materials.slow_sound_absorber

Slow-sound slit panels loaded with Helmholtz resonators (perfect absorbers).

A rigid panel perforated by a periodic array of thin closed slits, whose upper wall is loaded by an array of Helmholtz resonators (HRs), behaves as a deep-subwavelength, locally reacting sound absorber. The resonators slow the sound inside the slit, pulling the slit resonance down to the deep subwavelength regime, and the intrinsic visco-thermal losses can be tuned to exactly balance the leakage of the structure (critical coupling), giving perfect absorption at a chosen frequency and angle. The model follows the transfer-matrix treatment of Jimenez, Groby, Pagneux and Romero-Garcia (Iridescent Perfect Absorption in Critically-Coupled Acoustic Metamaterials Using the Transfer Matrix Method, Appl. Sci. 2017, 7, 618) together with the resonator model and end corrections detailed in the supplementary material of Jimenez, Huang, Romero-Garcia, Pagneux and Groby (Ultra-thin metamaterial for perfect and quasi-omnidirectional sound absorption, Appl. Phys. Lett. 2016, 109, 121902).

The building blocks, all in the e^{+j w t} convention used throughout phonometry (a passive medium has Im(k) < 0):

• Visco-thermal effective parameters. The slit of height h uses the narrow-channel effective density and bulk modulus (Appl. Sci. Eq. (6); Appl. Phys. Lett. Eqs. (A1)-(A2)):

  rho_s = rho0 [1 - tanh((h/2) G_rho) / ((h/2) G_rho)]^-1 and kappa_s = kappa0 [1 + (gamma - 1) tanh((h/2) G_kappa) / ((h/2) G_kappa)]^-1

  with G_rho = sqrt(j w rho0 / eta) and G_kappa = sqrt(j w Pr rho0 / eta). The square necks and cavities use the rectangular-duct series of Stinson (1991), reproduced as Appl. Sci. Eqs. (7)-(8) with the transverse wavenumbers alpha_k = (2k+1) pi / a and beta_m = (2m+1) pi / b. The duct series is printed in the opposite time convention of the source; it is returned conjugated here so the neck and cavity share the e^{+j w t} passivity of the slit. Both models are pinned in the tests to their exact limits: the effective density tends to rho0 and the bulk modulus to kappa0 as the boundary layers vanish, and j w rho tends to the Poiseuille flow resistivity of the channel as w -> 0 (12 eta / h^2 for the slit, 28.454 eta / w^2 for a square duct).

• Helmholtz-resonator impedance. Each resonator is a neck (length l_n, side w_n) over a closed cavity (length l_c, side w_c); its impedance follows Appl. Phys. Lett. Eq. (A23) with the neck-to-cavity radiation end correction of Eqs. (A24)-(A26).

• Transfer matrix. The panel is the chain M_dl (M_s M_HR M_s)... of half-lattice slit steps (Appl. Sci. Eq. (2)), resonators as point shunt scatterers (Eq. (3)) and the slit-radiation end correction (Eq. (3)/(A27)). The slit-radiation series impedance is printed in the sources as -i w dl_slit rho0 / (phi_t S0); like the duct series it is used conjugated here (+j w), so it acts as the added radiation mass it models and lowers the slit-panel resonance. The rigidly-backed reflection factor is R = (T11 cos(theta) - Z0 T21) / (T11 cos(theta) + Z0 T21) with Z0 = rho0 c0 / S0 (Eq. (4)), and alpha = 1 - |R|^2. Perfect absorption (critical coupling) is reached when the reflection zero sits on the real-frequency axis, i.e. Re(Z) cos(theta) = Z0 and Im(Z) = 0 with Z = T11 / T21 the acoustic surface impedance (Eq. (9)).

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critical_coupling_design

critical_coupling_design(
    target_frequency: float,
    resonator: HelmholtzResonator,
    *,
    lattice_step: float,
    period: float,
    angle: float = 0.0,
    slit_height_bounds: tuple[float, float] = (0.0002, 0.005),
    cavity_length_bounds: tuple[float, float] = (0.002, 0.2),
    end_correction: bool = True,
    slit_radiation: bool = True,
    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,
) -> CriticalCouplingResult

Solve resonator/slit geometry for perfect absorption at a frequency.

