Source: https://jmrplens.github.io/phonometry/reference/api/materials/porous-absorber/ Porous-material models and multilayer absorber prediction. Three complementary building blocks, all in the `e^{+j w t}` time convention with the forward wave carried by `e^{-j k x}` (so a passive medium has `Im(k) < 0`): * **Equivalent-fluid models** for the characteristic impedance `Zc` and the complex wavenumber `k` of a rigid-frame porous material: - the one-parameter **Delany-Bazley** power law in the absorber variable `X = rho0 f / sigma` (Mechel, *Formulas of Acoustics* 2e, Sect. G.11 Eqs. (1)-(2); Bies, Hansen & Howard, *Engineering Noise Control* 5e, Appendix D Eqs. (D.22)-(D.23) and Table D.1; Hopkins, *Sound Insulation*, Eqs. (1.171)-(1.174)), stated valid for `0.01 < X < 1.0` and porosity close to one. Table D.1 also provides coefficient sets fitted to polyester (Garai & Pompoli 2005) and to foams (Dunn & Davern 1986, Wu 1988), exposed here as presets. - the **Miki** modification, regressed on the same Delany-Bazley data under a positive-real (passivity) constraint so the model stays well behaved below the fit range (Miki 1990, *J. Acoust. Soc. Jpn (E)* 11(1), Eqs. (30)-(34), in the variable `f / sigma`). - the five-parameter **Johnson-Champoux-Allard (JCA)** semi-phenomenological model with flow resistivity, porosity, tortuosity and the viscous/thermal characteristic lengths (Cox & D'Antonio, *Acoustic Absorbers and Diffusers* 3e, Eqs. (6.19)-(6.25); Attenborough & Van Renterghem, *Predicting Outdoor Sound* 2e, Eqs. (5.13)-(5.14)). The returned equivalent-fluid density and bulk modulus are the surface-normalised quantities (they absorb the porosity), so `Zc = sqrt(rho_e K_e)` and `k = w sqrt(rho_e / K_e)` hold for every model. * **Transfer-matrix multilayer prediction**: each fluid layer contributes `[[cos(kx d), j Zx sin(kx d)], [j sin(kx d)/Zx, cos(kx d)]]` with the in-depth wavenumber `kx = sqrt(k^2 - k0^2 sin^2 theta)` from Snell's law and `Zx = Zc k / kx` (Cox & D'Antonio Eqs. (2.29)-(2.32); Bies Eq. (D.83); equivalent to the layer-recursion of Bies Eq. (D.95) and Mechel Sect. D.4). Thin resonant sheets (perforated plate, microperforated plate, limp membrane) enter as series transfer impedances `[[1, z],[0, 1]]`. The stack is closed by a rigid wall, by free air or by an arbitrary termination impedance, giving the surface impedance, the oblique reflection factor and `alpha(theta)`. This same layer transfer matrix underlies the critically-coupled perfect-absorber designs of Jiménez, Groby, Pagneux & Romero-García (2017, *Applied Sciences* 7(6), 618, doi:10.3390/app7060618) and, for a rigidly-backed high-porosity layer, Jiménez, Romero-García & Groby (2018, *Acta Acustica united with Acustica* 104(3), 396-409, doi:10.3813/AAA.919183), where the critical-coupling condition on the surface impedance yields total single-frequency absorption. * **Resonant sheets and random incidence**: the perforated-plate impedance uses the end-corrected air-plug mass and the visco-thermal surface resistance (Cox & D'Antonio Eqs. (7.6)/(7.12)/(7.21), end-correction variants of Table 7.1); the microperforated plate follows Maa's exact short-tube impedance (Maa 1998, *J. Acoust. Soc. Am.* 104(5), Eq. (2), with the Eq. (5) end corrections; reproduced as Cox & D'Antonio Eqs. (7.33)-(7.35) and built on the same Bessel kernel as Mechel Sect. G.3); the membrane is the limp surface mass `j w m` (Cox & D'Antonio Eq. (7.14); Bies Eq. (D.96)). The random-incidence (Paris) integral follows Mechel Sect. D.5 Eqs. (9)-(10), with the closed form for locally reacting surfaces implemented in [`statistical_absorption`](/phonometry/reference/api/materials/porous-absorber/#statistical_absorption) (its maximum over passive impedances is the published 0.951). > Auto-generated