Source: https://jmrplens.github.io/phonometry/materials/absorbers/porous-absorbers/ # Porous and Multilayer Absorbers Given a porous material's **flow resistivity** (the quantity the [flow rig](https://jmrplens.github.io/phonometry/materials/absorbers/airflow-resistance/) measures) the classical equivalent-fluid models predict its complex characteristic impedance and wavenumber, and a **transfer-matrix** stack of layers (porous blankets, air gaps, perforated and microperforated panels, membranes) predicts the absorption coefficient of the whole construction before anything is built. This page covers the three porous models (Delany–Bazley, Miki, Johnson–Champoux–Allard), the multilayer solver, the resonant sheet layers of Maa and the random-incidence (Paris) integral. The measurement counterparts live in the [impedance tube](https://jmrplens.github.io/phonometry/materials/absorbers/impedance-tube/) and [reverberation room](https://jmrplens.github.io/phonometry/materials/absorbers/absorption-measurement/) guides; rating the predicted spectrum lives in the latter's [ISO 11654 section](https://jmrplens.github.io/phonometry/materials/absorbers/absorption-measurement/#3-weighted-rating-and-absorption-class-iso-11654); and the resonant metamaterial relatives of these constructions, slow-sound slit panels that reach perfect absorption at critical coupling, have their own guide, [Metamaterial Absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/metamaterial-absorbers/). ## 1. Equivalent-fluid models of a porous material A rigid-frame porous material behaves like an *equivalent fluid* with a complex characteristic impedance $Z_\mathrm{c}$ and wavenumber $k$ (time convention $e^{+j\omega t}$, so a passive medium has $\mathrm{Im}(k) < 0$). **Delany–Bazley** (Mechel 2e Sect. G.11; Bies 5e Appendix D, Table D.1; Hopkins Eqs. 1.171–1.174) is the one-parameter power law in the absorber variable $X = \rho_0 f / \sigma$: $$ \frac{Z_\mathrm{c}}{\rho_0 c_0} = 1 + C_1 X^{-C_2} - j\,C_3 X^{-C_4}, \qquad \frac{k}{k_0} = 1 + C_5 X^{-C_6} - j\,C_7 X^{-C_8}, $$ with the classic rockwool/fibreglass coefficients $(0.0571,\,0.754,\,0.087,\,0.732,\,0.0978,\,0.700,\,0.189,\,0.595)$ and a stated fit range $0.01 < X < 1.0$ (porosity close to one). The library also ships the Table D.1 presets fitted to polyester (`"garai_pompoli"`) and to foams (`"dunn_davern"`, `"wu"`). Outside the fit range a `PorousAbsorberWarning` is raised and the extrapolated values are still returned; the classic failure is a *negative* real part of the layer input impedance at low frequency (Mechel Sect. G.12). **Miki** (1990) refitted the same Delany–Bazley data under a passivity (positive-real) constraint, so the model stays physically well behaved below the fit range; it is the usual choice when a one-parameter model must be evaluated broadband. **Johnson–Champoux–Allard (JCA)** is the five-parameter semi-phenomenological model (Cox & D'Antonio 3e Eqs. 6.19–6.25): flow resistivity $\sigma$, porosity $\phi$, tortuosity $\alpha_\infty$ and the viscous/thermal characteristic lengths $\Lambda$, $\Lambda'$ give the effective density and bulk modulus with the exact limits $j\omega\rho_\mathrm{e} \to \sigma$ at DC, $\rho_\mathrm{e} \to (\alpha_\infty \rho_0/\phi)(1 + (1-j)\,\delta_v/\Lambda)$ at high frequency, and the isothermal-to-adiabatic transition in $K_\mathrm{e}$. ```python import numpy as np from phonometry import materials f = np.geomspace(200.0, 4000.0, 200) db = materials.delany_bazley(f, 20000.0) # sigma in Pa s/m2 mk = materials.miki(f, 20000.0) jca = materials.johnson_champoux_allard( f, 20000.0, porosity=0.98, tortuosity=1.0, viscous_length=8.7e-5, thermal_length=8.7e-5, ) print(np.round(db.normalized_impedance[0], 3)) # (2.598-2.209j) print(np.round(mk.normalized_impedance[0], 3)) # (2.286-1.965j) print(np.round(jca.normalized_impedance[0], 3)) # (2.321-2.075j) db.plot() # normalised Zc and k components vs frequency ``` *The classical presentation of an equivalent-fluid model (Cox & D'Antonio 3e, Figs. 6.19–6.20): at low frequency the viscous forces dominate and the material looks stiff and lossy (all four components large); as frequency rises the components fall towards the free-air limits $Z_\mathrm{c} \to \rho_0 c_0$ and $k \to k_0$, so a thin layer only works where its thickness is a fair fraction of the wavelength inside the material.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials f = np.geomspace(100.0, 5000.0, 260) mk = materials.miki(f, 20000.0) # sigma = 20 kPa s/m^2 # One line: normalised Zc and k components on a log-log grid. mk.plot() plt.show() # By hand, from the result's fields: fig, ax = plt.subplots() ax.loglog(f, mk.normalized_impedance.real, label="Re(Zc)/rho c") ax.loglog(f, -mk.normalized_impedance.imag, "--", label="-Im(Zc)/rho c") ax.loglog(f, mk.normalized_wavenumber.real, label="Re(k)/k0") ax.loglog(f, -mk.normalized_wavenumber.imag, "--", label="-Im(k)/k0") ax.set(xlabel="Frequency [Hz]", ylabel="Normalised characteristic value") ax.legend() plt.show() ```