Critical coupling (perfect absorption) requires the acoustic surface impedance Z = T11 / T21 of the rigidly-backed panel to satisfy Re(Z) cos(theta) = Z0 and Im(Z) = 0 at target_frequency (Appl. Sci. 2017 Eq. (9)), i.e. the reflection zero lies on the real-frequency axis. Holding the neck geometry and cavity side of resonator fixed, this tunes the cavity length (which sets the resonance frequency) and the slit height (which sets the visco-thermal leakage balance) to meet both conditions, so alpha ~ 1 at the design point.

Parameters

Name	Description
target_frequency	Design frequency f0, in hertz.
resonator	Base geometry; its cavity_length is used as the initial guess and its neck and cavity side are held fixed.
lattice_step	Resonator lattice step a, in metres.
period	Slit array period d, in metres.
angle	Design angle of incidence theta, in radians.
slit_height_bounds	Search bounds for the slit height, in metres.
cavity_length_bounds	Search bounds for the cavity length, in metres.
end_correction	Include the resonator radiation end corrections.
slit_radiation	Include the slit-to-free-air radiation correction.
speed_of_sound	Speed of sound c0 in air, in m/s.
air_density	Air density rho0, 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 CriticalCouplingResult. A SlowSoundAbsorberWarning is emitted (via warnings.warn) if the solver does not reach perfect absorption within tolerance.

CriticalCouplingResult

CriticalCouplingResult(
    target_frequency: float,
    angle: float,
    resonator: HelmholtzResonator,
    slit_height: float,
    absorption: float,
    normalized_impedance: complex,
    converged: bool,
)

Outcome of a critical-coupling (perfect-absorption) design.

resonator and slit_height are the solved geometry that places the reflection zero on the real-frequency axis at target_frequency and angle; absorption is the modelled coefficient there (~1) and normalized_impedance the achieved Z cos(theta) / Z0 (~1). converged flags whether the root find met its tolerance.

helmholtz_resonator_impedance

helmholtz_resonator_impedance(
    frequency: ArrayLike,
    resonator: HelmholtzResonator,
    *,
    slit_height: float | None = None,
    lattice_step: float | None = None,
    end_correction: bool = True,
    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,
    sum_terms: int = 40,
) -> Complex

Acoustic impedance of a Helmholtz resonator with visco-thermal losses.

The neck and cavity use the square-duct effective parameters of rectangular_duct_properties; the impedance is Appl. Phys. Lett. 2016 Eq. (A23) with the neck-to-cavity radiation correction of Eq. (A24) and, when slit_height and lattice_step are supplied, the neck-to-slit correction of Eqs. (A25)-(A26) added to the total neck length correction:

`Z_HR = -j [cos(k_n l_n) cos(k_c l_c)

• Z_n k_n dl cos(k_n l_n) sin(k_c l_c) / Z_c
• Z_n sin(k_n l_n) sin(k_c l_c) / Z_c] / [sin(k_n l_n) cos(k_c l_c) / Z_n
• k_n dl sin(k_n l_n) sin(k_c l_c) / Z_c

• cos(k_n l_n) sin(k_c l_c) / Z_c]`

with Z_n = sqrt(kappa_n rho_n) / w_n^2, k_n = w sqrt(rho_n / kappa_n) (and likewise for the cavity), reducing to Eq. (A22) when dl = 0.

Parameters

Name	Description
frequency	Frequency vector f, in hertz.
resonator	The HelmholtzResonator geometry.
slit_height	Slit height h for the neck-to-slit correction; if None that correction is omitted.
lattice_step	Lattice step a for the neck-to-slit correction.
end_correction	Include the radiation end corrections (default True).
air_density	Air density rho0, 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.
sum_terms	Transverse modes kept per axis in the duct series.

