materials.absorbers.impedance_tube

Two-microphone transfer-function method in the impedance tube.

BS EN ISO 10534-2:2001: the complex reflection factor r at the sample surface is obtained from the measured transfer function H12 between two microphones flush-mounted in the wall of a tube terminated by the specimen, and from it the surface impedance and the normal-incidence absorption coefficient (Clause 7, Eqs. (17)-(20)). Time convention $e^{+ j ω t}$ ; the incident wave carries $e^{+ j k_{0} x}$ and the reflected wave $e^{- j k_{0} x}$ (Annex D, Eqs. (D.1)-(D.8)). The complex wavenumber is $k_{0} = k_{0}^{'} - j k_{0}^{''}$ with the attenuation constant $k_{0}^{''}$ (Clause 2.6, Annex A). Air properties from Clause 7.2, Eqs. (5)/(7), use temperature in kelvin.

The tube itself is described here as well - its cross-section, the hydraulic diameter a rectangular tube reports, the complex wavenumber, the lower-bound wall attenuation and the plane-wave working range - because a specimen is characterised only where the field in the tube is a plane wave, and because the four-microphone method measures in the same tube and reuses that arithmetic.

The other two standardised impedance-tube methods are their own modules, each kept in its own sign convention (they are not interchangeable): standing_wave for the probe-traverse standing-wave-ratio method of BS EN ISO 10534-1:2001, and four_microphone for the transmission transfer-matrix method of ASTM E2611-19, whose air properties are given in degrees Celsius and whose forward wave carries the opposite exponent sign.

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absorption_from_reflection

absorption_from_reflection(reflection: ArrayLike) -> Real

Normal-incidence absorption coefficient (ISO 10534-2, Eq. (18)).

$α = 1 - ∣ r ∣^{2}$ . This form is shared with ISO 10534-1 Eq. (9) and ASTM E2611-19 Eq. (28).

Parameters

Name	Description
`reflection`	Complex reflection factor `r`.

Returns: Absorption coefficient alpha (real).

air_density_iso

air_density_iso(
    temperature: ArrayLike,
    atmospheric_pressure: ArrayLike = 101.325,
) -> Real

Air density (ISO 10534-2:2001, Eq. (7)).

$ρ = ρ_{0} (p_{a} T_{0}) / (p_{0} T)$ with $ρ_{0} = 1.186$ kg/m3, $T_{0} = 293$ K and $p_{0} = 101.325$ kPa.

Parameters

Name	Description
`temperature`	Air temperature `T`, in kelvin.
`atmospheric_pressure`	Atmospheric pressure `pa`, in kilopascals (default 101,325 kPa).

Returns: Air density rho, in kilograms per cubic metre.

apply_mic_calibration

apply_mic_calibration(
    h12_uncorrected: ArrayLike,
    calibration_factor: ArrayLike,
) -> Complex

Apply the microphone calibration factor (ISO 10534-2, Eq. (13)).

$H_{12} = H_{12, uncorrected} / H_{c}$ .

Parameters

Name	Description
`h12_uncorrected`	Uncorrected measured transfer function.
`calibration_factor`	Calibration factor `Hc` from `mic_calibration_factor`.

Returns: Corrected transfer function H12.

characteristic_impedance

characteristic_impedance(density: float, speed_of_sound: float) -> float

Characteristic impedance of air rho c (rayls).

A convenience for both standards (ISO 10534-2 Clause 7.2; ASTM E2611-19 Clause 8.2/8.3): the real product of air density and speed of sound.

Parameters

Name	Description
`density`	Air density `rho`, in kg/m3.
`speed_of_sound`	Speed of sound `c`, in m/s.

Returns: Characteristic impedance rho c, in rayls.

hydraulic_diameter

hydraulic_diameter(width: float, height: float) -> float

Hydraulic diameter of a rectangular tube, $4 A / P$ (ISO 10534-2, A.2.1.5).

For a rectangular cross-section of side lengths w and h the ratio of four times the area to the perimeter reduces to $d_{h} = 2 w h / (w + h)$ ; a square tube gives d_h equal to the side length. This is the d the Eq. (A.18) attenuation estimate expects for rectangular tubes (see tube_attenuation_constant).

Parameters

Name	Description
`width`	Inner side length `w`, in metres.
`height`	Inner side length `h`, in metres.

Returns: Hydraulic diameter $d_{h} = 4 A / P$ , in metres.

ImpedanceTubeResult

ImpedanceTubeResult(
    frequency: Real,
    reflection: Complex,
    surface_impedance: Complex,
    normalized_impedance: Complex,
    absorption: Real,
    spacing: float | None = None,
    x1: float | None = None,
    diameter: float | None = None,
    shape: str | None = None,
)

Two-microphone impedance-tube result (ISO 10534-2:2001).

