materials.absorbers.biot

Biot poroelastic layers: the three waves and the 6x6 transfer matrix.

An equivalent fluid replaces a porous material by a single wave travelling in the pores while the skeleton stands still (rigid frame) or is merely dragged along by the pore fluid (limp frame, see limp_frame). Neither can carry a wave in the skeleton itself, so neither can produce a frame resonance. The Biot theory can: it treats the frame as an elastic solid coupled to the pore fluid through a potential coupling coefficient and an inertial coupling coefficient, and it predicts three waves in an isotropic porous layer, two compressional and one shear (Allard & Atalla, Propagation of Sound in Porous Media 2e, chapter 6).

This module implements that theory in the $e^{+j\omega t}$ convention of the rest of the package, exactly as printed:

• Elastic coefficients P, Q and R for the usual case of a frame built from a material far stiffer than the frame itself ($K_s\to \infty$), Eqs. (6.26)-(6.29) printed pp. 116-117: $R=\varphi K_f$, $Q=(1-\varphi )K_f$, $P=4N/3+K_b+(1-\varphi {)}^2{K}_f/\varphi$ with the frame bulk modulus $K_b=2N(1+\nu )/(3(1-2\nu ))$.
• Modified densities rho11, rho12, rho22 (Eq. (6.56), printed p. 120) built from the inertial coupling ${\rho }_a=\varphi {\rho }_0({\alpha }_{\infty }-1)$ and the visco-inertial term $j\sigma {\varphi }^{2}G(\omega )/\omega$. The latter is not re-derived here: it is read back from the equivalent-fluid model handed in, through the identity ${\rho }_{22}={\varphi }^2{\rho }_{\mathrm{eq}}$ stated on printed p. 253, so a Biot layer and a rigid-frame layer built from the same PorousMediumResult share one visco-thermal description by construction.
• The two compressional waves as the eigenvalues of Eq. (6.65) (Eqs. (6.67)-(6.69), printed p. 121), the shear wave of Eq. (6.83), and the fluid-to-frame velocity ratios mu1, mu2 (Eq. (6.71)) and mu3 (Eq. (6.84)).
• The closed-form surface impedance of a hard-backed layer at normal incidence, Eqs. (6.107)-(6.108) printed p. 128, in biot_surface_impedance, and the $\lambda /4$ frame resonance it develops, Eq. (6.110) printed p. 129, in frame_quarter_wave_resonance.
• The 6x6 layer matrix of chapter 11: the field vector $[v_1^s,v_3^s,v_3^f,s_{33}^s,s_{13}^s,s_{33}^f]$ (Eq. (11.26)) expressed through the wave-amplitude matrix $[\Gamma (x_3)]$ of Table 11.1 (printed p. 252), from which poroelastic_transfer_matrix returns $[T]=[\Gamma (-h)][\Gamma (0){]}^{-1}$ (Eq. (11.34)).

The layer is used through PoroelasticLayer inside layered_absorber, which switches to the global-matrix assembly of Sect. 11.5 as soon as a poroelastic layer is present, with the coupling matrices of Sect. 11.4 ($[I_{pf}]$ / $[J_{pf}]$ Eq. (11.73), $[I_{pp}]$ Eq. (11.67)) and the hard-wall conditions $[Y]$ of Eq. (11.81).

On oracles. Allard & Atalla publish no table of computed surface impedances, so the model is anchored on closed forms and on exact limits: the rigid-frame limit reproduces the already-anchored Johnson-Champoux-Allard equivalent fluid, the limp limit reproduces limp_frame, and the chapter 11 assembly reproduces the chapter 6 closed form Eq. (6.107) to machine precision. The book does print four output numbers for the fully specified glass wool of Table 6.1, and all four are reproduced: the airborne branch changes from $({\delta }_1,{\mu }_1)$ to $({\delta }_2,{\mu }_2)$ at 495 Hz, $|{\mu }_a|>40$ above 50 Hz, and mu_b runs from 1.0 at 50 Hz to 0.82 at 1500 Hz (all printed pp. 124-125), while the impedance peak of a 5.6 cm layer sits at 860 Hz (printed p. 129). The third of those is reproduced by $\mathrm{Re}({\mu }_b)$: the printed sentence calls it a modulus, but $|{\mu }_b|$ is 0.939 at 1500 Hz against the printed 0.82, and docs/ERRATA.md records why. See tests/materials/absorbers/test_biot.py.