Returns: Complex acoustic impedance Z_HR, in Pa s/m3, shaped like frequency.

HelmholtzResonator

HelmholtzResonator(
    neck_length: float,
    neck_side: float,
    cavity_length: float,
    cavity_side: float,
)

A square-cross-section Helmholtz resonator loading a slit.

neck_length l_n and neck_side w_n describe the neck, cavity_length l_c and cavity_side w_c the closed cavity; all lengths are in metres.

HelmholtzResonator.plot()

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

Draw the resonator cross-section to scale (dimensioned).

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

plot_helmholtz_resonator_geometry

plot_helmholtz_resonator_geometry(
    resonator: HelmholtzResonator,
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw a square-section Helmholtz resonator cross-section, to scale.

Neck opening upward into free air, cavity below, with the four defining dimensions (neck side and length, cavity side and length) dimensioned.

Parameters

Name	Description
resonator	A HelmholtzResonator.
ax	Existing axes, or None to create a figure.
language	Label language, "en" (default) or "es".
kwargs	Forwarded to the cavity rectangle.

Returns: The axes.

plot_slit_absorber_geometry

plot_slit_absorber_geometry(
    resonators: Sequence[HelmholtzResonator] | HelmholtzResonator,
    ax: Axes | None = None,
    *,
    slit_height: float,
    lattice_step: float,
    period: float,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw one period of the slit metamaterial absorber, to scale.

Side cut of the panel: the slit (height h) runs from the mouth at the left into the panel; N Helmholtz resonators load it from below at the lattice step a (total depth L = N a); the panel repeats vertically with period d; rigid back wall at the right.

Parameters

Name	Description
resonators	The resonator chain of slit_helmholtz_absorber (one per lattice step, or a single resonator reused for all steps).
ax	Existing axes, or None to create a figure.
slit_height	Slit height h, in metres.
lattice_step	Lattice step a, in metres.
period	Panel period d, in metres.
language	Label language, "en" (default) or "es".
kwargs	Forwarded to the slit rectangle.

Returns: The axes.

rectangular_duct_properties

rectangular_duct_properties(
    frequency: ArrayLike,
    *,
    side: float,
    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,
    sum_terms: int = 40,
) -> tuple[Complex, Complex]

Effective density and bulk modulus of a square duct of the given side.

The Stinson (1991) rectangular-duct series (Appl. Sci. 2017 Eqs. (7)-(8)),

rho = -rho0 a^2 b^2 / (64 G_rho^2 S_rho) and kappa = kappa0 / (gamma + 64 (gamma - 1) G_kappa^2 / (a^2 b^2) S_kappa),

with S = sum_k sum_m [alpha_k^2 beta_m^2 (alpha_k^2 + beta_m^2 - G^2)]^-1, alpha_k = (2k+1) pi / a, beta_m = (2m+1) pi / b, G_rho^2 = j w rho0 / eta and G_kappa^2 = j w Pr rho0 / eta. Here the duct is square (a = b = side). The series is transcribed in the source’s time convention and returned conjugated so the result is passive in the e^{+j w t} convention (Im(k) < 0). The normalising constant 64 is fixed by the exact limits rho -> rho0, kappa -> kappa0 as the boundary layers vanish and by the Poiseuille resistivity 28.454 eta / side^2 as w -> 0.

Parameters

Name	Description
frequency	Frequency vector f, in hertz.
side	Square-duct side length, in metres.
air_density	Air density rho0, 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.
sum_terms	Transverse modes kept per axis (default 40).

Returns: (rho, kappa) complex arrays shaped like frequency.

slit_effective_properties

slit_effective_properties(
    frequency: ArrayLike,
    *,
    slit_height: float,
    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,
) -> tuple[Complex, Complex]

Effective density and bulk modulus of a narrow slit of height h.

rho_s = rho0 [1 - tanh(x_rho) / x_rho]^-1 and kappa_s = kappa0 [1 + (gamma - 1) tanh(x_kappa) / x_kappa]^-1 with x_rho = (h/2) sqrt(j w rho0 / eta) and x_kappa = (h/2) sqrt(j w Pr rho0 / eta) (Appl. Sci. 2017 Eq. (6); Appl. Phys. Lett. 2016 Eqs. (A1)-(A2)). kappa0 = gamma P0.