All arrays share the shape of frequency. reflection is the complex reflection factor r at the sample surface (Eq. (17)), surface_impedance the absolute surface impedance Z in rayls (Eq. (19)), normalized_impedance the ratio $Z / (ρ c_{0})$ (Eq. (19)) and absorption the normal-incidence coefficient $α = 1 - ∣ r ∣^{2}$ (Eq. (18)).

The trailing fields retain the tube geometry the reduction was run with (microphone spacing s, distance x1 from the sample to the farther microphone, tube diameter and cross-section shape, stored canonically as "circular"/"rectangular" - a "square" input is kept as "rectangular"); they default to None when not supplied to two_microphone_impedance.

ImpedanceTubeResult.plot()

ImpedanceTubeResult.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.

ImpedanceTubeResult.plot_geometry()

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

Draw the two-microphone tube to scale (dimensioned side view).

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

Raises

Exception	When
ValueError	If the result does not retain its tube geometry (`spacing`/`x1`).

ImpedanceTubeResult.report()

ImpedanceTubeResult.report(
    path: str,
    *,
    metadata: ReportMetadata | None = None,
    engine: str = 'reportlab',
    verbose: bool = False,
    language: str = 'en',
) -> str

Render an ISO 10534-2 impedance-tube test-report fiche to a PDF.

Writes a one-page accredited normal-incidence report (BS EN ISO 10534-2:2001, two-microphone transfer-function method): the standard-basis line, an optional metadata header block (client, specimen, tube diameter d, microphone spacing s, the measured frequency range, mounting, climate …), a two-panel body with the per-frequency table (frequency, absorption alpha and the real/imaginary parts of the normalised surface impedance $z = Z / (ρ c_{0})$ ) beside the alpha(f) curve, and a footer with the fixed disclaimer. ISO 10534-2 is a characterisation, so there is no pass/fail verdict and no single-number rating (the random-incidence weighted alpha_w is an ISO 11654 / ISO 354 quantity, not comparable to the normal-incidence coefficient reported here).

Parameters

Name	Description
`path`	Destination path of the PDF file.
`metadata`	Optional `ReportMetadata`; `None` produces a body-and-disclaimer fiche whose header shows only the measured frequency range. The applicable descriptive/geometric fields are `client`, `manufacturer`, `specimen`, `tube_diameter`, `tube_shape`, `mic_spacing`, `mounting`, `test_room`, `test_date`, `temperature`, `pressure`, `measurement_standard`, `laboratory`, `operator`, `report_id` and `notes`. The `requirement` field is ignored (ISO 10534-2 has no verdict).
`engine`	Rendering back end; only `"reportlab"` is supported.
`verbose`	When `True`, the value table inserts the reflection-factor magnitude `\|r\|` column.
`language`	Fiche language: `"en"` (default, English, decimal point) or `"es"` (Spanish, decimal comma).

Returns: The written path as a str.

Raises

Exception	When
ValueError	If `engine` is not `"reportlab"`.
ImportError	If reportlab is not installed (`pip install phonometry[report]`), or matplotlib is missing for the embedded figure (`pip install phonometry[plot]`).

ImpedanceTubeWarning

Advisory for out-of-plane-wave-range impedance-tube frequencies.

mic_calibration_factor

mic_calibration_factor(
    h12_config1: ArrayLike,
    h12_config2: ArrayLike,
) -> Complex

Microphone-mismatch calibration factor Hc (ISO 10534-2, Eq. (10)).

$H_{c} = H_{12}^{I} / H_{12}^{I I}$ from a transfer function measured on an absorptive specimen in the standard configuration (I) and with the two microphones physically interchanged (II) - the cabling to the analyser is not swapped (Clause 7.5.2).

Parameters

Name	Description
`h12_config1`	Transfer function `H12^I` in the standard configuration.
`h12_config2`	Transfer function `H12^II` with microphones swapped.

Returns: Complex calibration factor Hc.

normalized_surface_admittance

normalized_surface_admittance(reflection: ArrayLike) -> Complex

Normalised surface admittance $Gρ c_{0}$ (ISO 10534-2, Eq. (20)).

$Gρ c_{0} = (ρ c_{0}) / Z = (1 - r) / (1 + r)$ .

Parameters

Name	Description
`reflection`	Complex reflection factor `r`.

Returns: Normalised surface admittance (complex).

normalized_surface_impedance

normalized_surface_impedance(reflection: ArrayLike) -> Complex

Normalised surface impedance $Z / (ρ c_{0})$ (ISO 10534-2, Eq. (19)).

$Z / (ρ c_{0}) = (1 + r) / (1 - r)$ .

Parameters

Name	Description
`reflection`	Complex reflection factor `r`.

Returns: Normalised surface impedance $Z / (ρ c_{0})$ (complex).

plane_wave_frequency_range

plane_wave_frequency_range(
    spacing: float,
    speed_of_sound: float,
    *,
    diameter: float | None = None,
    shape: str = 'circular',
) -> tuple[float, float]

Working plane-wave frequency range (f_l, f_u) (ISO 10534-2, 4.2-4.5).

