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 $Z_c=\sqrt{{\rho }_e{K}_e}$ and $k=\omega \sqrt{{\rho }_e/K_e}$ 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 AirProperties, 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.

Auto-generated from the source docstrings by scripts/generate_api_docs.py (make api-docs). Do not edit by hand.

AirProperties

AirProperties(
    speed_of_sound: float = 343.0,
    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,
)

State of the air the visco-thermal models propagate through.

The six quantities a narrow-channel model needs: the speed of sound c0 in m/s, the density ${\rho }_0$ in kg/m3, the dynamic viscosity $\eta$ in Pa s, the Prandtl number Pr, the ratio of specific heats $\gamma$ and the static pressure $P_0$ in Pa (the adiabatic bulk modulus is ${\kappa }_0=\gamma P_0$). The defaults are dry air at 20 degC, the values the models were published with.

Every field is validated on construction, so an impossible air state is rejected once, where it is written, rather than at each model that reads it.

Raises

Exception	When
ValueError	If any quantity is not positive and finite.

decoupling_frequency

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

Zwikker-Kosten decoupling frequency Fd of a porous frame.

$F_d=\sigma {\varphi }^2/(2\pi {\rho }_1)$ (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

Name	Description
flow_resistivity	Airflow resistivity sigma, in Pa s/m2 (> 0).
porosity	Open porosity phi (0 < phi <= 1).
frame_density	Bulk density of the frame rho1, in kg/m3 (> 0): the mass of solid per unit volume of material, i.e. the density of the sample as weighed, not the density of the material the fibres are made of.

Returns: The decoupling frequency Fd, in hertz.

Raises

Exception	When
ValueError	for a non-positive input or a porosity above 1.

DEFAULT_AIR

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

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).

$Z_c=\rho c(1+C_1{X}^{-C_2}-j{C}_3{X}^{-C_4})$ 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

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)

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).

$f_0=(c/2\pi )\sqrt{\epsilon /(t^{\prime }d)}$ 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

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)):

$${\rho }_e=\frac{T\rho }{\varphi }\left[1+\frac{\sigma \varphi }{j\omega \rho T}\sqrt{1+\frac{4j{T}^2\eta \rho \omega }{{\sigma }^2{L}^2{\varphi }^2}}\right]$$

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

$$K_e=\frac{\gamma P_0/\varphi }{\gamma -(\gamma -1){\left[1+\frac{8\eta }{j{L^{\prime }}^2\mathrm{Pr}\omega \rho }\sqrt{1+\frac{j\rho \omega \mathrm{Pr}{L^{\prime }}^2}{16\eta }}\right]}^{-1}}$$

with tortuosity T, porosity phi, viscous/thermal characteristic lengths L / L'; then $Z_c=\sqrt{K_e{\rho }_e}$ and $k=\omega \sqrt{{\rho }_e/K_e}$ (Eqs. (6.24)-(6.25)). Both quantities are surface-normalised (the $1/\varphi$ factors are included). The model has the exact limits $j\omega {\rho }_e\to \sigma$ as $\omega \to 0$ and ${\rho }_e\to (T\rho /\varphi )(1+(1-j){\delta }_v/L)$ as $\omega \to \infty$ (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 $T={\alpha }_{\infty }$ (>= 1).
viscous_length	Viscous characteristic length L, in metres.
thermal_length	Thermal characteristic length L', in metres (physically $L^{\prime }\ge L$).
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.

limp_frame

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):

$${\rho }_{\mathrm{limp}}=\frac{{\rho }_t{\rho }_{\mathrm{eq}}-{\rho }_0^2}{{\rho }_t+{\rho }_{\mathrm{eq}}-2{\rho }_0}$$

with rho_eq the rigid-frame effective density of medium, rho0 the density of the pore fluid and ${\rho }_t={\rho }_1+\varphi {\rho }_0$ 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)), ${\rho }_{\mathrm{limp}}\to {\rho }_t$, 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

Name	Description
medium	A rigid-frame PorousMediumResult (its effective_density is rho_eq and its bulk_modulus is kept).
frame_density	Bulk density of the frame rho1, in kg/m3 (> 0).
porosity	Open porosity phi (0 < phi <= 1, Default: 1,0, the high-porosity assumption of the one-parameter models).

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

Raises

Exception	When
ValueError	for a non-positive input or a porosity above 1.

limp_frame_applicable

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 $|K_c/K_f|<0.05$, and the frame structural interaction study of Doutres et al. (2007) relaxes it to $|K_c/K_f|<0.2$. With K_f taken as the isothermal bulk modulus of air, $P_0=101.3$ 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

Name	Description
frame_bulk_modulus	Bulk modulus of the frame in vacuum K_c, in Pa (>= 0; pass abs(K_c) for a complex modulus).
criterion	Key into LIMP_FRAME_CRITERIA, "doutres" (Default, 0,2) or "beranek" (0,05).
fluid_bulk_modulus	Bulk modulus of the pore fluid K_f, in Pa (Default: 101 325, the isothermal value for air).

Returns: True when $|K_c/K_f|$ does not exceed the threshold.

Raises

Exception	When
ValueError	for a negative modulus or an unknown criterion.

LIMP_FRAME_CRITERIA

Constant (dict).

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

membrane_impedance

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

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.

$f_0=(1/2\pi )\sqrt{\rho c^2/(md)}$ 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

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.

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):

$$z_1=j\omega \rho t{\left[1-\frac{2}{x\sqrt{-j}}\frac{J_1(x\sqrt{-j})}{J_0(x\sqrt{-j})}\right]}^{-1}$$

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

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.

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)): $Z_c=\rho c(1+0.070Y^{-0.632}-j0.107Y^{-0.632})$ 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

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)):

$$m=\frac{\rho }{\epsilon }\left[t+2\delta a+\sqrt{\frac{8\nu }{\omega }}\left(1+\frac{t}{2a}\right)\right]$$

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

$$r=\frac{\rho }{\epsilon }\sqrt{8\nu \omega }\left(1+\frac{t}{2a}\right)$$

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

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.

perforation_end_correction

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):

$$\delta =0.85(1-1.47{\epsilon }^{1/2}+0.47{\epsilon }^{3/2})$$

Each orifice end adds $\delta a$ of air-plug length, and $\delta \to 0.85$ 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.

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 ($\mathrm{Im}(k)<0$ for the $e^{+j\omega t}$ convention), effective_density $=Z_{c}k/\omega$ and bulk_modulus $=Z_c\omega /k$ the surface-normalised equivalent-fluid density and bulk modulus, so that $Z_c=\sqrt{{\rho }_e{K}_e}$ and $k=\omega \sqrt{{\rho }_e/K_e}$ 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 $k_0=\omega /c$.

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.