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 (its maximum over passive impedances is the published 0.951).

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AirLayer

AirLayer(thickness: float)

A plain air gap of thickness metres inside the stack.

delany_bazley

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 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 ("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.

DELANY_BAZLEY_COEFFICIENTS

Constant (dict).

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).

DELANY_BAZLEY_VALIDITY = (0.01, 1.0)

diffuse_field_absorption

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 (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.
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.
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.

DiffuseFieldAbsorptionResult

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()

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

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 of eps.
speed_of_sound	Speed of sound c in air, in m/s.

Returns: Resonance frequency f0, in hertz.

johnson_champoux_allard

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.

layered_absorber

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, 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, MicroperforatedPlateLayer, 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.

LayeredAbsorberResult

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()

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()

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

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

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

MembraneLayer(surface_density: float, resistance: float = 0.0)

A limp impervious membrane (see membrane_impedance).

microperforated_plate_impedance

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

MicroperforatedPlateLayer(
    thickness: float,
    hole_radius: float,
    open_area: float,
    end_correction: float = 0.85,
)

A microperforated plate (see microperforated_plate_impedance).

miki

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 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.

MIKI_VALIDITY

Constant (tuple).

MIKI_VALIDITY = (0.01, 1.0)

perforated_plate_impedance

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 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 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

PerforatedPlateLayer(
    thickness: float,
    hole_radius: float,
    open_area: float,
    end_correction: float | None = None,
)

A rigid perforated plate (see perforated_plate_impedance).

perforation_end_correction

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

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, 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

PorousLayer(thickness: float, medium: PorousMediumResult)

A porous layer of thickness metres described by medium.

medium is a PorousMediumResult (from delany_bazley, miki, johnson_champoux_allard, or built directly from measured Zc/k data) evaluated on the same frequency vector that is passed to layered_absorber.

PorousMediumResult

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()

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

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.