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biot_surface_impedance

biot_surface_impedance(waves: BiotWavesResult, thickness: float) -> Complex

Surface impedance of a hard-backed Biot layer at normal incidence.

The closed form of Allard & Atalla 2e, Eqs. (6.107)-(6.108) (printed p. 128), obtained by writing the four compressional-wave amplitudes against the three boundary conditions of a layer glued to an impervious rigid wall (zero frame and fluid velocity at the wall, Eq. (6.95); continuity of pressure and of the total normal stress at the free face, Eqs. (6.97)-(6.99); conservation of the volume flow, Eq. (6.100)):

$$Z=-j(Z_1^s{Z}_2^f{\mu }_2-Z_2^s{Z}_1^f{\mu }_1)/D$$ $$D=(1-\varphi +\varphi {\mu }_2)[Z_1^s-(1-\varphi )Z_1^f{\mu }_1]\tan ({\delta }_{2}l)+(1-\varphi +\varphi {\mu }_1)[(1-\varphi )Z_2^f{\mu }_2-Z_2^s]\tan ({\delta }_{1}l)$$

with the four characteristic impedances of Eqs. (6.74)-(6.77), $Z_i^f=(R+Q/{\mu }_i){\delta }_i/(\varphi \omega )$ and $Z_i^s=(P+Q{\mu }_i){\delta }_i/\omega$.

This is an independent derivation of the same physics as the chapter 11 transfer-matrix assembly reached through PoroelasticLayer; the two agree to machine precision, which is one of the anchors of this module. Unlike the transfer matrix, it is restricted to normal incidence, to a single layer and to a glued rigid backing.

Parameters

Name	Description
waves	The BiotWavesResult of the material.
thickness	Layer thickness l, in metres (> 0).

Returns: The complex surface impedance Z, in Pa s/m.

Raises

Exception	When
ValueError	for a non-positive thickness.

biot_waves

biot_waves(
    medium: PorousMediumResult,
    *,
    porosity: float,
    tortuosity: float,
    frame_density: float,
    shear_modulus: complex,
    poisson_ratio: float = 0.0,
) -> BiotWavesResult

The two compressional waves and the shear wave of a Biot layer.

medium supplies the visco-thermal description of the pore fluid, through the two identities of Allard & Atalla printed p. 253, ${\rho }_{22}={\varphi }^2{\rho }_{\mathrm{eq}}$ and $R={\varphi }^2{K}_{\mathrm{eq}}$, where rho_eq and K_eq are the surface-normalised effective density and bulk modulus that every model in porous returns. Pass the rigid-frame johnson_champoux_allard result: the frame motion is the Biot model’s business, not the equivalent fluid’s, so the rigid-frame effective density is the correct input (a limp-corrected one would count the frame inertia twice).

From it, with the inertial coupling ${\rho }_a=\varphi {\rho }_0({\alpha }_{\infty }-1)$ (Eq. (6.44)):

• ${\rho }_{22}=\varphi {\rho }_0+{\rho }_a-j\sigma {\varphi }^{2}G(\omega )/\omega$, ${\rho }_{12}=-{\rho }_a+j\sigma {\varphi }^{2}G(\omega )/\omega$ and ${\rho }_{11}={\rho }_1+{\rho }_a-j\sigma {\varphi }^{2}G(\omega )/\omega$ (Eq. (6.56), printed p. 120), the visco-inertial term being recovered as $\varphi {\rho }_0{\alpha }_{\infty }-{\rho }_{22}$;
• $K_f=\varphi K_{\mathrm{eq}}$, then $R=\varphi K_f$, $Q=(1-\varphi )K_f$ and $P=4N/3+K_b+(1-\varphi {)}^2{K}_f/\varphi$ (Eqs. (6.26)-(6.28));
• ${\delta }_1^2$ and ${\delta }_2^2$ from Eqs. (6.67)-(6.69), ${\delta }_3^2$ from Eq. (6.83), mu1, mu2 from Eq. (6.71) and mu3 from Eq. (6.84).

Parameters

Name	Description
medium	Rigid-frame PorousMediumResult for the pore fluid, evaluated on the frequency vector of interest.
porosity	Open porosity phi (0 < phi <= 1).
tortuosity	High-frequency tortuosity a_inf (>= 1).
frame_density	Bulk density of the frame rho1, in kg/m3 (> 0).
shear_modulus	Complex shear modulus N of the frame, in Pa ($\mathrm{Im}(N)\ge 0$; a structural loss factor eta gives $N=N^{\prime }(1+j\eta )$).
poisson_ratio	Poisson coefficient nu of the frame (Default: 0, the value Allard & Atalla use for their glass wool).