The three models agree closely over the Delany–Bazley fit range (Cox & D'Antonio Figs. 6.19–6.21 make the same comparison); JCA extends the prediction physically outside it. A `PorousMediumResult` built from measured data (for example the $Z_\mathrm{c}$, $k$ recovered by the [ASTM E2611 transfer-matrix reduction](https://jmrplens.github.io/phonometry/materials/absorbers/impedance-tube/)) plugs into the layer solver exactly like a modelled one. ### Limp frames: when the skeleton moves (Allard & Atalla 11.3.4) Every model above assumes the frame stands still. That is only true above the **decoupling frequency** of Zwikker and Kosten, $$ F_\mathrm{d} = \frac{\sigma\,\phi^2}{2\pi\rho_1}, $$ with $\rho_1$ the bulk density of the frame (the density of the sample as weighed, not of the material the fibres are made of). Below $F_\mathrm{d}$ the visco-inertial coupling is strong enough for the wave in the pores to drag the frame along, and a light frame (aeronautic-grade fibreglass, felts, thin screens) has real inertia to contribute. Neglecting the *stiffness* of the frame but not its *mass* in the Biot mixed pressure-displacement formulation leaves an equivalent fluid with the same bulk modulus and a corrected effective density (Allard & Atalla 2e Eqs. 11.53-11.55, after Panneton 2007): $$ \tilde\rho_{\text{limp}} = \frac{\rho_\mathrm{t}\,\tilde\rho_{\text{eq}} - \rho_0^2} {\rho_\mathrm{t} + \tilde\rho_{\text{eq}} - 2\rho_0}, \qquad \rho_\mathrm{t} = \rho_1 + \phi\rho_0, $$ where $\tilde\rho_{\text{eq}}$ is the rigid-frame effective density of any of the three models above and $\rho_\mathrm{t}$ is the apparent total density of the material. What anchors the correction is the printed equation itself, transcribed term by term against the page; the book tabulates no computed limp density anywhere. Two limits it states in prose are exact as well and are checked, but they corroborate rather than pin the form: neither constrains the $\rho_0$ terms, and a sign-flipped variant of the same equation satisfies both. The limits are: - **heavy frame**: as $\rho_1$ grows the correction vanishes and the rigid-frame result comes back; - **low frequency**: the rigid-frame density diverges as $\sigma/(j\omega)$, while the limp one converges on the finite, real $\rho_\mathrm{t}$. A rigid frame forbids rigid-body motion of the sample; a limp one allows it, which is why the limp model is the right one for an unconstrained specimen in an impedance tube. `limp_frame` takes a `PorousMediumResult` and returns another one, so it drops straight into a `PorousLayer` in the stack. *The same 50 mm soft fibrous layer under both models (Allard & Atalla Table 11.2, the input set behind their Fig. 11.2). Above the decoupling frequency the two predictions are indistinguishable, which is the plot's own check on the correction; below it the rigid-frame effective density runs away and the limp one settles on $\rho_\mathrm{t}/\rho_0 = 25.9$.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import ( PorousLayer, decoupling_frequency, johnson_champoux_allard, layered_absorber, limp_frame, limp_frame_applicable, ) # Allard & Atalla Table 11.2: soft fibrous layer, 50 mm. f = np.linspace(1.0, 2000.0, 800) rigid = johnson_champoux_allard( f, 25e3, porosity=0.98, tortuosity=1.02, viscous_length=90e-6, thermal_length=180e-6, ) limp = limp_frame(rigid, frame_density=30.0, porosity=0.98) print(round(decoupling_frequency(25e3, porosity=0.98, frame_density=30.0), 1)) # 127.4 print(round(float(limp.effective_density[0].real), 1)) # 31.2 = rho_t print(limp_frame_applicable(20e3), limp_frame_applicable(25e3)) # True False limp.plot() # normalised Zc and k of the corrected medium plt.show() # By hand, the Fig. 11.2 view: rho0 = rigid.air_density fig, ax = plt.subplots() for medium, style in ((rigid, "--"), (limp, "-")): ax.plot(f, medium.effective_density.real / rho0, style) ax.plot(f, medium.effective_density.imag / rho0, style) ax.set(xlabel="Frequency [Hz]", ylabel="rho_e / rho_0", ylim=(-30, 30)) plt.show() ```

