materials.diffusers.metadiffuser

Metadiffusers: deep-subwavelength Schroeder-like sound diffusers.

A metadiffuser is a rigidly backed slotted panel whose slits are each loaded by an array of Helmholtz resonators (Jimenez, Cox, Romero-Garcia and Groby, Metadiffusers: Deep-subwavelength sound diffusers, Sci. Rep. 7, 5389, 2017). The resonators slow the sound inside each slit, so the slit reaches its quarter-wavelength condition at a fraction of the depth a plain well would need; by giving every slit a different geometry the panel presents a spatially dependent complex reflection coefficient R_n(f) along its face. Tuning that profile to a Schroeder phase grating reproduces the scattering of a quadratic-residue or primitive-root diffuser from a panel 1/46 to 1/20 of the design wavelength thick, and driving single slits to critical coupling adds the perfectly absorbing 0 state that ternary sequences require.

Each slit is modelled with the transfer-matrix chain of slit_helmholtz_absorber (visco-thermal effective parameters, resonator end corrections and slit radiation correction); the panel is locally reacting, so the wells do not couple internally and the far field follows from the Fraunhofer integral of the per-well reflection sequence (predict_diffuser_polar_response) reduced to the ISO 17497-2 directional diffusion coefficient.

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metadiffuser_diffusion_spectrum

metadiffuser_diffusion_spectrum(
    frequencies: ArrayLike,
    wells: Sequence[MetadiffuserWell | None],
    *,
    depth: float,
    period: float,
    angles: ArrayLike = (-90, -85, -80, -75, -70, -65, -60, -55, -50, -45, -40, -35, -30, -25, -20, -15, -10, -5, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90),
    source_angle: float = 0.0,
    periods: int = 1,
    resonator_geometry: str = 'slit',
    air: AirProperties = ...,
) -> DiffusionSpectrum

Normalized diffusion-coefficient spectrum d_n(f) of a metadiffuser.

Evaluates the far-field polar response at each frequency with metadiffuser_polar_response, forms the ISO 17497-2 directional diffusion coefficient band by band and normalises it against a flat rigid reference of the same footprint (all wells $R=1$) with normalized_diffusion_coefficient, exactly as the paper reports delta_n.

Parameters

Name	Description
frequencies	Frequencies of the spectrum, in hertz (1-D).
wells	Sequence of MetadiffuserWell (or None for a flat rigid strip) describing one period of the panel face.
depth	Panel depth L common to all slits, in metres.
period	Well pitch d along the panel face, in metres.
angles	Receiver reflection angles theta, in degrees.
source_angle	Angle of incidence psi, in degrees.
periods	Number of repetitions N_p of the single period.
resonator_geometry	"slit" (default) for the paper’s two-dimensional resonators, "square" for square-duct necks and cavities.
air	State of the air the panel radiates into and the slits and resonators are filled with (AirProperties); its speed of sound c0 also carries the far field.

Returns: A DiffusionSpectrum carrying the raw d(f) and the normalised d_n(f).

metadiffuser_polar_response

metadiffuser_polar_response(
    frequency: float,
    wells: Sequence[MetadiffuserWell | None],
    *,
    depth: float,
    period: float,
    angles: ArrayLike = (-90, -85, -80, -75, -70, -65, -60, -55, -50, -45, -40, -35, -30, -25, -20, -15, -10, -5, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90),
    source_angle: float = 0.0,
    periods: int = 1,
    resonator_geometry: str = 'slit',
    air: AirProperties = ...,
) -> DiffuserPolarResponse

Far-field polar response of a metadiffuser at one frequency.

Computes the per-well complex reflection sequence at frequency with metadiffuser_reflection (the panel is locally reacting, so the slit chains see the incidence angle source_angle) and evaluates the Fraunhofer far field and ISO 17497-2 directional diffusion coefficient with predict_diffuser_polar_response.

Parameters

Name	Description
frequency	Frequency of the prediction f, in hertz.
wells	Sequence of MetadiffuserWell (or None for a flat rigid strip) describing one period of the panel face.
depth	Panel depth L common to all slits, in metres.
period	Well pitch d along the panel face, in metres; it is the well_width of the far-field model.
angles	Receiver reflection angles theta, in degrees.
source_angle	Angle of incidence psi of the source, in degrees; also applied to the local slit reflection.
periods	Number of repetitions N_p of the single period; the grating lobes of a Schroeder-like design require periods >= 2.
resonator_geometry	"slit" (default) for the paper’s two-dimensional resonators, "square" for square-duct necks and cavities.
air	State of the air the panel radiates into and the slits and resonators are filled with (AirProperties); its speed of sound c0 also carries the far field.

