r| = 0.5$ and $\alpha = 0.75$.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials level_diff = np.linspace(0.5, 40.0, 300) # L_max - L_min [dB] swr = materials.standing_wave_ratio_from_level(level_diff) fig, ax = plt.subplots() ax.plot(level_diff, materials.standing_wave_absorption(swr), label="Absorption coefficient alpha") ax.plot(level_diff, materials.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() ```

```python from phonometry import materials s = float(materials.standing_wave_ratio_from_level(9.542)) # level difference [dB] -> SWR print(round(s, 2)) # 3.0 print(round(float(materials.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. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_standing_wave_tube.webm) ```python import numpy as np from phonometry import materials f = np.array([500.0, 1000.0, 1800.0]) x1, spacing, c0 = 0.12, 0.03, 343.2 k0 = materials.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 = materials.reflection_factor(h12, spacing=spacing, x1=x1, wavenumber=k0) print(np.round(materials.absorption_from_reflection(r), 3)) # [0.75 0.75 0.75] print(np.round(materials.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. *A 50 mm porous absorber measured over the working band of a 100 mm tube: the absorption climbs as the layer thickness grows against the wavelength, and $|r|$ falls as its mirror image ($\alpha = 1 - |r|^2$).*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials # A 50 mm porous absorber (Miki, sigma = 20 kPa s/m^2) in a 100 mm tube with # 100 mm spacing (working band ~170 Hz to 1.5 kHz): the layer model supplies # the true reflection factor, from which the transfer function H12 follows. f = np.linspace(200.0, 1500.0, 260) med = materials.miki(f, 20000.0) layer = materials.layered_absorber(f, [materials.PorousLayer(0.05, med)]) spacing, x1, c0 = 0.10, 0.20, 343.2 k0 = materials.tube_wavenumber(f, c0) x2 = x1 - spacing r_true = layer.reflection h12 = (np.exp(1j*k0*x2) + r_true*np.exp(-1j*k0*x2)) / \ (np.exp(1j*k0*x1) + r_true*np.exp(-1j*k0*x1)) result = materials.two_microphone_impedance( h12, frequency=f, spacing=spacing, x1=x1, speed_of_sound=c0, characteristic_impedance=407.0, diameter=0.10, ) # One line: alpha(f) with |r| overlaid. result.plot() plt.show() # By hand, from the result's fields: fig, ax = plt.subplots() ax.plot(f, result.absorption, label="Absorption alpha") ax.plot(f, np.abs(result.reflection), "--", label="Reflection factor |r|") ax.set_xlabel("Frequency [Hz]") ax.set_ylabel("Coefficient") ax.legend() plt.show() ```

### ISO 10534-2 report (`.report()`) `ImpedanceTubeResult.report(path)` renders a one-page PDF fiche laid out like an accredited normal-incidence impedance-tube test report (ISO 10534-2:2001): a standard-basis line, a metadata header block, the per-frequency table (the absorption coefficient $\alpha$ and the real and imaginary parts of the normalised surface impedance $z = Z/\rho c_0$) beside the $\alpha(f)$ curve (the result's own `.plot()`, on a continuous logarithmic frequency axis), a boxed characterisation headline and a footer with the fixed disclaimer. ISO 10534-2 is a characterisation, so the fiche carries no pass/fail verdict and no single-number rating; the normal-incidence coefficient is not comparable to the random-incidence $\alpha_s$/$\alpha_w$ of ISO 354 / ISO 11654. Setting `verbose=True` inserts the reflection-factor magnitude $|r|$ column. It uses the same `ReportMetadata` container and rendering engine as the other fiches. The measured frequency range is taken from the result; the applicable descriptive and geometric `ReportMetadata` fields are `client`, `manufacturer`, `specimen`, `tube_diameter`, `tube_shape`, `mic_spacing`, `mounting`, `test_room`, `test_date`, `temperature`, `pressure`, `measurement_standard`, `laboratory`, `operator`, `report_id` and `notes` (`tube_diameter` and `mic_spacing` are given in metres and printed in millimetres). The `requirement` field is ignored (ISO 10534-2 has no verdict). Rendering needs reportlab (`pip install phonometry[report]`); only `engine="reportlab"` is supported. Pass `language="es"` for a Spanish fiche (translated fixed strings and a comma decimal separator). ```python from phonometry import materials, ReportMetadata result = materials.two_microphone_impedance( h12, frequency=freqs, spacing=0.05, x1=0.10, speed_of_sound=c0, characteristic_impedance=rho_c, diameter=0.10, ) result.report( "alpha_fiche.pdf", metadata=ReportMetadata( specimen="Resistive facing over an 86 mm rigidly-backed air cavity", tube_diameter=0.10, # m (printed as 100 mm) mic_spacing=0.05, # m (printed as 50 mm) measurement_standard="ISO 10534-2", laboratory="Phonometry Reference Laboratory", ), ) # normal-incidence alpha and impedance over the tube band ``` The example fiche, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso10534_impedance_tube_example.pdf) *Normal-incidence impedance-tube fiche (`ImpedanceTubeResult.report`), the $\alpha$ spectrum with the surface impedance.