The correction is only worth applying where the frame really is limp. Beranek (1947) asked for $|K_c/K_\mathrm{f}| < 0.05$ between the bulk modulus of the frame in vacuum and that of the fluid in the pores; the frame-structural-interaction study of Doutres et al. (2007) relaxed that to $0.2$, which for air ($K_\mathrm{f} \approx P_0 = 101.3$ kPa) is the rule of thumb that the frame must be softer than about 20 kPa. `limp_frame_applicable` applies either threshold. Neither accounts for mounting: a thin light foam decoupled from a vibrating structure by an air gap behaves limply well above the limit, and a material bonded to a vibrating structure should not be treated as rigid-framed at all. The practical consequence, for the 50 mm layer above with a rigid backing, is a lower absorption coefficient in the bottom two thirds of an octave and a slightly higher one either side of 500 Hz: ```python import numpy as np from phonometry import ( PorousLayer, johnson_champoux_allard, layered_absorber, limp_frame, ) bands = np.array([100, 125, 160, 200, 250, 315, 400, 500, 1000], dtype=float) rigid = johnson_champoux_allard( bands, 25e3, porosity=0.98, tortuosity=1.02, viscous_length=90e-6, thermal_length=180e-6, ) limp = limp_frame(rigid, frame_density=30.0, porosity=0.98) for medium in (rigid, limp): print(layered_absorber(bands, [PorousLayer(0.05, medium)]).absorption.round(2)) # [0.07 0.11 0.17 0.24 0.32 0.43 0.54 0.64 0.88] rigid frame # [0.04 0.08 0.15 0.24 0.36 0.48 0.61 0.71 0.91] limp frame ``` **Honest note on validation.** Allard & Atalla contains exactly one table of computed numbers in the whole book, and it is not a surface impedance: every prediction-versus-measurement pair in the porous chapters, Fig. 11.2 included, is a figure. No published source checked (Allard & Atalla itself, Cox & D'Antonio, Mechel, and the round-robin and Biot literature) tabulates the quantity either. The anchor is therefore the printed Eq. 11.55 itself, transcribed term by term against the printed page. The two exact limits the book states in prose (heavy frame and $\omega \to 0$) are checked as well, but they are weaker than they look: a sign-flipped variant of the equation satisfies both, and reproduces the $1/\rho_1$ decay of the heavy-frame residual too, so the limits cannot tell the printed form from that variant. The decoupling frequency evaluated on the fully specified glass wool of their Table 6.1, where pure arithmetic gives 43.27 Hz, is independent of all this. ### Elastic frames: the full Biot layer (Allard & Atalla 6 and 11) The limp model throws the frame stiffness away. Keep it, and the porous layer stops being a fluid: the Biot theory treats the skeleton as an elastic solid coupled to the pore fluid through a potential coupling coefficient $Q$ and an inertial coupling coefficient, and predicts **three** waves in an isotropic material instead of one. Two are compressional and one is shear (Allard & Atalla 2e ch. 6). The stress-strain relations are $$ \sigma^s_{ij} = \left[(P - 2N)\theta^s + Q\theta^f\right]\delta_{ij} + 2N e^s_{ij}, \qquad \sigma^f_{ij} = -\phi p\,\delta_{ij} = (Q\theta^s + R\theta^f)\delta_{ij}, $$ and for the usual case of a frame whose solid grains are much stiffer than the frame they build ($K_\mathrm{s} \to \infty$, true of glass, rock and polymer frames) the three elastic coefficients follow from the shear modulus $N$, the Poisson coefficient $\nu$ and the bulk modulus $K_\mathrm{f}$ of the fluid in the pores (Eqs. 6.26-6.29): $$ R = \phi K_\mathrm{f}, \qquad Q = (1-\phi)K_\mathrm{f}, \qquad P = \tfrac{4}{3}N + K_\mathrm{b} + \frac{(1-\phi)^2}{\phi}K_\mathrm{f}, \qquad K_\mathrm{b} = \frac{2N(1+\nu)}{3(1-2\nu)}. $$ With the modified densities $\tilde\rho_{11}$, $\tilde\rho_{12}$, $\tilde\rho_{22}$ of Eq. 6.56 the two compressional wavenumbers come out as the eigenvalues of a 2x2 problem (Eqs. 6.67-6.69) and the shear wavenumber from Eq. 6.83. `biot_waves` returns all three, together with the ratios $\mu$ of the fluid displacement over the frame displacement that say which medium each wave travels in. The wave whose $|\mu|$ is large is the **airborne** wave, the one an equivalent fluid already models; the other is the **frame-borne** wave, which has no equivalent-fluid counterpart at all. `biot_waves` takes the *rigid-frame* equivalent fluid of the pores, normally a `johnson_champoux_allard` result. That is deliberate: the frame inertia is the Biot model's own business, so handing it a `limp_frame` medium would count that inertia twice. The consequence a designer cares about is a resonance. A layer glued to a rigid wall holds its frame still at the wall and free at the front face, so the frame-borne wave resonates a quarter wavelength inside the layer, at $$ f_\mathrm{r} = \frac{1}{4l}\sqrt{\frac{\mathrm{Re}(K_c)}{\rho_1}}, \qquad K_c = \frac{2(1-\nu)N}{1-2\nu} $$ (Eqs. 6.109-6.111), where $K_c$ is the longitudinal elastic coefficient of the frame **in vacuum** and $\rho_1$ its bulk density. `frame_quarter_wave_resonance` evaluates it. Nothing in the rigid-frame or limp-frame models can produce that peak. *The glass wool of Allard & Atalla Table 6.1, 100 mm glued to a rigid wall, under both models. Away from the resonance the two are indistinguishable, which is the plot's own check on the Biot layer; around it the poroelastic prediction develops the dip-and-peak in the real part and the sharp maximum in the imaginary part that the book measures and plots in its Fig. 6.10.