Returns: A BiotWavesResult.

Raises

Exception	When
ValueError	for an out-of-range porosity, tortuosity, frame density, shear modulus or Poisson coefficient.

BiotWavesResult

BiotWavesResult(
    frequency: Real,
    porosity: float,
    tortuosity: float,
    frame_density: float,
    shear_modulus: complex,
    poisson_ratio: float,
    elastic_p: Complex,
    elastic_q: Complex,
    elastic_r: Complex,
    density_11: Complex,
    density_12: Complex,
    density_22: Complex,
    compressional_wavenumber_1: Complex,
    compressional_wavenumber_2: Complex,
    shear_wavenumber: Complex,
    velocity_ratio_1: Complex,
    velocity_ratio_2: Complex,
    velocity_ratio_3: Complex,
)

The three Biot waves of an isotropic air-saturated porous material.

All arrays share the shape of frequency. compressional_wavenumber_1 and compressional_wavenumber_2 are delta1 and delta2 of Eqs. (6.67)-(6.68) (the branch with $-\sqrt{\Delta }$ first, as printed, with $\sqrt{\Delta }$ taken on the root with non-positive real part so that the numbering matches the book’s own example), shear_wavenumber is delta3 of Eq. (6.83), all in rad/m and all taken on the root with non-negative real part. velocity_ratio_1, velocity_ratio_2 and velocity_ratio_3 are the ratios mu of the fluid displacement over the frame displacement (Eqs. (6.71) and (6.84)). elastic_p, elastic_q and elastic_r are the Biot elastic coefficients and density_11, density_12, density_22 the modified densities of Eq. (6.56).

The airborne wave is the one whose $|\mu |$ is the larger (the pore fluid moves far more than the frame); the frame-borne wave is the other. Which of delta1 / delta2 plays which role swaps with frequency, so use the airborne_* and frame_borne_* properties rather than the numbered ones.

BiotWavesResult.airborne_is_second

property

Whether the airborne wave is $({\delta }_2,{\mu }_2)$ at each frequency.

Neither labelling is continuous in general. delta1 and delta2 are the two branches of one square root (Eqs. (6.67) and (6.68)), so compressional_wavenumber_1 and compressional_wavenumber_2 swap wherever the discriminant crosses the cut of numpy.sqrt; on the Table 6.1 glass wool that happens at 495.99 Hz, exactly where the book puts the change of root, and the two wavenumbers jump past each other by 24 rad/m there. This $|\mu |$ sorting is the physical labelling of Sect. 6.5.4 and it removes that jump where the two events coincide, as they do there, but it introduces one of its own wherever $|{\mu }_1|$ and $|{\mu }_2|$ cross away from the cut: on a sweep of 864 parameter sets, 30 left a visible step in the sorted airborne wavenumber.

Nothing downstream depends on which root is called which. The closed form Eq. (6.107) and the $[\Gamma ]$ of Table 11.1 are both invariant under the permutation of the two compressional waves and under a sign flip of either, so the surface impedance is continuous across the crossing even where the labels are not.

BiotWavesResult.airborne_velocity_ratio

property

Fluid-over-frame velocity ratio mu_a of the airborne wave.

BiotWavesResult.airborne_wavenumber

property

Wavenumber delta_a of the airborne compressional wave, rad/m.

BiotWavesResult.frame_borne_velocity_ratio

property

Fluid-over-frame velocity ratio mu_b of the frame-borne wave.

BiotWavesResult.frame_borne_wavenumber

property

Wavenumber delta_b of the frame-borne wave, in rad/m.

BiotWavesResult.plot()

BiotWavesResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot the three Biot wavenumbers against frequency.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

frame_bulk_modulus

frame_bulk_modulus(shear_modulus: complex, poisson_ratio: float) -> complex

Bulk modulus Kb of the frame in vacuum from N and nu.

$K_b=2N(\nu +1)/(3(1-2\nu ))$ (Allard & Atalla 2e, Eq. (6.29), printed p. 116). Kb is the quantity the jacketed “gedanken experiment” of Eq. (6.7) measures, and the one the limp-frame rules of thumb of limp_frame_applicable compare with the bulk modulus of the pore fluid.