from the source docstrings by `scripts/generate_api_docs.py` (`make api-docs`). Do not edit by hand. ## AirLayer ```python AirLayer(thickness: float) ``` A plain air gap of `thickness` metres inside the stack. ## delany_bazley ```python delany_bazley( frequency: ArrayLike, flow_resistivity: float, *, coefficients: str | tuple[float, ...] = 'delany_bazley', speed_of_sound: float = 343.0, air_density: float = 1.205, ) -> PorousMediumResult ``` Delany-Bazley one-parameter porous model (power laws in `X`). `Zc = rho c (1 + C1 X^-C2 - j C3 X^-C4)` and `k = (w/c)(1 + C5 X^-C6 - j C7 X^-C8)` with `X = rho f / sigma` (Mechel 2e Sect. G.11 Eqs. (1)-(2); Bies 5e Eqs. (D.22)-(D.23) with the Table D.1 coefficients; Hopkins Eqs. (1.171)-(1.173)). A [`PorousAbsorberWarning`](/phonometry/reference/api/materials/porous-absorber/#porousabsorberwarning) is raised when any `X` leaves the stated `0.01 < X < 1.0` validity range (Hopkins Eq. (1.174)); the values are still returned. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `flow_resistivity` | Airflow resistivity `sigma`, in Pa s/m2. | | `coefficients` | Preset name from [`DELANY_BAZLEY_COEFFICIENTS`](/phonometry/reference/api/materials/porous-absorber/#delany_bazley_coefficients) (`"delany_bazley"` rockwool/fibreglass default, `"garai_pompoli"` polyester, `"dunn_davern"` / `"wu"` foams) or an explicit `(C1..C8)` tuple. | | `speed_of_sound` | Speed of sound `c` in air, in m/s. | | `air_density` | Air density `rho`, in kg/m3. | **Returns:** A [`PorousMediumResult`](/phonometry/reference/api/materials/porous-absorber/#porousmediumresult). ## DELANY_BAZLEY_COEFFICIENTS *Constant* (`dict`). ```python DELANY_BAZLEY_COEFFICIENTS = {'delany_bazley': (0.0571, 0.754, 0.087, 0.732, 0.0978, 0.7, 0.189, 0.595), 'garai_pompoli': (0.078, 0.623, 0.074, 0.66, 0.159, 0.571, 0.121, 0.53), 'dunn_davern': (0.114, 0.369, 0.0985, 0.758, 0.168, 0.715, 0.136, 0.491), 'wu': (0.212, 0.455, 0.105, 0.607, 0.163, 0.592, 0.188, 0.544)} ``` ## DELANY_BAZLEY_VALIDITY *Constant* (`tuple`). ```python DELANY_BAZLEY_VALIDITY = (0.01, 1.0) ``` ## diffuse_field_absorption ```python diffuse_field_absorption( frequency: ArrayLike, layers: list[Layer] | tuple[Layer, ...], *, angle_limit: float = 1.5707963267948966, quadrature_points: int = 64, termination: str | complex | ArrayLike = 'rigid', speed_of_sound: float = 343.0, air_density: float = 1.205, viscosity: float = 1.84e-05, ) -> DiffuseFieldAbsorptionResult ``` Random-incidence absorption by the Paris integral (Mechel Sect. D.5). `alpha_dif = (2 / sin^2 theta_lim) * int_0^theta_lim alpha(theta) cos(theta) sin(theta) d(theta)` (Mechel 2e Sect. D.5 Eq. (9)), evaluated with fixed-order Gauss-Legendre quadrature over the bulk-reacting `alpha(theta)` of [`layered_absorber`](/phonometry/reference/api/materials/porous-absorber/#layered_absorber) (Sect. D.6 notes the bulk integral generally must be evaluated numerically). Some references truncate the integral at 75-87 degrees instead of 90 (Sect. D.5); set `angle_limit` accordingly. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `layers` | Layer stack, as in [`layered_absorber`](/phonometry/reference/api/materials/porous-absorber/#layered_absorber). | | `angle_limit` | Upper integration angle `theta_lim`, in radians (0 \< theta_lim \<= pi/2; default pi/2). | | `quadrature_points` | Gauss-Legendre order (default 64). | | `termination` | As in [`layered_absorber`](/phonometry/reference/api/materials/porous-absorber/#layered_absorber). | | `speed_of_sound` | Speed of sound `c` in air, in m/s. | | `air_density` | Air density `rho`, in kg/m3. | | `viscosity` | Dynamic viscosity of air, in Pa s. | **Returns:** A [`DiffuseFieldAbsorptionResult`](/phonometry/reference/api/materials/porous-absorber/#diffusefieldabsorptionresult). ## DiffuseFieldAbsorptionResult ```python DiffuseFieldAbsorptionResult( frequency: Real, absorption: Real, angle_limit: float, ) ``` Random-incidence (Paris-integral) absorption of a layered absorber. `absorption` is `alpha_dif(f)` from Mechel 2e Sect. D.5 Eq. (9): the plane-wave `alpha(theta)` weighted by `cos(theta) sin(theta)` and normalised by `sin^2(theta_limit)`. ### DiffuseFieldAbsorptionResult.plot() ```python DiffuseFieldAbsorptionResult.plot( ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Plot the random-incidence absorption spectrum `alpha_dif(f)`. Requires matplotlib (`pip install phonometry[plot]`); returns the `Axes`. ## helmholtz_resonance_frequency ```python helmholtz_resonance_frequency( *, cavity_depth: float, plate_thickness: float, hole_radius: float, open_area: float, end_correction: float | None = None, speed_of_sound: float = 343.0, ) -> float ``` Resonance of a perforated sheet over a shallow cavity (closed form). `f0 = (c / 2 pi) sqrt(eps / (t' d))` with the end-corrected plug length `t' = t + 2 delta a` (Cox & D'Antonio 3e, Eqs. (7.4)/(7.6), valid for `k d << 1`). **Parameters** | Name | Description | | :--- | :--- | | `cavity_depth` | Cavity depth `d`, in metres. | | `plate_thickness` | Plate thickness `t`, in metres. | | `hole_radius` | Hole radius `a`, in metres. | | `open_area` | Fractional open area `eps` (0..1). | | `end_correction` | End-correction factor `delta` per end; default [`perforation_end_correction`](/phonometry/reference/api/materials/porous-absorber/#perforation_end_correction) of `eps`. | | `speed_of_sound` | Speed of sound `c` in air, in m/s. | **Returns:** Resonance frequency `f0`, in hertz. ## johnson_champoux_allard ```python johnson_champoux_allard( frequency: ArrayLike, flow_resistivity: float, *, porosity: float, tortuosity: float, viscous_length: float, thermal_length: float, speed_of_sound: float = 343.0, air_density: float = 1.205, viscosity: float = 1.84e-05, prandtl_number: float = 0.71, heat_capacity_ratio: float = 1.4, atmospheric_pressure: float = 101325.0, ) -> PorousMediumResult ``` Johnson-Champoux-Allard five-parameter rigid-frame model. Effective density (Cox & D'Antonio 3e, Eq. (6.19)): `rho_e = (T rho / phi) [1 + (sigma phi / (j w rho T)) sqrt(1 + 4 j T^2 eta rho w / (sigma^2 L^2 phi^2))]` and effective bulk modulus (Eq. (6.20)): `K_e = (gamma P0 / phi) / (gamma - (gamma - 1) [1 + (8 eta / (j L'^2 Pr w rho)) sqrt(1 + j rho w Pr L'^2 / (16 eta))]^-1)` with tortuosity `T`, porosity `phi`, viscous/thermal characteristic lengths `L` / `L'`; then `Zc = sqrt(K_e rho_e)` and `k = w sqrt(rho_e / K_e)` (Eqs. (6.24)-(6.25)). Both quantities are surface-normalised (the `1/phi` factors are included). The model has the exact limits `j w rho_e -> sigma` as `w -> 0` and `rho_e -> (T rho / phi)(1 + (1 - j) delta_v / L)` as `w -> inf` (Johnson et al. 1987), pinned in the tests. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `flow_resistivity` | Airflow resistivity `sigma`, in Pa s/m2. | | `porosity` | Open porosity `phi` (0 \< phi \<= 1). | | `tortuosity` | High-frequency tortuosity `T = alpha_inf` (>= 1). | | `viscous_length` | Viscous characteristic length `L`, in metres. | | `thermal_length` | Thermal characteristic length `L'`, in metres (physically `L' >= L`). | | `speed_of_sound` | Speed of sound `c` in air, in m/s. | | `air_density` | Air density `rho`, in kg/m3. | | `viscosity` | Dynamic viscosity `eta` of air, in Pa s. | | `prandtl_number` | Prandtl number `Pr` of air. | | `heat_capacity_ratio` | Ratio of specific heats `gamma`. | | `atmospheric_pressure` | Static pressure `P0`, in Pa. | **Returns:** A [`PorousMediumResult`](/phonometry/reference/api/materials/porous-absorber/#porousmediumresult). ## layered_absorber ```python layered_absorber( frequency: ArrayLike, layers: list[Layer] | tuple[Layer, ...], *, angle: float = 0.0, termination: str | complex | ArrayLike = 'rigid', speed_of_sound: float = 343.0, air_density: float = 1.205, viscosity: float = 1.84e-05, ) -> LayeredAbsorberResult ``` Transfer-matrix prediction of a layered absorber at one angle. The *layers* list is ordered from the sound-incidence side towards the *termination*. Fluid layers ([`AirLayer`](/phonometry/reference/api/materials/porous-absorber/#airlayer), [`PorousLayer`](/phonometry/reference/api/materials/porous-absorber/#porouslayer)) contribute the oblique chain matrix of Cox & D'Antonio 3e Eq. (2.29) (equivalently the impedance recursion of Bies 5e Eq. (D.95) and the scheme of Mechel 2e Sect. D.4); sheet layers ([`PerforatedPlateLayer`](/phonometry/reference/api/materials/porous-absorber/#perforatedplatelayer), [`MicroperforatedPlateLayer`](/phonometry/reference/api/materials/porous-absorber/#microperforatedplatelayer), [`MembraneLayer`](/phonometry/reference/api/materials/porous-absorber/#membranelayer)) enter as locally reacting series impedances. The chain is closed by a rigid wall (`termination="rigid"`), by radiation into free air behind (`termination="free"`, `Z_L = rho c / cos(theta)`) or by an arbitrary complex impedance. The reflection factor is `R = (Zs cos(theta) - rho c) / (Zs cos(theta) + rho c)` and `alpha = 1 - |R|^2` (Mechel 2e Sect. D.3 Eq. (2)). `Zs`, `R` and `alpha` are evaluated with the numerically robust admittance recursion (algebraically identical to the chain product but immune to the `e^{|Im(kx)| d}` overflow of the raw matrix entries for extremely attenuating layers); the raw chain matrix is still returned in `transfer_matrix` and may overflow in such extreme cases. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `layers` | Layer stack from the incidence side to the termination. | | `angle` | Polar angle of incidence `theta`, in radians (`0 <= theta < pi/2 - 1e-6`; grazing incidence is excluded). | | `termination` | `"rigid"` (default), `"free"`, or a non-zero complex impedance (scalar or per-frequency array), in Pa s/m. | | `speed_of_sound` | Speed of sound `c` in air, in m/s. | | `air_density` | Air density `rho`, in kg/m3. | | `viscosity` | Dynamic viscosity of air, in Pa s (sheet layers). | **Returns:** A [`LayeredAbsorberResult`](/phonometry/reference/api/materials/porous-absorber/#layeredabsorberresult). ## LayeredAbsorberResult ```python LayeredAbsorberResult( frequency: Real, angle: float, surface_impedance: Complex, normalized_impedance: Complex, reflection: Complex, absorption: Real, transfer_matrix: Complex, layers: tuple[Layer, ...] | None = None, ) ``` Oblique-incidence prediction of a layered absorber. All arrays share the shape of `frequency`. `surface_impedance` is the specific impedance `Zs = p / u_n` at the front face (may be `inf` for a lossless-sheet stack over a rigid wall), `reflection` the complex plane-wave reflection factor `R(theta)`, `absorption` the coefficient `alpha(theta) = 1 - |R|^2` and `transfer_matrix` the total chain matrix with shape `(2, 2, len(frequency))` (unimodular: every layer is reciprocal). `layers` retains the layer sequence the stack was solved with (front layer first) so `plot_geometry` can draw the cross-section; it is appended after the original fields and defaults to `None` for hand-built results. ### LayeredAbsorberResult.plot() ```python LayeredAbsorberResult.plot( ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Plot the absorption spectrum `alpha(f)` with `|R|` overlaid. Requires matplotlib (`pip install phonometry[plot]`); returns the `Axes`. ### LayeredAbsorberResult.plot_geometry() ```python LayeredAbsorberResult.plot_geometry( ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Draw the