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import ( PoroelasticLayer, PorousLayer, biot_waves, frame_elastic_coefficient, frame_quarter_wave_resonance, johnson_champoux_allard, layered_absorber, ) # Allard & Atalla Table 6.1: glass wool "Domisol Coffrage", 100 mm, glued. f = np.linspace(200.0, 1500.0, 1301) shear = 2.2e6 * (1 + 0.1j) # 220 N/cm2, loss factor 0.1 med = johnson_champoux_allard( f, 40e3, porosity=0.94, tortuosity=1.06, viscous_length=0.56e-4, thermal_length=1.1e-4, ) print(frame_elastic_coefficient(shear, 0.0)) # (4400000+440000j) print(round(frame_quarter_wave_resonance( 0.10, shear_modulus=shear, poisson_ratio=0.0, frame_density=130.0), 1)) # 459.9 waves = biot_waves(med, porosity=0.94, tortuosity=1.06, frame_density=130.0, shear_modulus=shear) print(round(float(abs(waves.airborne_velocity_ratio[800])), 1)) # 42.4 print(np.round(waves.frame_borne_velocity_ratio[-1], 3)) # (0.811+0.473j) biot = layered_absorber(f, [PoroelasticLayer(0.10, med, 0.94, 1.06, 130.0, shear)]) rigid = layered_absorber(f, [PorousLayer(0.10, med)]) fig, ax = plt.subplots() for res, style in ((rigid, "--"), (biot, "-")): ax.plot(f, res.normalized_impedance.real, style) ax.plot(f, res.normalized_impedance.imag, style) ax.set(xlabel="Frequency [Hz]", ylabel="Zs / rho0 c0", ylim=(-3, 3)) plt.show() waves.plot() # the three Biot wavenumbers against frequency plt.show() ```

Inside a stack, a `PoroelasticLayer` carries the same six variables the theory needs, $[v_1^s, v_3^s, v_3^f, \sigma^s_{33}, \sigma^s_{13}, \sigma^f_{33}]$, so `layered_absorber` switches from the two-variable chain to the global-matrix assembly of Allard & Atalla Sect. 11.5: fluid and sheet layers keep their 2x2 matrices, each poroelastic layer enters through the six wave amplitudes of its own $[\Gamma]$ blocks rather than through a transfer matrix, and the two are joined by the printed coupling matrices of Sect. 11.4. Solving for the amplitudes directly avoids inverting $[\Gamma(0)]$, which is what makes a very soft or very thick frame tractable at all. Two adjacent poroelastic layers are coupled as **bonded** frames (their Eq. 11.67); a sheet next to one is coupled as a free, mechanically decoupled screen. The returned `transfer_matrix` is filled with `nan` for such a stack, because a 2x2 chain matrix does not exist for it; the surface impedance, reflection factor and absorption are unaffected. For the same 100 mm layer the resonance moves absorption by up to 0.21 at its sharpest; on third-octave centres the largest shift is the 0.12 at 500 Hz: ```python import numpy as np from phonometry import ( PoroelasticLayer, PorousLayer, johnson_champoux_allard, layered_absorber, ) bands = np.array([250, 315, 400, 500, 630, 800, 1000], dtype=float) med = johnson_champoux_allard( bands, 40e3, porosity=0.94, tortuosity=1.06, viscous_length=0.56e-4, thermal_length=1.1e-4, ) shear = 2.2e6 * (1 + 0.1j) for layer in (PorousLayer(0.10, med), PoroelasticLayer(0.10, med, 0.94, 1.06, 130.0, shear)): print(layered_absorber(bands, [layer]).absorption.round(2)) # [0.55 0.58 0.62 0.65 0.69 0.74 0.78] rigid frame # [0.53 0.56 0.61 0.77 0.71 0.74 0.78] Biot poroelastic ``` The layer is worth the extra five parameters when the frame is stiff and heavy enough to resonate in the band of interest (dense mineral wool, structural foams), when the material is bonded to a plate that shakes it directly, or when a measured impedance shows a feature no equivalent fluid can explain. A light, soft blanket in free air is better served by `limp_frame`. **Honest note on validation.** There is no published table of $Z_\mathrm{s}(f)$ or $\alpha(f)$ for a fully specified Biot layer, here or anywhere else checked, so this model cannot be pinned digit by digit the way the standards-based modules are. What it is anchored on instead, in decreasing strength: the **rigid-frame limit**, where making the frame infinitely stiff and heavy must reproduce the JCA equivalent fluid, whose own conformance *is* pinned on published digits, and does so with a residual that falls exactly as the inverse of the stiffness over eight decades, at four angles of incidence, with and without a rigid backing; the **limp limit**, where taking the stiffness to zero reproduces `limp_frame`; the agreement to machine precision between the **two independent derivations** the book gives, the ch. 6 closed form Eq. 6.107 and the ch. 11 global-matrix assembly; and the three computed numbers the book does print in prose for the Table 6.1 glass wool, all reproduced: the airborne wave changes root at 495 Hz, $|\mu_\mathrm{a}| > 40$ above 50 Hz, and $\mu_\mathrm{b}$ falls from 1.0 at 50 Hz to 0.82 at 1500 Hz. That last one is matched by $\mathrm{Re}(\mu_\mathrm{b})$, not by $|\mu_\mathrm{b}|$, even though the printed sentence says "the ratio modulus": the model gives $\mu_\mathrm{b}(1500) = 