Returns: A DiffuserPolarResponse.

metadiffuser_reflection

metadiffuser_reflection(
    frequency: ArrayLike,
    wells: Sequence[MetadiffuserWell | None],
    *,
    depth: float,
    period: float,
    angle: float = 0.0,
    resonator_geometry: str = 'slit',
    air: AirProperties = ...,
) -> MetadiffuserResult

Per-well reflection spectra of a metadiffuser panel (Sci. Rep. Eq. (6)).

Runs the rigidly backed slit transfer-matrix chain of slit_helmholtz_absorber once per well: each MetadiffuserWell becomes a slit of height h_n and depth L loaded by its M resonators on the lattice $a=L/M$, and None wells are flat rigid strips with $R=1$. The panel is locally reacting, so a well’s reflection does not depend on its neighbours and the incidence angle enters only through the front air impedance.

Parameters

Name	Description
frequency	Frequency vector f, in hertz.
wells	Sequence of MetadiffuserWell (or None for a flat rigid strip) describing one period of the panel face.
depth	Panel depth L common to all slits, in metres.
period	Well pitch d along the panel face, in metres.
angle	Polar angle of incidence theta, in radians.
resonator_geometry	"slit" (default) for the paper’s two-dimensional resonators, "square" for square-duct necks and cavities.
air	State of the air the panel radiates into and the slits 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 MetadiffuserResult with one reflection row per well.

MetadiffuserResult

MetadiffuserResult(
    frequency: Real,
    reflection: Complex,
    absorption: Real,
    well_absorption: Real,
    wells: tuple[MetadiffuserWell | None, ...] | None = None,
    depth: float | None = None,
    period: float | None = None,
)

Spectra of a metadiffuser panel, one reflection row per well.

reflection has shape (N, len(frequency)) with the complex pressure reflection factor of each well (flat strips are exactly 1), absorption is the face-averaged energy absorption $\alpha (f)=1-{\mathrm{mean}}_n|R_n{|}^2$ and well_absorption the per-well ${\alpha }_n=1-|R_n{|}^2$. The trailing fields retain the geometry the prediction was run with (wells, depth, period) so plot_geometry can draw the panel section; they default to None for hand-built results.

MetadiffuserResult.plot()

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

Plot the per-well and face-averaged absorption spectra.

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

MetadiffuserResult.plot_geometry()

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

Draw the panel cross-section to scale (slits and resonators).

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

Raises

Exception	When
ValueError	If the result does not retain its geometry.

MetadiffuserWell

MetadiffuserWell(
    slit_height: float,
    resonators: tuple[HelmholtzResonator, ...],
)

One slit of a metadiffuser panel.

slit_height is the slit opening h_n along the panel face and resonators the Helmholtz resonators loading the slit, ordered from the panel face towards the rigid backing; the resonator lattice step is $a=L/M$ for a panel of depth L and M resonators. All lengths are in metres. A None entry in a well sequence stands for a flat rigid strip of the panel face ($R=1$), the +1 state of ternary-sequence designs.

plot_metadiffuser_panel_geometry

plot_metadiffuser_panel_geometry(
    wells: Sequence[Any],
    ax: Axes | None = None,
    *,
    depth: float,
    period: float,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw one period of a metadiffuser panel, to scale.

Side cut of the slotted panel (Sci. Rep. 7:5389 Fig. 1(b)): the face runs along the top with the sound arriving from above, the wells repeat horizontally at the pitch d, and each well is a slit of height h_n descending the panel depth, loaded by its resonators at the lattice step a = L / M shelved sideways into the septum; None wells are flat rigid strips; rigid back wall underneath.

Parameters

Name	Description
wells	The well sequence of metadiffuser_reflection (MetadiffuserWell or None per well).
ax	Existing axes, or None to create a figure.
depth	Panel depth L, in metres.
period	Well pitch d, in metres.
language	Label language, "en" (default) or "es".
kwargs	Forwarded to the slit rectangles.

Returns: The axes.