solved stack cross-section to scale (dimensioned). Requires matplotlib (`pip install phonometry[plot]`); returns the `Axes`. **Raises** | Exception | When | | :--- | :--- | | ValueError | If the result does not retain its `layers`. | ## membrane_impedance ```python membrane_impedance( frequency: ArrayLike, *, surface_density: float, resistance: float = 0.0, ) -> Complex ``` Transfer impedance of a limp impervious membrane. `z = r + j w m` - the surface-mass reactance (Cox & D'Antonio 3e, Eq. (7.14); Bies 5e Eq. (D.96)) plus an optional empirical resistance for the internal/fixing losses. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `surface_density` | Mass per unit area `m`, in kg/m2. | | `resistance` | Series flow resistance `r`, in Pa s/m (default 0). | **Returns:** Complex transfer impedance `z`, in Pa s/m. ## membrane_resonance_frequency ```python membrane_resonance_frequency( *, surface_density: float, cavity_depth: float, isothermal: bool = False, speed_of_sound: float = 343.0, air_density: float = 1.205, ) -> float ``` Mass-spring resonance of a membrane over a shallow cavity. `f0 = (1 / 2 pi) sqrt(rho c^2 / (m d))` for an adiabatic air spring - numerically the classical `f0 = 60 / sqrt(m d)` (Cox & D'Antonio 3e, Eq. (7.9)). With `isothermal=True` the spring stiffness drops by `gamma`, giving `~50 / sqrt(m d)` (Eq. (7.10)), the porous-filled cavity case below about 500 Hz. **Parameters** | Name | Description | | :--- | :--- | | `surface_density` | Membrane mass per unit area `m`, in kg/m2. | | `cavity_depth` | Cavity depth `d`, in metres. | | `isothermal` | Use the isothermal air-spring stiffness. | | `speed_of_sound` | Speed of sound `c` in air, in m/s. | | `air_density` | Air density `rho`, in kg/m3. | **Returns:** Resonance frequency `f0`, in hertz. ## MembraneLayer ```python MembraneLayer(surface_density: float, resistance: float = 0.0) ``` A limp impervious membrane (see [`membrane_impedance`](/phonometry/reference/api/materials/porous-absorber/#membrane_impedance)). ## microperforated_plate_impedance ```python microperforated_plate_impedance( frequency: ArrayLike, *, thickness: float, hole_radius: float, open_area: float, end_correction: float = 0.85, air_density: float = 1.205, viscosity: float = 1.84e-05, ) -> Complex ``` Transfer impedance of a microperforated plate (Maa's exact model). The specific impedance of one submillimetre hole is the exact short-tube result (Maa 1998, Eq. (2); reproduced as Cox & D'Antonio 3e Eq. (7.33) and the same Bessel kernel as Mechel 2e Sect. G.3): `z1 = j w rho t [1 - (2 / (x sqrt(-j))) J1(x sqrt(-j)) / J0(x sqrt(-j))]^-1` with the perforate constant `x = a sqrt(rho w / eta)`. Dividing by the open area and adding Maa's Eq. (5) end corrections - the Rayleigh/Ingard surface resistance `sqrt(2 w rho eta) / (2 eps)` and the piston end-correction reactance `j w rho (2 delta a) / eps` (`0.85 d` total for the default `delta = 0.85` per end) - gives the sheet transfer impedance (Cox & D'Antonio Eq. (7.35)). **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `thickness` | Plate thickness `t`, in metres. | | `hole_radius` | Hole radius `a`, in metres (submillimetre for a genuine microperforated design). | | `open_area` | Fractional open area `eps` (0..1). | | `end_correction` | End-correction factor `delta` per end (default 0.85, the isolated-orifice value used by Maa). | | `air_density` | Air density `rho`, in kg/m3. | | `viscosity` | Dynamic viscosity `eta` of air, in Pa s. | **Returns:** Complex transfer impedance `z`, in Pa s/m. ## MicroperforatedPlateLayer ```python MicroperforatedPlateLayer( thickness: float, hole_radius: float, open_area: float, end_correction: float = 0.85, ) ``` A microperforated plate (see [`microperforated_plate_impedance`](/phonometry/reference/api/materials/porous-absorber/#microperforated_plate_impedance)). ## miki ```python miki( frequency: ArrayLike, flow_resistivity: float, *, speed_of_sound: float = 343.0, air_density: float = 1.205, ) -> PorousMediumResult ``` Miki (1990) positive-real modification of the Delany-Bazley model. In the variable `Y = f / sigma` (Miki 1990, Eqs. (30)-(34)): `Zc = rho c (1 + 0.070 Y^-0.632 - j 0.107 Y^-0.632)` and, from the propagation constant `gamma = alpha + j beta` via `k = beta - j alpha`, `k = (w/c)(1 + 0.109 Y^-0.618 - j 0.160 Y^-0.618)`. The regression was constrained to be positive real, so the surface impedance of a hard-backed layer keeps a non-negative real part even below the Delany-Bazley range; a [`PorousAbsorberWarning`](/phonometry/reference/api/materials/porous-absorber/#porousabsorberwarning) still flags `Y` outside the fit range `0.01 < f/sigma < 1.0` (paper Sect. 4.1). **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `flow_resistivity` | Airflow resistivity `sigma`, in Pa s/m2. | | `speed_of_sound` | Speed of sound `c` in air, in m/s. | | `air_density` | Air density `rho`, in kg/m3. | **Returns:** A [`PorousMediumResult`](/phonometry/reference/api/materials/porous-absorber/#porousmediumresult). ## MIKI_VALIDITY *Constant* (`tuple`). ```python MIKI_VALIDITY = (0.01, 1.0) ``` ## perforated_plate_impedance ```python perforated_plate_impedance( frequency: ArrayLike, *, thickness: float, hole_radius: float, open_area: float, end_correction: float | None = None, air_density: float = 1.205, viscosity: float = 1.84e-05, ) -> Complex ``` Transfer impedance of a rigid perforated plate with circular holes. Acoustic mass with both end corrections and the boundary-layer term (Cox & D'Antonio 3e, Eq. (7.6)): `m = (rho/eps)[t + 2 delta a + sqrt(8 nu / w)(1 + t/(2a))]` and visco-thermal surface resistance (Eq. (7.12)): `r = (rho/eps) sqrt(8 nu w) (1 + t/(2a))`, giving `z = r + j w m` (the series impedance added on top of the backing, Eq. (7.21)). Assumes hole radii well above the boundary-layer thickness; use [`microperforated_plate_impedance`](/phonometry/reference/api/materials/porous-absorber/#microperforated_plate_impedance) for submillimetre holes. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `thickness` | Plate thickness `t`, in metres. | | `hole_radius` | Hole radius `a`, in metres. | | `open_area` | Fractional open area `eps` (0..1). | | `end_correction` | End-correction factor `delta` per end; default [`perforation_end_correction`](/phonometry/reference/api/materials/porous-absorber/#perforation_end_correction) of `eps`. | | `air_density` | Air density `rho`, in kg/m3. | | `viscosity` | Dynamic viscosity `eta` of air, in Pa s. | **Returns:** Complex transfer impedance `z`, in Pa s/m. ## PerforatedPlateLayer ```python PerforatedPlateLayer( thickness: float, hole_radius: float, open_area: float, end_correction: float | None = None, ) ``` A rigid perforated plate (see [`perforated_plate_impedance`](/phonometry/reference/api/materials/porous-absorber/#perforated_plate_impedance)). ## perforation_end_correction ```python perforation_end_correction(open_area: float) -> float ``` End-correction factor `delta` of a circular perforation. `delta = 0.85 (1 - 1.47 eps^1/2 + 0.47 eps^3/2)` - the Fok-function interaction correction for circular holes (Cox & D'Antonio 3e, Table 7.1, Nesterov row; no open-area limit). Each orifice end adds `delta a` of air-plug length, and `delta -> 0.85` for an isolated hole. **Parameters** | Name | Description | | :--- | :--- | | `open_area` | Fractional open area `eps` of the sheet (0..1). | **Returns:** End-correction factor `delta` (dimensionless, per end). ## plot_absorber_stack ```python plot_absorber_stack( layers: Sequence[Layer] | Layer, ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Draw