0.811 + 0.473j$, whose real part is 1.1 % from the printed value while its modulus, 0.939, is 14.5 % away. That is recorded in [Errata](https://jmrplens.github.io/phonometry/reference/errata/), along with the parameter sweep that fails to bring $|\mu_\mathrm{b}|$ anywhere near 0.82. The impedance peak of the book's thinner sample is printed as 860 Hz and comes out at 863.5 Hz under the $\mathrm{Im}(Z_\mathrm{s})$ peak rule. Everything else, including the whole oblique-incidence behaviour beyond its rigid-frame limit, rests on closed forms and on structural identities, not on published digits. ## 2. Multilayer prediction by transfer matrices Each fluid layer of thickness $d$ contributes the chain matrix (Cox & D'Antonio Eq. 2.29; equivalently the impedance recursion of Bies Eq. D.95 and Mechel Sect. D.4) $$ \begin{bmatrix} p \\ u \end{bmatrix}_{\text{front}} = \begin{bmatrix} \cos(k_x d) & j Z_x \sin(k_x d) \\ j \sin(k_x d)/Z_x & \cos(k_x d) \end{bmatrix} \begin{bmatrix} p \\ u \end{bmatrix}_{\text{back}}, $$ with the in-depth wavenumber $k_x = \sqrt{k^2 - k_0^2 \sin^2\theta}$ from Snell's law and $Z_x = Z_\mathrm{c} k / k_x$. Thin resonant sheets enter as series impedances $[[1, z], [0, 1]]$. Closing the chain with a rigid wall (or free air, or any impedance) gives the surface impedance, the reflection factor $R(\theta)$ and $\alpha(\theta) = 1 - |R|^2$. A single hard-backed porous layer reduces to the textbook closed form $Z_\mathrm{s} = -j Z_\mathrm{c} \cot(k d)$ (Mechel Sect. D.3, Eq. 1). Before the matrices, the physical picture: a hard-backed porous layer is an equivalent fluid wrapped in five measurable numbers. The 50 mm layer below, evaluated with the five-parameter JCA model those numbers belong to, absorbs 91 % of a normal-incidence 1 kHz wave; the code that follows runs the same layer through the one-parameter Miki fit instead, which lands slightly higher (0.937), a fair picture of the spread between the two models. ```python import numpy as np from phonometry import materials f = np.geomspace(200.0, 4000.0, 300) med = materials.miki(f, 20000.0) res = materials.layered_absorber(f, [materials.PorousLayer(0.05, med)]) i = np.argmin(np.abs(f - 1000.0)) print(round(res.absorption[i], 3)) # 0.937 at 1 kHz res.plot() # alpha(f) with |R| overlaid ``` The solver evaluates the physical quantities through a numerically robust admittance recursion (immune to the $e^{|\mathrm{Im}(k_x)| d}$ overflow of raw matrix entries for extremely attenuating layers) and still exposes the full chain matrix (reciprocal by construction, $\det T = 1$) in `transfer_matrix`, ready for the [ASTM E2611 machinery](https://jmrplens.github.io/phonometry/materials/absorbers/impedance-tube/) (`TransferMatrix`). *Four constructions, one 50 mm budget: the porous layer works broadband but fades at low frequency; the microperforated, perforated and membrane designs trade bandwidth for a resonant peak placed ever lower. The dotted lines are the shallow-cavity closed forms; the full model sits below them because the viscous plug mass and the finite cavity depth are not negligible.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials as m f = np.geomspace(50.0, 5000.0, 500) med = m.miki(f, 20000.0) med_light = m.miki(f, 10000.0) designs = { "Porous 50 mm": [m.PorousLayer(0.05, med)], "MPP + cavity": [m.MicroperforatedPlateLayer(0.5e-3, 0.15e-3, 0.008), m.AirLayer(0.048)], "Perforated + porous": [m.PerforatedPlateLayer(0.006, 0.0025, 0.05), m.PorousLayer(0.025, med), m.AirLayer(0.019)], "Membrane + porous": [m.MembraneLayer(2.0), m.AirLayer(0.01), m.PorousLayer(0.038, med_light)], } fig, ax = plt.subplots() for label, layers in designs.items(): ax.semilogx(f, m.layered_absorber(f, layers).absorption, label=label) ax.set(xlabel="Frequency [Hz]", ylabel="Absorption coefficient") ax.legend() plt.show() ```

Behind every one of those curves there is a plain layer list, read front to back in the order the incident wave meets it. `plot_absorber_stack` draws that list to scale before any physics runs, and a solved result retains its layers, so `materials.layered_absorber(f, layers).plot_geometry()` draws the same cross-section from the result itself. *The layer list as the wave meets it: microperforated plate, air cavity, porous layer, rigid backing. Drawing the stack to scale before computing anything catches the classic metres-versus-millimetres slip in a thickness at a glance.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials f = np.linspace(200.0, 4000.0, 100) layers = [ materials.MicroperforatedPlateLayer(0.001, 0.0002, 0.01), materials.AirLayer(0.03), materials.PorousLayer(0.05, materials.miki(f, 20000.0)), ] # The free function draws any layer list; a solved result retains its # layers, so this draws the same cross-section: # materials.layered_absorber(f, layers).plot_geometry() materials.plot_absorber_stack(layers) plt.show() ```

