materials.absorbers.slow_sound

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 $\mathrm{Im}(k)<0$):

• 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)):

  ${\rho }_s={\rho }_0[1-\tanh ((h/2)G_{\rho })/((h/2)G_{\rho }){]}^{-1}$ and ${\kappa }_s={\kappa }_0[1+(\gamma -1)\tanh ((h/2)G_{\kappa })/((h/2)G_{\kappa }){]}^{-1}$

  with $G_{\rho }=\sqrt{j\omega {\rho }_0/\eta }$ and $G_{\kappa }=\sqrt{j\omega \mathrm{Pr}{\rho }_0/\eta }$. 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 $e^{+j\omega t}$ 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 $j\omega \rho$ tends to the Poiseuille flow resistivity of the channel as $\omega \to 0$ ($12\eta /h^2$ 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 $R=(T_{11}\cos (\theta )-Z_0{T}_{21})/(T_{11}\cos (\theta )+Z_0{T}_{21})$ with $Z_0={\rho }_0{c}_0/S_0$ (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. $\mathrm{Re}(Z)\cos (\theta )=Z_0$ and $\mathrm{Im}(Z)=0$ with $Z=T_{11}/T_{21}$ 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,
    air: AirProperties = ...,
) -> CriticalCouplingResult

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

Critical coupling (perfect absorption) requires the acoustic surface impedance $Z=T_{11}/T_{21}$ of the rigidly-backed panel to satisfy $\mathrm{Re}(Z)\cos (\theta )=Z_0$ and $\mathrm{Im}(Z)=0$ 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.
air	State of the air the panel is designed for (AirProperties): its speed of sound c0, density rho0, viscosity eta, Prandtl number Pr, ratio of specific heats gamma and static pressure P0.

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,
    geometry: str = 'square',
    air: AirProperties = ...,
    sum_terms: int = 40,
) -> Complex

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

With the default geometry="square" the neck and cavity are square ducts using the 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:

$$Z_{\mathrm{HR}}=-j\frac{\cos (k_n{l}_n)\cos (k_c{l}_c)-Z_n{k}_n\mathrm{dl}\cos (k_n{l}_n)\sin (k_c{l}_c)/Z_c-Z_n\sin (k_n{l}_n)\sin (k_c{l}_c)/Z_c}{\sin (k_n{l}_n)\cos (k_c{l}_c)/Z_n-k_n\mathrm{dl}\sin (k_n{l}_n)\sin (k_c{l}_c)/Z_c+\cos (k_n{l}_n)\sin (k_c{l}_c)/Z_c}$$

with $Z_n=\sqrt{{\kappa }_n{\rho }_n}/w_n^2$, $k_n=\omega \sqrt{{\rho }_n/{\kappa }_n}$ (and likewise for the cavity), reducing to Eq. (A22) when dl = 0.

With geometry="slit" the resonator is two-dimensional (the neck and cavity are slit-like ducts spanning the lattice step): the effective parameters come from slit_effective_properties with the neck and cavity widths, the duct sections are w_n a and w_c a, and the end corrections are the 2-D fits of Sci. Rep. 7:5389 Eqs. (11)-(12); both slit_height and lattice_step are then required.

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 ("square" only).
lattice_step	Lattice step a for the neck-to-slit correction.
end_correction	Include the radiation end corrections (default True).
geometry	"square" (default) for square-duct necks and cavities, "slit" for the two-dimensional resonator model.
air	State of the air in the neck and cavity (AirProperties); the density rho0, viscosity eta, Prandtl number Pr, ratio of specific heats gamma and static pressure P0 are read from it.
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: AirProperties = ...,
    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)),

$$\rho =\frac{-{\rho }_0{a}^2{b}^2}{64G_{\rho }^2{S}_{\rho }}$$ $$\kappa =\frac{{\kappa }_0}{\gamma +64(\gamma -1)G_{\kappa }^2/(a^2{b}^2)S_{\kappa }}$$

with $S={\sum }_k{\sum }_m[{\alpha }_k^2{\beta }_m^2({\alpha }_k^2+{\beta }_m^2-G^2){]}^{-1}$, ${\alpha }_k=(2k+1)\pi /a$, ${\beta }_m=(2m+1)\pi /b$, $G_{\rho }^2=j\omega {\rho }_0/\eta$ and $G_{\kappa }^2=j\omega \mathrm{Pr}{\rho }_0/\eta$. Here the duct is square ($a=b$, both equal to side). The series is transcribed in the source’s time convention and returned conjugated so the result is passive in the $e^{+j\omega t}$ convention ($\mathrm{Im}(k)<0$). The normalising constant 64 is fixed by the exact limits $\rho \to {\rho }_0$, $\kappa \to {\kappa }_0$ as the boundary layers vanish and by the Poiseuille resistivity $28.454\eta /{\text{side}}^2$ as $\omega \to 0$.

Parameters

Name	Description
frequency	Frequency vector f, in hertz.
side	Square-duct side length, in metres.
air	State of the air in the duct (AirProperties); the density rho0, viscosity eta, Prandtl number Pr, ratio of specific heats gamma and static pressure P0 are read from it.
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: AirProperties = ...,
) -> tuple[Complex, Complex]

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

$${\rho }_s={\rho }_0{\left[1-\frac{\tanh (x_{\rho })}{x_{\rho }}\right]}^{-1}$$ $${\kappa }_s={\kappa }_0{\left[1+(\gamma -1)\frac{\tanh (x_{\kappa })}{x_{\kappa }}\right]}^{-1}$$

with $x_{\rho }=(h/2)\sqrt{j\omega {\rho }_0/\eta }$ and $x_{\kappa }=(h/2)\sqrt{j\omega \mathrm{Pr}{\rho }_0/\eta }$ (Appl. Sci. 2017 Eq. (6); Appl. Phys. Lett. 2016 Eqs. (A1)-(A2)). ${\kappa }_0=\gamma P_0$.

Parameters

Name	Description
frequency	Frequency vector f, in hertz.
slit_height	Slit height h, in metres.
air	State of the air in the slit (AirProperties); the density rho0, viscosity eta, Prandtl number Pr, ratio of specific heats gamma and static pressure P0 are read from it.

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,
    resonator_geometry: str = 'square',
    air: AirProperties = ...,
) -> 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 $T=M_{\mathrm{dl}}(M_s{M}_{\mathrm{HR}}{M}_s)\cdots$ over the N resonators, where each resonator sits between two half-lattice slit steps; the rigidly-backed reflection factor is $R=(T_{11}\cos (\theta )-Z_0{T}_{21})/(T_{11}\cos (\theta )+Z_0{T}_{21})$ with $Z_0={\rho }_0{c}_0/S_0$, $S_0=da$, 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 $L=Na$.
period	Slit array period d along the face, in metres ($d\ge h$).
angle	Polar angle of incidence theta, in radians ($0\le \theta <\pi /2-10^{-6}$).
end_correction	Include the resonator radiation end corrections.
slit_radiation	Include the slit-to-free-air radiation correction.
air	State of the air the panel radiates into and the slit and resonators are filled with (AirProperties): its speed of sound c0, density rho0, viscosity eta, Prandtl number Pr, ratio of specific heats gamma and static pressure P0.

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 $Z=T_{11}/T_{21}$ in Pa s/m3 of the rigidly backed panel, normalized_impedance its ratio to $Z_0={\rho }_0{c}_0/S_0$, 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.