materials.absorbers.porous

Porous-material models and resonant sheet impedances.

Two complementary building blocks, all in the $e^{+j\omega t}$ time convention with the forward wave carried by $e^{-jkx}$ (so a passive medium has $\mathrm{Im}(k)<0$):

• 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 $X={\rho }_{0}f/\sigma$ (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 $Z_{\mathrm{c}}=\sqrt{{\rho }_{\mathrm{e}}{K}_{\mathrm{e}}}$ and $k=\omega \sqrt{{\rho }_{\mathrm{e}}/K_{\mathrm{e}}}$ hold for every model.
  • the limp-frame correction of any of the three rigid-frame models (Allard & Atalla, Propagation of Sound in Porous Media 2e, Sect. 11.3.4, Eqs. (11.53)-(11.55), printed pp. 251-253): a light frame is dragged along by the pore fluid, so its inertia has to be carried by the equivalent fluid. Only the effective density changes; the bulk modulus is the rigid-frame one. See limp_frame and decoupling_frequency.

• Resonant sheets: 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 $j\omega m$ (Cox & D’Antonio Eq. (7.14); Bies Eq. (D.96)). Each sheet is closed by the shallow-cavity resonance it is designed around, helmholtz_resonance_frequency for a perforate and membrane_resonance_frequency for a membrane.

The air all of them propagate through is described by Fluid, which carries the six quantities a visco-thermal model can need (speed of sound, density, viscosity, Prandtl number, ratio of specific heats and static pressure) with the values these models were published with. The narrow-channel models of slow_sound and metadiffuser take it as a single argument.

These are the elements a multilayer absorber is assembled from; declaring a stack of them and solving it with the transfer matrix is the subject of layered.

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airflow_resistivity_from_bulk_density(
    bulk_density_kg_m3: float,
    *,
    fit: FibreResistivityFit,
) -> float

Airflow resistivity of a mineral wool from its bulk density (Eq. 1.165).

$r=k_1{\rho }_{\mathrm{bulk}}^{1+k_2}/d_{\mathrm{fibre}}^2$, the empirical relation Hopkins gives after Bies (1988) and Nichols (1947), with the fibre diameter in micrometres.

The relation is a straight line through measured points on a log-log plot, not a law, so it belongs to the material the points came from. Passing a bulk density outside the range the fit was made over emits a PorousAbsorberWarning naming both.

Parameters

Returns: The airflow resistivity r, in Pa s/m2.

Raises

decoupling_frequency(
    flow_resistivity: float,
    *,
    porosity: float,
    frame_density: float,
) -> float

Zwikker-Kosten decoupling frequency Fd of a porous frame.

$F_{\mathrm{d}}=\sigma {\varphi }^2/(2\pi {\rho }_1)$ (Allard & Atalla 2e, Sect. 11.3.4, printed p. 251; the same closed form as their Eq. (6.90), printed p. 126). Above Fd the visco-inertial coupling between the pore fluid and the frame is too weak for the acoustic wave to shake the frame, so the rigid-frame equivalent fluid of johnson_champoux_allard applies; below it the frame moves and the limp correction of limp_frame matters.

Parameters

Returns: The decoupling frequency Fd, in hertz.

Raises

delany_bazley(
    frequency: ArrayLike,
    flow_resistivity: float,
    *,
    coefficients: str | tuple[float, ...] = 'delany_bazley',
    fluid: Fluid = ...,
) -> PorousMediumResult

Delany-Bazley one-parameter porous model (power laws in X).

$Z_{\mathrm{c}}=\rho c(1+C_1{X}^{-C_2}-j{C}_3{X}^{-C_4})$ and $k=(\omega /c)(1+C_5{X}^{-C_6}-j{C}_7{X}^{-C_8})$ 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

Returns: A PorousMediumResult.

Constant (mapping).

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)}

Constant (tuple).

DELANY_BAZLEY_VALIDITY = (0.01, 1.0)

fibre_characteristic_lengths(
    fibre_radius_m: float,
    *,
    bulk_density_kg_m3: float,
    fibre_density_kg_m3: float,
) -> FibreCharacteristicLengths

Both characteristic lengths of a fibrous layer, from its geometry.