## 3. Resonant sheets: perforated, microperforated, membrane **Perforated panel.** The air plugs in the holes are the mass of a Helmholtz resonator: $m = (\rho_0/\varepsilon)\,[t + 2\delta a + \sqrt{8\nu/\omega}\,(1 + t/2a)]$ with the open area $\varepsilon$, the end-correction factor $\delta$ per orifice end and the visco-thermal resistance $r = (\rho_0/\varepsilon)\sqrt{8\nu\omega}\,(1 + t/2a)$ (Cox & D'Antonio Eqs. 7.6/7.12). The default end correction is the Fok-function interaction fit $\delta = 0.85\,(1 - 1.47\sqrt{\varepsilon} + 0.47\varepsilon^{3/2})$ (Table 7.1), valid for any open area. For a shallow cavity the resonance is $f_0 = (c_0/2\pi)\sqrt{\varepsilon/(t'\,d)}$ (Eq. 7.4). **Microperforated panel (MPP).** With submillimetre holes the viscous boundary layer fills the orifice and the panel absorbs *without any porous material*. The library implements Maa's exact short-tube impedance (Maa 1998, Eq. 2), $$ z_1 = j\omega\rho_0 t \left[ 1 - \frac{2}{x\sqrt{-j}}\,\frac{J_1(x\sqrt{-j})}{J_0(x\sqrt{-j})} \right]^{-1}, \qquad x = a\sqrt{\rho_0 \omega/\eta}, $$ plus the Eq. 5 end corrections (surface resistance $\tfrac{1}{2}\sqrt{2\omega\rho_0\eta}$ and piston reactance $0.85\,d$ total), divided by the open area. The perforate constant $x$ (proportional to the hole radius over the viscous boundary-layer thickness) governs everything: at the resonance $\omega_0 m = \cot(\omega_0 D/c_0)$ the peak absorption is $4r/(1+r)^2$ and the half-absorption bandwidth is $f_2/f_1 = \pi/\mathrm{arccot}(1+r) - 1$ (Maa Eqs. 9–21, Table I). ```python import numpy as np from phonometry import materials # Maa (1998) Fig. 5: d = t = 0.2 mm, holes every 2.5 mm, cavity 6 cm. eps = (np.pi / 4.0) * (0.2 / 2.5) ** 2 f = np.linspace(100.0, 4000.0, 2000) res = materials.layered_absorber( f, [materials.MicroperforatedPlateLayer(0.2e-3, 0.1e-3, eps), materials.AirLayer(0.06)], ) i = np.argmax(res.absorption) print(f"peak alpha = {res.absorption[i]:.2f} at {f[i]:.0f} Hz") # peak alpha = 0.96 at 677 Hz res.plot() # alpha(f) with |R| overlaid: the resonant MPP peak ``` *Maa's own Fig. 5 design, no porous material anywhere: the viscous losses in the submillimetre holes damp the panel-cavity resonance into a broad absorption peak. The narrow feature near 2.9 kHz is the half-wave cavity resonance, where the cavity presents a pressure node to the panel and the absorption collapses before the next resonance restores it.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials # Maa (1998) Fig. 5: d = t = 0.2 mm, holes every 2.5 mm, cavity 6 cm. eps = (np.pi / 4.0) * (0.2 / 2.5) ** 2 f = np.linspace(100.0, 4000.0, 1200) res = materials.layered_absorber( f, [materials.MicroperforatedPlateLayer(0.2e-3, 0.1e-3, eps), materials.AirLayer(0.06)], ) # One line: alpha(f) with |R| overlaid. res.plot() plt.show() # By hand, from the result's fields: fig, ax = plt.subplots() ax.semilogx(f, res.absorption, label="Absorption alpha") ax.semilogx(f, np.abs(res.reflection), "--", label="Reflection factor |R|") ax.set(xlabel="Frequency [Hz]", ylabel="Coefficient") ax.legend() plt.show() ```

**Membrane.** A limp impervious sheet is the surface mass $z = j\omega m$ (Cox Eq. 7.14; Bies Eq. D.96); over a cavity it resonates at the classical $f_0 \approx 60/\sqrt{m d}$ (adiabatic; $\approx 50/\sqrt{m d}$ when the cavity is porous-filled and isothermal, Cox Eqs. 7.9/7.10). The closed forms are exposed as `helmholtz_resonance_frequency` and `membrane_resonance_frequency`; the full frequency response comes from the same layer stack. ## 4. Oblique and random incidence `layered_absorber(..., angle=theta)` evaluates the full bulk-reacting stack at any polar angle; sheets are locally reacting (angle-independent), fluid layers refract per Snell's law; for an MPP over a cavity this reproduces Maa's oblique closed form (Eq. 23) exactly. The random-incidence coefficient is the Paris integral (Mechel Sect. D.5, Eq. 9) $$ \alpha_\mathrm{dif} = \frac{2}{\sin^2\theta_\mathrm{lim}} \int_0^{\theta_\mathrm{lim}} \alpha(\theta)\,\cos\theta\,\sin\theta\,\mathrm{d}\theta, $$ evaluated by Gauss–Legendre quadrature in `diffuse_field_absorption` (``angle_limit`` defaults to 90°; truncations at 75–87° are in use). For a *locally reacting* surface with known normalised impedance the integral has the closed form of Mechel Eq. 10, exposed as `statistical_absorption`; its maximum over all passive impedances is the published **0.951** (at $z \approx 1.57$). ```python import numpy as np from phonometry import materials f = np.array([250.0, 500.0, 1000.0, 2000.0]) med = materials.miki(f, 20000.0) layers = [materials.PorousLayer(0.05, med)] normal = materials.layered_absorber(f, layers) diffuse = materials.diffuse_field_absorption(f, layers) print(np.round(normal.absorption, 2)) # [0.26 0.62 0.94 0.95] print(np.round(diffuse.absorption, 2)) # [0.37 0.68 0.9 0.95] diffuse.plot() # alpha_dif(f); overlay normal.absorption to compare print(round(float(materials.statistical_absorption(1.567 + 0j)), 3)) # 0.951 ``` *Why the reverberation room reads higher than the tube: the Paris integral weights the oblique angles, whose waves travel a longer path inside the layer, so $\alpha_\mathrm{dif}$ exceeds the normal-incidence $\alpha(0°)$ exactly where the layer is thin against the wavelength. This is the model-side counterpart of the tube-versus-reverberation-room discussion of [Sound Absorption Measurement and Rating](https://jmrplens.github.io/phonometry/materials/absorbers/absorption-measurement/).