* **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: $$ \mathrm{TL} = 20\log_{10}\left|\frac{T_{11} + T_{12}/\rho c + \rho c\,T_{21} + T_{22}}{2}\right|. $$ To scale it looks like this. A `TransferMatrix` recovered by the two-load or one-load solvers retains `l1`, `s1`, `l2`, `s2` and the specimen thickness, so `tm.plot_geometry()` redraws the tube a measurement was taken in. *Where the four microphones of the transfer-matrix method actually sit around a 50 mm specimen in a 100 mm tube: $s_1 = s_2$ = 50 mm, $l_1$ = 100 mm, $l_2$ = 200 mm, and the changeable termination that provides the second load. The ASTM working range for this geometry is 69 to 2011 Hz.*

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```python import matplotlib.pyplot as plt from phonometry import materials materials.plot_transmission_tube_geometry( l1=0.10, s1=0.05, l2=0.20, s2=0.05, thickness=0.05, diameter=0.10, ) plt.show() # A TransferMatrix from transfer_matrix_two_load / _one_load retains its # geometry, so tm.plot_geometry() redraws the tube it was measured in. ```

```python import numpy as np from phonometry import materials # 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 = materials.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, and its `.plot()` draws the transmission loss with the hard-backed absorption overlaid (a matrix built by the solvers retains $\rho c$ and, when supplied, the frequency vector, so only a hand-built matrix needs them as arguments). The matrix does not have to come from a measurement: the [multilayer solver](https://jmrplens.github.io/phonometry/guides/porous-absorbers/) exposes the chain matrix of any modelled stack in the same convention, so a predicted specimen can be read out exactly like a measured one. ```python import numpy as np from phonometry import materials # The chain matrix of a modelled 50 mm porous layer, read back through the # ASTM E2611 machinery: TL and hard-backed absorption of the same specimen. f = np.linspace(200.0, 1600.0, 300) med = materials.miki(f, 20000.0) layer = materials.layered_absorber(f, [materials.PorousLayer(0.05, med)]) chain = layer.transfer_matrix # shape (2, 2, len(f)) tm = materials.TransferMatrix(t11=chain[0, 0], t12=chain[0, 1], t21=chain[1, 0], t22=chain[1, 1]) print(np.round(float(tm.transmission_loss(407.0)[-1]), 1)) # 9.7 dB at 1.6 kHz tm.plot(f, 407.0) # TL(f) with the hard-backed absorption overlaid ``` *The same four-pole entries answer two different questions: how much sound the free-standing layer lets through (the transmission loss, Eq. (26)) and how much the rigidly-backed layer absorbs (Eq. (28)).*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials f = np.linspace(200.0, 1600.0, 300) med = materials.miki(f, 20000.0) layer = materials.layered_absorber(f, [materials.PorousLayer(0.05, med)]) chain = layer.transfer_matrix # shape (2, 2, len(f)) tm = materials.TransferMatrix(t11=chain[0, 0], t12=chain[0, 1], t21=chain[1, 0], t22=chain[1, 1]) # One line: TL(f) on the left axis, hard-backed absorption on the right. tm.plot(f, 407.0) plt.show() # By hand, from the matrix methods: fig, ax = plt.subplots() ax.plot(f, tm.transmission_loss(407.0), label="Transmission loss TL_n") twin = ax.twinx() twin.plot(f, tm.absorption_hard_backed(407.0), "--", color="gray", label="Hard-backed absorption alpha") ax.set_xlabel("Frequency [Hz]") ax.set_ylabel("Transmission loss TL_n [dB]") twin.set_ylabel("Hard-backed absorption alpha") plt.show() ```