Allard & Atalla Eqs. (5.29) and (5.30), PDF page 91 (printed p. 81), model the fibres as infinitely long cylinders of radius R and give, for a porosity close to 1,

where L is the total length of fibre per unit volume. Nobody measures L, so it is eliminated here through its own definition, which for cylinders is ${\rho }_{\mathrm{bulk}}={\rho }_{\mathrm{fibre}}\pi R^{2}L$, leaving $\Lambda =R{\rho }_{\mathrm{fibre}}/(2{\rho }_{\mathrm{bulk}})$ from three numbers a table does print. That substitution is arithmetic on the definition and not a second model.

This is a different model from viscous_characteristic_length, not a second opinion on the same one, and on Hopkins’ own rock wool the two drift apart with density: the cylinder model is the lower estimate throughout, by a factor of 1,4 at 38 kg/m3 and 1,9 at 155 kg/m3. Which to prefer is a question about the material, and neither is a substitute for measuring it.

Parameters

Returns: The viscous and thermal lengths, in metres.

Raises

FibreCharacteristicLengths(
    viscous_length_m: ForwardRef('float'),
    thermal_length_m: ForwardRef('float'),
)

The two characteristic lengths of a fibrous layer, in metres.

FibreResistivityFit(
    k1: ForwardRef('float'),
    k2: ForwardRef('float'),
    fibre_diameter_um: ForwardRef('float'),
    bulk_density_range_kg_m3: ForwardRef('tuple[float, float]'),
    direction: ForwardRef('str'),
    source: ForwardRef('str'),
)

A published k1, k2 pair of Hopkins Eq. (1.165), and its range.

The pair is not a property of mineral wool in general: k1 belongs to a material manufactured in a particular way and k2 to how its fibres are oriented, so the fit carries the fibre diameter it was made at and the bulk-density range it was fitted over, and using it outside that range is announced rather than silent.

Attributes

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).

$f_0=(c/2\pi )\sqrt{\epsilon /(t^{\prime }d)}$ with the end-corrected plug length $t^{\prime }=t+2\delta a$ (Cox & D’Antonio 3e, Eqs. (7.4)/(7.6), valid for $kd\ll 1$).

Parameters

Returns: Resonance frequency f0, in hertz.

johnson_champoux_allard(
    frequency: ArrayLike,
    flow_resistivity: float,
    *,
    porosity: float,
    tortuosity: float,
    viscous_length: float,
    thermal_length: float,
    fluid: Fluid = ...,
) -> PorousMediumResult

Johnson-Champoux-Allard five-parameter rigid-frame model.

Effective density (Cox & D’Antonio 3e, Eq. (6.19)):

and effective bulk modulus (Eq. (6.20)):

with tortuosity T, porosity phi, viscous/thermal characteristic lengths L / L'; then $Z_{\mathrm{c}}=\sqrt{K_{\mathrm{e}}{\rho }_{\mathrm{e}}}$ and $k=\omega \sqrt{{\rho }_{\mathrm{e}}/K_{\mathrm{e}}}$ (Eqs. (6.24)-(6.25)). Both quantities are surface-normalised (the $1/\varphi$ factors are included). The model has the exact limits $j\omega {\rho }_{\mathrm{e}}\to \sigma$ as $\omega \to 0$ and ${\rho }_{\mathrm{e}}\to (T\rho /\varphi )(1+(1-j){\delta }_v/L)$ as $\omega \to \infty$ (Johnson et al. 1987), pinned in the tests.

Parameters

Returns: A PorousMediumResult.

limp_frame(
    medium: PorousMediumResult,
    frame_density: float,
    *,
    porosity: float = 1.0,
) -> PorousMediumResult

Limp-frame correction of a rigid-frame equivalent fluid (A&A 11.3.4).

A light frame (aeronautic-grade fibreglass, felts, screens) is dragged along by the pore fluid instead of standing still, and the rigid-frame models of delany_bazley, miki and johnson_champoux_allard have no way to carry that inertia. Neglecting the stiffness of the frame altogether 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), printed pp. 252-253, after Panneton 2007):

with rho_eq the rigid-frame effective density of medium, rho0 the density of the pore fluid and ${\rho }_{\mathrm{t}}={\rho }_1+\varphi {\rho }_0$ the apparent total density of the material. What anchors this expression is the printed equation itself, transcribed term by term; Allard & Atalla tabulate no computed limp density anywhere, so there are no published digits to check against. The book also states two exact limits in prose, and both are verified, but they are weaker than they look: neither pins the ${\rho }_0^2$ and $2{\rho }_0$ terms, since a sign-flipped variant of Eq. (11.55) satisfies both of them (and even the $1/{\rho }_1$ decay of the heavy-frame residual). They corroborate the transcription rather than determine it:

• heavy frame: as ${\rho }_1\to \infty$ the correction vanishes and the rigid-frame result is recovered (the book’s own reading of Eq. (11.55));
• low frequency: since ${\rho }_{\mathrm{eq}}\to \sigma /(j\omega )$ as $\omega \to 0$ (Eq. (5.37)), ${\rho }_{\mathrm{limp}}\to {\rho }_{\mathrm{t}}$, a finite real density, where the rigid-frame model diverges. The rigid frame forbids rigid-body motion of the sample; the limp one allows it, which is why the two differ mainly at low frequency and why the limp model is the right one for an unconstrained sample in an impedance tube.

The corrected medium is a drop-in PorousMediumResult, so it can be handed to PorousLayer inside layered_absorber exactly like the rigid-frame one.

Use decoupling_frequency to see where the frame stops following the fluid and limp_frame_applicable for the published bulk-modulus rule of thumb on when the frame may be treated as limp at all.

Parameters

Returns: A PorousMediumResult with model "limp_frame(<base model>)".

Raises

limp_frame_applicable(
    frame_bulk_modulus: float,
    *,
    criterion: str = 'doutres',
    fluid_bulk_modulus: float = 101325.0,
) -> bool

Whether the limp-frame model may be used, by published rule of thumb.

Both published criteria compare the bulk modulus of the frame in vacuum K_c with that of the fluid in the pores K_f (Allard & Atalla 2e, printed pp. 253-254): Beranek (1947) requires $|K_c/K_{\mathrm{f}}|<0.05$, and the frame structural interaction study of Doutres et al. (2007) relaxes it to $|K_c/K_{\mathrm{f}}|<0.2$. With K_f taken as the isothermal bulk modulus of fluid, $P_0=101.3$ kPa, the relaxed criterion is the book’s statement that “the limp model is applicable for materials having a bulk modulus lower than 20 kPa”. Neither criterion accounts for boundary or mounting conditions, and the book notes that a thin light foam decoupled from a vibrating structure by an air gap behaves limply well above the limit.

Parameters

Returns: True when $|K_c/K_{\mathrm{f}}|$ does not exceed the threshold.

Raises

Constant (mapping).

LIMP_FRAME_CRITERIA = {'beranek': 0.05, 'doutres': 0.2}

membrane_impedance(
    frequency: ArrayLike,
    *,
    surface_density: float,
    resistance: float = 0.0,
) -> Complex

Transfer impedance of a limp impervious membrane.

$z=r+j\omega m$ - 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

Returns: Complex transfer impedance z, in Pa s/m.

membrane_resonance_frequency(
    *,
    surface_density: float,
    cavity_depth: float,
    isothermal: bool = False,
    fluid: Fluid = ...,
) -> float

Mass-spring resonance of a membrane over a shallow cavity.

$f_0=(1/2\pi )\sqrt{\rho c^2/(md)}$ for an adiabatic air spring - numerically the classical $f_0=60/\sqrt{md}$ (Cox & D’Antonio 3e, Eq. (7.9)). With isothermal=True the spring stiffness drops by gamma, giving $\sim 50/\sqrt{md}$ (Eq. (7.10)), the porous-filled cavity case below about 500 Hz.

Parameters

Returns: Resonance frequency f0, in hertz.

microperforated_plate_impedance(
    frequency: ArrayLike,
    *,
    thickness: float,
    hole_radius: float,
    open_area: float,
    end_correction: float = 0.85,
    fluid: Fluid = ...,
) -> 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):

with the perforate constant $x=a\sqrt{\rho \omega /\eta }$. Dividing by the open area and adding Maa’s Eq. (5) end corrections - the Rayleigh/Ingard surface resistance $\sqrt{2\omega \rho \eta }/(2\epsilon )$ and the piston end-correction reactance $j\omega \rho (2\delta a)/\epsilon$ ($0.85d$ total for the default $\delta =0.85$ per end) - gives the sheet transfer impedance (Cox & D’Antonio Eq. (7.35)).