*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials f = np.geomspace(125.0, 4000.0, 200) layers = [materials.PorousLayer(0.05, materials.miki(f, 20000.0))] normal = materials.layered_absorber(f, layers) diffuse = materials.diffuse_field_absorption(f, layers) # One line, then the normal-incidence overlay on the same axes: ax = diffuse.plot() ax.plot(f, normal.absorption, "--", label="Normal incidence alpha(0)") ax.legend() plt.show() ```

The classical layers above absorb with bulk or with a quarter-wave cavity. Their resonant metamaterial relatives, rigid panels of sub-millimetre slits loaded by Helmholtz resonators that slow the sound and reach perfect absorption ($\alpha = 1$) at critical coupling in panels only $\lambda/38$ deep, moved to their own guide: [Metamaterial Absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/metamaterial-absorbers/) covers the transfer-matrix model, the critical-coupling design solver and the FDTD cross-check of the meshed cell. ## Practical notes **Parameters are inputs.** Nothing on this page measures anything: the five JCA parameters, the frame's shear modulus and Poisson coefficient and the sheet geometries all enter as numbers the caller supplies. Of their characterisation methods only the flow resistivity has a guide ([Airflow Resistance](https://jmrplens.github.io/phonometry/materials/absorbers/airflow-resistance/)); an inverse fit of the remaining parameters against a measured impedance is a `scipy.optimize` exercise, not a library function. **Fit ranges.** Delany–Bazley warns (and extrapolates) outside $0.01 < X < 1$ and Miki outside $0.01 < f/\sigma < 1$; treat sub-range values as qualitative. JCA needs four extra parameters but behaves physically everywhere; with $\Lambda = \Lambda' = \sqrt{8\alpha_\infty\eta/(\phi\sigma)}$ and $\alpha_\infty = 1$ it tracks Delany–Bazley over the fit range. **Rigid or limp frame.** Every equivalent-fluid model above assumes a motionless frame, which only holds above the decoupling frequency $F_\mathrm{d} = \sigma\phi^2/2\pi\rho_1$. Below it a light frame moves with the pore fluid, and `limp_frame` adds its inertia (Allard & Atalla Eqs 11.53-11.55). Use it for felts, screens and light fibreglass, for anything measured unconstrained in an impedance tube, and wherever the frame in vacuum is softer than about 20 kPa (`limp_frame_applicable`); do not use it, or the rigid model either, for a material bonded to a vibrating structure. When the frame is stiff and heavy enough to resonate in the band of interest, neither equivalent fluid will do at all: use a `PoroelasticLayer` and the full Biot theory. **Local vs. bulk reaction.** The layer solver is bulk-reacting (sound refracts and travels inside the layers). `statistical_absorption` assumes local reaction, a good approximation for high flow resistivity, partitioned cavities or thin resonant facings; for thick, light porous layers integrate the bulk model with `diffuse_field_absorption` instead (Mechel Sect. D.6). Either way the prediction is for a laterally infinite sample: the finite-size edge effect that lets a measured ISO 354 $\alpha_\mathrm{s}$ exceed the predicted $\alpha_\mathrm{dif}$, and even exceed 1.0, is not modelled here. **Where the numbers were checked.** The models are pinned digit-exact to the printed coefficient tables (Bies Table D.1, Miki Eqs. 30–34), the solver to the closed forms above and to the `TransferMatrix` recovery of the [impedance-tube page](https://jmrplens.github.io/phonometry/materials/absorbers/impedance-tube/), the MPP to Maa's own approximation (stated ~6 % agreement with the exact Eq. 2), design example and Table I, and the Paris integral to its locally reacting closed form. Misprints found in the sources during this work are recorded in the [errata registry](https://jmrplens.github.io/phonometry/reference/errata/). ## See also - [Airflow Resistance](https://jmrplens.github.io/phonometry/materials/absorbers/airflow-resistance/), [Impedance Tube](https://jmrplens.github.io/phonometry/materials/absorbers/impedance-tube/) and [Sound Absorption Measurement and Rating](https://jmrplens.github.io/phonometry/materials/absorbers/absorption-measurement/): the