**Mounting pitfalls.** Most bad tube data are made at the sample holder, long before the signal processing. The recurring failures: - **Perimeter gaps.** A specimen cut slightly undersize leaves an air sliver along the tube wall. Sound short-circuits around and behind the sample, and the gap itself resonates, so the measured absorption grows a spurious low-to-mid-frequency hump. Cut for a snug slide fit and seal the rim (a thin film of petroleum jelly is the classic remedy) without loading the front face. - **Compression.** A specimen cut oversize and forced in is denser than the product it is supposed to represent: the flow resistivity rises, a limp frame is stiffened, and the absorption curve shifts and flattens. The result is repeatable and wrong. - **Hidden back cavity.** If the specimen does not sit flush on the rigid backing, the unintended air layer acts as a quarter-wavelength cavity and moves the absorption peak down in frequency, flattering the material. Backing air gaps are perfectly legitimate mountings, but only when they are deliberate, dimensioned and reported with the result. - **Face not plane.** A bulging, tilted or torn front face scatters into cross-sectional modes below the nominal cut-on and breaks the normal-incidence assumption the equations rest on. Cut with a sharp tool; do not tear fibrous materials to size. - **One specimen is not the material.** Porous products are inhomogeneous at the scale of a tube sample. Measure several cuts and average; a spread between specimens is product variance worth reporting, not measurement noise to hide. ## 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 $\alpha_s$: 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), $\alpha_s$ usually comes out higher. For a locally reacting surface the two are linked by Paris' angular average, which defines the statistical absorption coefficient $$ \alpha_{st} = \int_0^{\pi/2} \alpha(\theta)\,\sin 2\theta\,\mathrm{d}\theta, $$ an integral that weights the oblique angles most heavily; evaluating it needs the angle-dependent $\alpha(\theta)$ from the measured surface impedance, not just the normal-incidence coefficient itself. **Why ISO 354 values exceed 1.** A ratio of absorbed to incident energy cannot exceed one, yet reverberation-room reports of $\alpha_s = 1.05$ 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 $\alpha_s = A/S$ 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 1). Prediction inputs get no such silent clipping in this library: each [reverberation-time estimator](https://jmrplens.github.io/phonometry/reference/api/rooms/reverberation-prediction/) 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](https://jmrplens.github.io/phonometry/reference/api/rooms/enclosed-space-absorption/) 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](https://jmrplens.github.io/phonometry/reference/api/rooms/enclosed-space-absorption/) and the $\alpha_w$ 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 of section 2 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 $\alpha_s$: feeding normal-incidence coefficients into a Sabine or EN 12354-6 budget systematically underpredicts the installed absorption. ## 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 and Ratings](https://jmrplens.github.io/phonometry/guides/insulation-field/)). **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`.

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```python 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() ```

```python 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 # 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 ``` ## References - 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 behind the measured quantities: airflow resistivity as a model input, and the surface impedance and absorption the tube recovers. - 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 measurement and design in practice: the tube and room methods side by side, mountings, and the edge-effect discussion behind the tube-or-reverberation-room section. - International Organization for Standardization. (2003). *Acoustics — Measurement of sound absorption in a reverberation room* (ISO 354:2003). [iso.org catalogue](https://www.iso.org/standard/34545.html). The reverberation-room method that produces the $\alpha_s$ spectra rated by ISO 11654, including the Annex B specimen mountings. - 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, the edition implemented here; 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 and its plane-wave frequency-range limits. - 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 method behind the transmission-loss helpers. - 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 method of section 2, with the through-origin regression and the 0.5 mm/s reference velocity. ## 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](https://jmrplens.github.io/phonometry/reference/conformance/) 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.