Parameters

Returns: Complex transfer impedance z, in Pa s/m.

miki(
    frequency: ArrayLike,
    flow_resistivity: float,
    *,
    fluid: Fluid = ...,
) -> PorousMediumResult

Miki (1990) positive-real modification of the Delany-Bazley model.

In the variable $Y=f/\sigma$ (Miki 1990, Eqs. (30)-(34)): $Z_{\mathrm{c}}=\rho c(1+0.070Y^{-0.632}-j0.107Y^{-0.632})$ and, from the propagation constant $\gamma =\alpha +j\beta$ via $k=\beta -j\alpha$, $k=(\omega /c)(1+0.109Y^{-0.618}-j0.160Y^{-0.618})$. 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

Returns: A PorousMediumResult.

Constant (tuple).

MIKI_VALIDITY = (0.01, 1.0)

perforated_plate_impedance(
    frequency: ArrayLike,
    *,
    thickness: float,
    hole_radius: float,
    open_area: float,
    end_correction: float | None = None,
    fluid: Fluid = ...,
) -> 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)):

and visco-thermal surface resistance (Eq. (7.12)):

giving $z=r+j\omega m$ (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

Returns: Complex transfer impedance z, in Pa s/m.

perforation_end_correction(open_area: float) -> float

End-correction factor delta of a circular perforation.

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 $\delta \to 0.85$ for an isolated hole.

Parameters

Returns: End-correction factor delta (dimensionless, per end).

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

Returns: The axes.

porosity_from_bulk_density(
    bulk_density_kg_m3: float,
    *,
    fibre_density_kg_m3: float,
) -> float

Porosity of a fibrous material from its two densities (Eq. 1.160).

$\varphi =1-{\rho }_{\mathrm{bulk}}/{\rho }_{\mathrm{fibre}}$, which holds when the fibres are solid and whatever binds them together has negligible mass. Hopkins states both conditions.

It closes on its own data: at the 2 600 kg/m3 fibre density and the 31 to 155 kg/m3 bulk-density range Hopkins prints for his rock wool, this returns 0,99 and 0,94, which is the porosity range he prints beside them.

Parameters

Returns: The open porosity phi, between 0 and 1.

Raises

Advisory for porous-model use outside the published fit range.

PorousMediumResult(
    frequencies: 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 frequencies. 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 ($\mathrm{Im}(k)<0$ for the $e^{+j\omega t}$ convention), effective_density $=Z_{\mathrm{c}}k/\omega$ and bulk_modulus $=Z_{\mathrm{c}}\omega /k$ the surface-normalised equivalent-fluid density and bulk modulus, so that $Z_{\mathrm{c}}=\sqrt{{\rho }_{\mathrm{e}}{K}_{\mathrm{e}}}$ and $k=\omega \sqrt{{\rho }_{\mathrm{e}}/K_{\mathrm{e}}}$ for every model.

property

Characteristic impedance normalised by $\rho c$ of fluid.

property

Wavenumber normalised by the free-air wavenumber $k_0=\omega /c$.

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.

Constant (phonometry.fluids.Fluid).

Constant (float).

ROCK_WOOL_FIBRE_DENSITY_KG_M3 = 2600.0

Constant (phonometry.materials.absorbers.porous.FibreResistivityFit).

Constant (phonometry.materials.absorbers.porous.FibreResistivityFit).

viscous_characteristic_length(
    flow_resistivity_pa_s_m2: float,
    *,
    porosity: float,
    tortuosity: float,
    fluid: Fluid = ...,
    shape_factor: float = 1.0,
) -> float

The viscous characteristic length from the three measured parameters.

$\Lambda =(8\eta {\alpha }_{\infty }/(\sigma \varphi ){)}^{1/2}/c$, Allard & Atalla Eq. (5.25), PDF page 90 (printed p. 80), after Johnson et al. (1986), with c close to 1.

How close is the question the shape factor exists for, and it is worth saying what “close” buys. Against the twenty-three specimens Allard & Atalla print with all four columns, c = 1 puts $\Lambda$ within a factor of two of the printed length for eighteen of them, with the middle of the set near 1,2. The five it misses are the two carpets, the woven screen and two rows whose lengths the book itself took from a different model, and there the same arithmetic wants a c between 2 and 9,5.

So estimating this length rather than measuring it is worth a factor of two on a bulk fibrous or open-cell absorber, and nothing at all on a floor covering or a screen.

Parameters

Returns: The viscous characteristic length Lambda, in metres.

Raises