measurement standards these prediction models connect to, namely ISO 9053-1/-2 (flow resistivity), ISO 10534-1/-2 and ASTM E2611 (impedance tube), and ISO 354 / ISO 11654 (random-incidence absorption and rating). - [Metamaterial Absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/metamaterial-absorbers/): the slow-sound slit panels and the critical-coupling condition that push perfect absorption into the deep-subwavelength regime. - [Diffusers and Their Coefficients](https://jmrplens.github.io/phonometry/materials/diffusers/diffusers/): the diffuser half of Cox & D'Antonio, implemented separately — this guide covers only the absorber half of the book. - Theory: [Acoustic material characterisation](https://jmrplens.github.io/phonometry/reference/theory/materials-surfaces/#acoustic-material-characterisation-iso-11654-iso-9053-12-iso-10534-12-astm-e2611): the characterisation quantities the empirical and phenomenological models take as input. ## References - Mechel, F. P. (Ed.). (2008). *Formulas of acoustics* (2nd ed.). Springer. [doi:10.1007/978-3-540-76833-3](https://doi.org/10.1007/978-3-540-76833-3). Sections D.3–D.6 (layer reflection, multilayer scheme, diffuse-field integrals) and G.11 (empirical porous relations). - 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). Chapter 6: the Biot theory of an elastic-framed porous material, its elastic coefficients, its two compressional waves and its shear wave, and the surface impedance and quarter-wave frame resonance of a hard-backed layer (Eqs 6.107-6.111). Chapter 11: the transfer-matrix method, the six-variable poroelastic layer matrix (Table 11.1), the coupling matrices (Sect. 11.4) and the global assembly (Sect. 11.5). Section 11.3.4: the rigid and limp frame limits of the Biot theory, the decoupling frequency and the limp effective density (Eqs 11.53-11.55, after Panneton 2007). - Bies, D. A., Hansen, C. H., & Howard, C. Q. (2017). *Engineering noise control* (5th ed.). CRC Press. [doi:10.1201/9781351228152](https://doi.org/10.1201/9781351228152). Appendix D: porous-material properties, Table D.1 coefficient sets and the layered-construction recursions D.91–D.99. - Cox, T. J., & D'Antonio, P. (2017). *Acoustic absorbers and diffusers: Theory, design and application* (3rd ed.). CRC Press. [doi:10.1201/9781315369211](https://doi.org/10.1201/9781315369211). Transfer-matrix modelling (Sect. 2.6), porous models (Sect. 6.5) and resonant-absorber design equations (Sects. 7.3/7.5). - Attenborough, K., & Van Renterghem, T. (2021). *Predicting outdoor sound* (2nd ed.). CRC Press. [doi:10.1201/9780429470806](https://doi.org/10.1201/9780429470806). Chapter 5: ground-impedance models, including the JCA family. - Hopkins, C. (2007). *Sound insulation*. Butterworth-Heinemann. [doi:10.4324/9780080550473](https://doi.org/10.4324/9780080550473). Section 1.3.2.2: the equivalent-gas model and the Delany–Bazley SI form. - Miki, Y. (1990). Acoustical properties of porous materials — Modifications of Delany–Bazley models. *Journal of the Acoustical Society of Japan (E)*, 11(1), 19–24. [doi:10.1250/ast.11.19](https://doi.org/10.1250/ast.11.19). The positive-real regression implemented in `miki`. - Maa, D.-Y. (1998). Potential of microperforated panel absorber. *Journal of the Acoustical Society of America*, 104(5), 2861–2866. [doi:10.1121/1.423870](https://doi.org/10.1121/1.423870). The exact MPP impedance (Eq. 2), end corrections, design formulas and the Fig. 5 example pinned in the tests. - Johnson, D. L., Koplik, J., & Dashen, R. (1987). Theory of dynamic permeability and tortuosity in fluid-saturated porous media. *Journal of Fluid Mechanics*, 176, 379–402. [doi:10.1017/S0022112087000727](https://doi.org/10.1017/S0022112087000727). The dynamic-tortuosity model behind the JCA effective density. - Delany, M. E., & Bazley, E. N. (1970). Acoustical properties of fibrous absorbent materials. *Applied Acoustics*, 3(2), 105–116. [doi:10.1016/0003-682X(70)90031-9](https://doi.org/10.1016/0003-682X(70)90031-9). The original empirical relations and their stated validity. ## Standards No standard governs these prediction models; they are textbook and journal methods (Mechel; Bies, Hansen & Howard; Cox & D'Antonio; Attenborough & Van Renterghem; Miki 1990; Maa 1998; Johnson et al. 1987) implemented clean-room from the cited sources. The measurement standards they connect to (ISO 9053-1/-2 for [flow resistivity](https://jmrplens.github.io/phonometry/materials/absorbers/airflow-resistance/), ISO 10534-1/-2 and ASTM E2611 for the [impedance tube](https://jmrplens.github.io/phonometry/materials/absorbers/impedance-tube/), ISO 354 / ISO 11654 for [random-incidence absorption and rating](https://jmrplens.github.io/phonometry/materials/absorbers/absorption-measurement/)) live in their own guides.