Source: https://jmrplens.github.io/phonometry/materials/absorbers/impedance-tube/ # Impedance Tube The impedance tube is the materials laboratory shrunk to a bench: a rigid duct a few centimetres across in which a loudspeaker drives a single plane wave against a small sample, and everything the sample does to that wave is read back from microphones in the tube wall. Because there is one wave and one angle, the tube recovers what a reverberation room cannot: the **complex** reflection factor and surface impedance at normal incidence, magnitude and phase, from a specimen the size of a coffee-cup lid. Three standards share the hardware and differ in method, and the library keeps their helpers separate and never mixes them: the **standing-wave-ratio** method of ISO 10534-1, the two-microphone **transfer-function** method of ISO 10534-2, and the four-microphone **transfer-matrix** method of ASTM E2611 that adds the transmission loss. The closing section runs the whole instrument, sample included, inside the [FDTD solver](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/) and recovers the analytic answer through the same reduction chains. ## 1. The working frequency range Everything the tube reports (ISO 10534-2, Clause 4) 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. That function implements only the Eq. (A.18) estimate, which the standard itself calls a lower limit; the measured determination of Eqs. (A.13)/(A.14) from the empty tube is not implemented. The frequency-range check is also the only condition the library verifies: the near-field distances of Clause 4.3, the position tolerances of Clause 4.5, the signal-to-noise floor of Clause 7.3 and the warm-up of Clause 5 are the operator's, and nothing here checks that any of them was met. The whole tube, sample included, can also be simulated: section 5 runs this exact measurement virtually inside the FDTD solver and recovers the analytic absorption of the modelled sample through the same reduction chain. ## 2. 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 ``` The ratio method predates the FFT analyser and survives because it is self-contained. The apparatus is a **probe microphone** on a graduated carriage: a thin tube sliding along the axis, sampling the interior field point by point while the loudspeaker holds a single pure tone. At each frequency the operator (or the stepper motor) locates one pressure maximum and the adjacent minimum, reads the level difference, and moves on to the next tone. Slow, but there is nothing to calibrate against anything else: one microphone measures both levels, so its sensitivity cancels exactly, and there is no inter-channel phase mismatch because there is only one channel. That is why the standing-wave method remains the arbitration method when two transfer-function tubes disagree. The closed forms above use only the level *difference*; the *positions* of the minima carry the rest of the information. With $x$ measured from the sample face towards the source, the interior field of a wave with reflection factor $r = |r|\,e^{j\Phi}$ is $$ p(x) = A\left(e^{jkx} + |r|\,e^{j\Phi}e^{-jkx}\right), \qquad |p|^2 = A^2\left(1 + |r|^2 + 2|r|\cos(2kx - \Phi)\right), $$ so pressure minima sit where the cosine is $-1$, at $$ 2k\,x_{\min,n} - \Phi = (2n - 1)\,\pi, \qquad \Phi = \frac{4\pi\,x_{\min,1}}{\lambda} - \pi , $$ with $x_{\min,1}$ the minimum nearest the sample. A minimum a quarter wavelength from the face means $\Phi = 0$ (a hard-backed sample at resonance); a minimum right at the face means $\Phi = -\pi$ (a rigid wall). Magnitude from the ratio, phase from the position: together they give the same complex $r$, and hence the same normalised impedance $Z/\rho c_0 = (1+r)/(1-r)$, that the transfer-function method of section 3 computes from $H_{12}$ in one shot. ```python import numpy as np from phonometry import materials # One 500 Hz reading: 9.54 dB between max and min, first minimum 12 cm out. f, c0 = 500.0, 343.2 swr = float(materials.standing_wave_ratio_from_level(9.542)) r_mag = float(materials.standing_wave_reflection_magnitude(swr)) phase = 4 * np.pi * 0.12 * f / c0 - np.pi # Phi from x_min,1 refl = r_mag * np.exp(1j * phase) print(round(r_mag, 2), round(np.degrees(phase), 1)) # 0.5 -54.1 print(np.round((1 + refl) / (1 - refl), 2)) # Z / rho c0: (1.13-1.22j) ``` Two practical cautions from the standard's own text. The minima far from the sample are shallower than the theory above says, because the travelling waves decay along the tube (viscous and thermal losses at the wall); ISO 10534-1 has the operator read the minimum **nearest** the sample and, for precision work, extrapolate the minimum levels to the sample face — that extrapolation, like the standard's probe-body corrections, stays with the operator: the closed forms here consume the levels as read. And the method leans on the purity of the tone: any harmonic distortion from a hard-driven loudspeaker puts energy at frequencies whose minima sit elsewhere, partially filling the notch being measured, so the analyser must be narrowband around the drive frequency rather than a broadband level meter. ## 3. 