Parameters

Name	Description
shear_modulus	Complex shear modulus N of the frame, in Pa ($\mathrm{Im}(N)\ge 0$ for a lossy frame in the $e^{+j\omega t}$ convention).
poisson_ratio	Poisson coefficient nu of the frame ($-1<\nu <0.5$).

Returns: The complex bulk modulus Kb of the frame in vacuum, in Pa.

Raises

Exception	When
ValueError	for a non-positive or non-finite N, a negative $\mathrm{Im}(N)$ or a nu outside $(-1,0.5)$.

frame_elastic_coefficient

frame_elastic_coefficient(
    shear_modulus: complex,
    poisson_ratio: float,
) -> complex

Longitudinal elastic coefficient Kc of the frame in vacuum.

$$K_c=\lambda +2\mu =K_b+4N/3=2(1-\nu )N/(1-2\nu )$$

(Allard & Atalla 2e, Eqs. (1.76) and (6.111), printed pp. 12 and 130). A compressional wave in the frame in vacuum travels at $\sqrt{K_c/{\rho }_1}$, which is what sets the $\lambda /4$ frame resonance of frame_quarter_wave_resonance.

Parameters

Name	Description
shear_modulus	Complex shear modulus N of the frame, in Pa.
poisson_ratio	Poisson coefficient nu of the frame.

Returns: The complex elastic coefficient Kc, in Pa.

Raises

Exception	When
ValueError	as frame_bulk_modulus.

frame_quarter_wave_resonance

frame_quarter_wave_resonance(
    thickness: float,
    *,
    shear_modulus: complex,
    poisson_ratio: float,
    frame_density: float,
) -> float

Quarter-wavelength resonance of the frame-borne wave (closed form).

A porous layer glued to a rigid wall holds the frame still at the wall and free at the front face, so the frame-borne compressional wave resonates where $l\mathrm{Re}({\delta }_b)=\pi /2$ (Allard & Atalla 2e, Eq. (6.109), printed p. 129). Since delta_b stays close to the frame-in-vacuum wavenumber $\omega \sqrt{{\rho }_1/K_c}$ of Eq. (6.88), the resonance sits at

$$f_r=\frac{1}{4l}\sqrt{\mathrm{Re}(K_c)/{\rho }_1}\text{\hspace{1em}(Eq. 6.110)}$$

This is the frequency at which the peak that no equivalent-fluid model can produce appears in the surface impedance of biot_surface_impedance; Eq. (6.110) is an approximation to it (delta_b is not exactly the in-vacuum wavenumber), so the peak lands a few per cent above fr.

Parameters

Name	Description
thickness	Layer thickness l, in metres (> 0).
shear_modulus	Complex shear modulus N of the frame, in Pa.
poisson_ratio	Poisson coefficient nu of the frame.
frame_density	Bulk density of the frame rho1, in kg/m3 (> 0): the mass of solid per unit volume of material.

Returns: The resonance frequency fr, in hertz.

Raises

Exception	When
ValueError	for a non-positive thickness or frame density, or an invalid N / nu.

poroelastic_transfer_matrix

poroelastic_transfer_matrix(
    waves: BiotWavesResult,
    thickness: float,
    *,
    transverse_wavenumber: ArrayLike = 0.0,
) -> Complex

The 6x6 transfer matrix $[T^p]$ of a Biot poroelastic layer.

$[T^p]=[\Gamma (-h)][\Gamma (0){]}^{-1}$ (Allard & Atalla 2e, Eq. (11.34), printed p. 249), relating the field vector $[v_1^s,v_3^s,v_3^f,s_{33}^s,s_{13}^s,s_{33}^f]$ (Eq. (11.26)) just inside the front face of the layer to the same vector just inside its back face, $V(M)=[T^p]V(M^{\prime })$. The wave-amplitude matrix $[\Gamma ]$ of Table 11.1 behind it is built by _gamma.

The layer solver of layered_absorber does not use this matrix: it assembles the wave amplitudes directly, which avoids inverting $[\Gamma (0)]$ and is far better conditioned for a very soft or a very thick frame. The matrix is exposed because it is the object chapter 11 is written around and the one an external multilayer chain expects.

Parameters

Name	Description
waves	The BiotWavesResult of the material.
thickness	Layer thickness h, in metres (> 0).
transverse_wavenumber	In-plane wavenumber $k_t=k\sin (\theta )$, in rad/m (Default: 0, normal incidence). Scalar or one value per frequency.

Returns: The transfer matrix with shape (len(frequency), 6, 6).

Raises

Exception	When
ValueError	for a non-positive thickness.