a layered-absorber cross-section to scale, rigid backing at right. Sound arrives from the left; each layer is drawn with its material fill and its thickness dimensioned below the stack. A membrane (no physical depth) is drawn as a thin sheet. **Parameters** | Name | Description | | :--- | :--- | | `layers` | The layer sequence of [`layered_absorber`](/phonometry/reference/api/materials/porous-absorber/#layered_absorber), front layer first, or a single layer. | | `ax` | Existing axes, or `None` to create a figure. | | `language` | Label language, `"en"` (default) or `"es"`. | | `kwargs` | Forwarded to the front-layer rectangle. | **Returns:** The axes. ## PorousAbsorberWarning Advisory for porous-model use outside the published fit range. ## PorousLayer ```python PorousLayer(thickness: float, medium: PorousMediumResult) ``` A porous layer of `thickness` metres described by *medium*. `medium` is a [`PorousMediumResult`](/phonometry/reference/api/materials/porous-absorber/#porousmediumresult) (from [`delany_bazley`](/phonometry/reference/api/materials/porous-absorber/#delany_bazley), [`miki`](/phonometry/reference/api/materials/porous-absorber/#miki), [`johnson_champoux_allard`](/phonometry/reference/api/materials/porous-absorber/#johnson_champoux_allard), or built directly from measured `Zc`/`k` data) evaluated on the same frequency vector that is passed to [`layered_absorber`](/phonometry/reference/api/materials/porous-absorber/#layered_absorber). ## PorousMediumResult ```python PorousMediumResult( frequency: Real, characteristic_impedance: Complex, wavenumber: Complex, effective_density: Complex, bulk_modulus: Complex, model: str, flow_resistivity: float, speed_of_sound: float, air_density: float, ) ``` Equivalent-fluid characterisation of a porous material. All arrays share the shape of `frequency`. `characteristic_impedance` is the complex characteristic impedance `Zc` in Pa s/m as seen from the material surface, `wavenumber` the complex wavenumber `k` in rad/m (`Im(k) < 0` for the `e^{+j w t}` convention), `effective_density = Zc k / w` and `bulk_modulus = Zc w / k` the surface-normalised equivalent-fluid density and bulk modulus, so that `Zc = sqrt(rho_e K_e)` and `k = w sqrt(rho_e / K_e)` for every model. ### PorousMediumResult.normalized_impedance *property* Characteristic impedance normalised by `rho c` of air. ### PorousMediumResult.normalized_wavenumber *property* Wavenumber normalised by the free-air wavenumber `k0 = w / c`. ### PorousMediumResult.plot() ```python PorousMediumResult.plot( ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Plot the normalised `Zc` and `k` components against frequency. Requires matplotlib (`pip install phonometry[plot]`); returns the `Axes`. ## statistical_absorption ```python statistical_absorption( normalized_impedance: ArrayLike, *, angle_limit: float = 1.5707963267948966, ) -> Real ``` Closed-form Paris integral for a locally reacting plane. With the normalised surface admittance `Z0 G = g1 + j g2 = 1/z` (Mechel 2e Sect. D.5 Eq. (10)): `alpha_dif = (8 g1 / sin^2 T) [1 - cos T + ((g1^2 - g2^2)/g2)(arctan((1 + g1)/g2) - arctan((g1 + cos T)/g2)) + g1 ln((g1^2 + g2^2 + 2 g1 cos T + cos^2 T)/(1 + g1^2 + g2^2 + 2 g1))]` reducing for `T = pi/2` to Eq. (4) and, for real admittance, to the printed `g2 = 0` special case. The maximum over passive impedances is 0.951 (the published bound for locally reacting absorbers, Sect. D.5). **Parameters** | Name | Description | | :--- | :--- | | `normalized_impedance` | Normalised surface impedance `z = Zs / (rho c)` (complex scalar or array), with `Re(z) > 0`. | | `angle_limit` | Upper integration angle `theta_lim`, in radians (0 \< theta_lim \<= pi/2; default pi/2). | **Returns:** Statistical absorption coefficient `alpha_dif`.