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)): $$ r = \frac{H_{12} - H_\mathrm{I}}{H_\mathrm{R} - H_{12}}\,e^{\,2 j k_0 x_1}, \qquad \alpha = 1 - |r|^2, \qquad \frac{Z}{\rho c_0} = \frac{1+r}{1-r}, $$ with $H_\mathrm{I} = e^{-j k_0 s}$, $H_\mathrm{R} = e^{+j k_0 s}$, 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 $\alpha_\mathrm{s}$/$\alpha_\mathrm{w}$ 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 and, for the figure the fiche embeds, matplotlib (`pip install "phonometry[report,plot]"`); only `engine="reportlab"` is supported. The fiche renders in English by default; pass `language="es"` for a Spanish fiche (translated fixed strings and a comma decimal separator). ```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.* ## 4. 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/materials/absorbers/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. ## 5. The virtual tube: the solver measured by the standards A three-row rigid-walled domain of the [2D FDTD solver](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/) is a plane-wave tube, and with the per-cell `damping` map a porous sample becomes an equivalent fluid (density, speed and loss maps). That closes a remarkable loop: the FDTD probe histories can be reduced through the library's own ISO 10534-2 and ASTM E2611 chains as if they were measurements, and the recovered spectra agree with the exact analytic answer for the same lossy layer to within 0.035 in absorption and 0.1 dB in transmission loss (the `tests/simulation` cross-checks run this on every commit). ```python import numpy as np from phonometry.materials import two_microphone_impedance from phonometry.simulation import FDTD2D c0, rho0, dx = 343.0, 1.2, 0.005 nx, d_cells = 280, 20 # 1.4 m tube, 10 cm sample c = np.full((3, nx), c0); rho = np.full((3, nx), rho0) sigma = np.zeros((3, nx)) c[:, -d_cells:] = 0.6 * c0 # the sample: slower, rho[:, -d_cells:] = 3.0 * rho0 # denser and lossy sigma[:, -d_cells:] = 600.0 sim = FDTD2D(c, dx, rho=rho, damping=sigma, edge_impedance={"left": rho0 * c0}) # anechoic source end sim.add_plane_wave("right", center=0.35, width=0.05) mics = (219, 229) # the ISO microphone pair records = np.zeros((2, 9000)) for n in range(records.shape[1]): sim.step() records[:, n] = sim.p[1, mics] spec = np.fft.rfft(records, axis=1) freqs = np.fft.rfftfreq(records.shape[1], sim.dt) band = (freqs > 300.0) & (freqs < 1200.0) result = two_microphone_impedance( spec[1, band] / spec[0, band], frequency=freqs[band], spacing=0.05, x1=(nx - d_cells) * dx - (mics[0] + 0.5) * dx, speed_of_sound=c0, characteristic_impedance=rho0 * c0) result.plot() # the absorption the virtual tube "measured" ``` The two clips below run exactly this experiment. In the impedance tube a sustained plane tone builds the standing wave the two microphones read: against the rigid end the minima are deep, in front of the sample they stay shallow. In the transmission tube a carrier packet crosses an anechoic duct: the empty tube passes it unchanged, the lossy layer splits it into a reflection and an attenuated transmission that the four ASTM microphones resolve. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_impedance_tube.webm) [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_transmission_tube.webm) ## See also - [Sound Absorption Measurement and Rating](https://jmrplens.github.io/phonometry/materials/absorbers/absorption-measurement/): the random-incidence reverberation-room coefficient, its ISO 11654 rating, and when the tube or the room is the right instrument. - [Airflow Resistance](https://jmrplens.github.io/phonometry/materials/absorbers/airflow-resistance/): the ISO 9053 flow resistivity that anchors the models a tube measurement is fitted against. - [Porous and Multilayer Absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/porous-absorbers/): the equivalent-fluid models and the multilayer solver whose chain matrix reads out through the same ASTM E2611 machinery as a measured specimen. - [FDTD Simulation](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/): the 2D solver behind the virtual tubes of section 5. - [Calibration and dBFS](https://jmrplens.github.io/phonometry/signals/metrology/calibration/): microphone calibration ahead of the two-microphone transfer function. - API reference: [`materials.absorbers.impedance_tube`](https://jmrplens.github.io/phonometry/reference/api/materials/impedance-tube/). ## 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: the surface impedance and absorption the tube recovers, and the models fitted to them. - International Organization for Standardization. (1996). *Acoustics — Determination of sound absorption coefficient and impedance in impedance tubes — Part 1: Method using standing wave ratio* (ISO 10534-1:1996; the edition implemented here is its European adoption BS EN ISO 10534-1:2001). [iso.org catalogue](https://www.iso.org/standard/18603.html). The standing-wave-ratio method of section 2. - 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. ## Standards BS EN ISO 10534-1:2001 (standing-wave-ratio method); BS EN ISO 10534-2:2001 (transfer-function method); ASTM E2611-19 (four-microphone transmission loss). Every equation is derived from the standard text; the [conformance report](https://jmrplens.github.io/phonometry/reference/conformance/) validates the library against closed-form identities and synthetic round-trips, and the virtual tubes of section 5 cross-check the FDTD solver against the same reduction chains on every commit.