Parameters

Name	Description
frequency	Frequency vector f, in hertz.
slit_height	Slit height h, in metres.
air_density	Air density rho0, 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: (rho_s, kappa_s) complex arrays shaped like frequency.

slit_helmholtz_absorber

slit_helmholtz_absorber(
    frequency: ArrayLike,
    resonators: HelmholtzResonator | list[HelmholtzResonator] | tuple[HelmholtzResonator, ...],
    *,
    slit_height: float,
    lattice_step: float,
    period: float,
    angle: float = 0.0,
    end_correction: bool = True,
    slit_radiation: bool = True,
    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,
) -> SlitResonatorAbsorberResult

Transfer-matrix prediction of a slit panel loaded with resonators.

The panel is a periodic array (period d along the panel face) of thin closed slits of height h, each loaded from its upper wall by the given resonators spaced by the lattice step a (Appl. Sci. 2017, Section 2). The total chain matrix is T = M_dl (M_s M_HR M_s) ... over the N resonators, where each resonator sits between two half-lattice slit steps; the rigidly-backed reflection factor is R = (T11 cos(theta) - Z0 T21) / (T11 cos(theta) + Z0 T21) with Z0 = rho0 c0 / S0, S0 = d a, and alpha = 1 - |R|^2 (Eq. (4)). The structure is locally reacting, so the internal chain does not depend on theta; only the front air impedance carries cos(theta).

Parameters

Name	Description
frequency	Frequency vector f, in hertz.
resonators	One HelmholtzResonator or a sequence of them, ordered from the panel face towards the rigid backing.
slit_height	Slit height h, in metres.
lattice_step	Resonator lattice step a along the slit, in metres; the slit depth is L = N a.
period	Slit array period d along the face, in metres (d >= h).
angle	Polar angle of incidence theta, in radians (0 <= theta < pi/2 - 1e-6).
end_correction	Include the resonator radiation end corrections.
slit_radiation	Include the slit-to-free-air radiation correction.
speed_of_sound	Speed of sound c0 in air, in m/s.
air_density	Air density rho0, 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 SlitResonatorAbsorberResult.

SlitResonatorAbsorberResult

SlitResonatorAbsorberResult(
    frequency: Real,
    angle: float,
    surface_impedance: Complex,
    normalized_impedance: Complex,
    reflection: Complex,
    absorption: Real,
    effective_wavenumber: Complex,
    effective_impedance: Complex,
    transfer_matrix: Complex,
    resonators: tuple[HelmholtzResonator, ...] | None = None,
    slit_height: float | None = None,
    lattice_step: float | None = None,
    period: float | None = None,
)

Prediction of a slit panel loaded with Helmholtz resonators.

All spectra share the shape of frequency. surface_impedance is the acoustic surface impedance Z = T11 / T21 in Pa s/m3 of the rigidly backed panel, normalized_impedance its ratio to Z0 = rho0 c0 / S0, reflection the plane-wave reflection factor R(theta), absorption the coefficient alpha = 1 - |R|^2, effective_wavenumber and effective_impedance the retrieved k_eff and Z_eff (Appl. Sci. 2017 Eq. (5)), and transfer_matrix the total 2x2 chain matrix with shape (2, 2, len(frequency)).

The trailing fields retain the panel geometry the prediction was run with (resonators, slit_height, lattice_step, period) so plot_geometry can draw the cross-section; they are appended after the original fields and default to None for hand-built results.

SlitResonatorAbsorberResult.plot()

SlitResonatorAbsorberResult.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.

SlitResonatorAbsorberResult.plot_geometry()

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

Draw one period of the panel 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 geometry.

SlowSoundAbsorberWarning

Advisory for slow-sound absorber use outside the modelled regime.