The upper limit is the smaller of the microphone-spacing bound $f_{u} s < 0.45 c_{0}$ (Eq. (4)) and, when the tube diameter is given, the cut-on bound $f_{u} d < 0.58 c_{0}$ for a circular tube (Eq. (2)) or $f_{u} d < 0.50 c_{0}$ for a rectangular tube (Eq. (3)). The lower limit uses the Clause 4.2 guideline that the spacing exceed 5 % of the wavelength, i.e. $f_{l} = c_{0} / (20 s)$ .

Parameters

Name	Description
`spacing`	Microphone spacing `s`, in metres.
`speed_of_sound`	Speed of sound `c0`, in metres per second.
`diameter`	Tube diameter (circular) or maximum lateral dimension (rectangular/square) `d`, in metres; `None` applies only the spacing bound.
`shape`	`"circular"`, `"rectangular"` or `"square"` (a square tube is the rectangular bound with `d` the side length).

Returns: Tuple (f_l, f_u) of the lower and upper frequency limits, in Hz.

plot_impedance_tube_geometry

plot_impedance_tube_geometry(
    ax: Axes | None = None,
    *,
    spacing: float,
    x1: float,
    diameter: float | None = None,
    shape: str | None = 'circular',
    sample_thickness: float | None = None,
    speed_of_sound: float = 343.2,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the ISO 10534-2 two-microphone tube to scale.

Side view: loudspeaker at the left (three tube diameters before the farther microphone, the Clause 4.3 margin), the two flush microphones at x1 and x1 - s from the sample face, the sample against its rigid backing at the right, the cross-section emblem, and the plane-wave working range of plane_wave_frequency_range.

Parameters

Name	Description
`ax`	Existing axes, or `None` to create a figure.
`spacing`	Microphone spacing `s`, in metres.
`x1`	Distance from the sample face to the farther microphone, in metres.
`diameter`	Inner diameter (circular) or lateral dimension (rectangular/square), in metres; `None` draws a nominal bore and omits the bore dimension and the cut-on bound.
`shape`	`"circular"`, `"rectangular"`, `"square"` or `None`.
`sample_thickness`	Drawn sample thickness, in metres; `None` draws a 50 mm nominal sample.
`speed_of_sound`	Speed of sound for the working range, in m/s.
`language`	Label language, `"en"` (default) or `"es"`.
`kwargs`	Forwarded to the tube-bore rectangle.

Returns: The axes.

plot_transmission_tube_geometry

plot_transmission_tube_geometry(
    ax: Axes | None = None,
    *,
    l1: float,
    s1: float,
    l2: float,
    s2: float,
    thickness: float,
    diameter: float | None = None,
    shape: str | None = 'circular',
    speed_of_sound: float = 343.2,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the ASTM E2611 four-microphone transmission tube to scale.

Side view: loudspeaker, the two upstream microphones at l1 and l1 + s1 from the front face of the specimen, the specimen spanning its thickness, the two downstream microphones at l2 and l2 + s2 (measured from the front face, the module’s locked convention), and the changeable termination of the two-load method, with the ASTM working range of plane_wave_frequency_range_astm.

Parameters

Name	Description
`ax`	Existing axes, or `None` to create a figure.
`l1`	Front face to the nearer upstream microphone, in metres.
`s1`	Upstream microphone spacing, in metres.
`l2`	Front face to the nearer downstream microphone, in metres.
`s2`	Downstream microphone spacing, in metres.
`thickness`	Specimen thickness, in metres; must be smaller than `l2` (the downstream microphones sit past the back face).
`diameter`	Inner diameter (circular) or largest section dimension (rectangular/square), in metres; `None` draws a nominal bore and omits the bore dimension and the cut-on bound.
`shape`	`"circular"`, `"rectangular"`, `"square"` or `None`.
`speed_of_sound`	Speed of sound for the working range, in m/s.
`language`	Label language, `"en"` (default) or `"es"`.
`kwargs`	Forwarded to the tube-bore rectangle.

Returns: The axes.

reflection_factor

reflection_factor(
    h12: ArrayLike,
    *,
    spacing: float,
    x1: float,
    wavenumber: ArrayLike,
) -> Complex

Complex reflection factor at the sample surface (ISO 10534-2, Eq. (17)).

r = \frac{H _{12} - H _{I}}{H _{R} - H _{12}} e^{+ 2 j k_{0} x_{1}}

with the incident- and reflected-wave transfer functions $H_{I} = e^{- j k_{0} s}$ (Eq. (D.5)) and $H_{R} = e^{+ j k_{0} s}$ (Eq. (D.6)), s the microphone spacing and x1 the distance from the sample to the farther microphone (Clause 7.7).

Parameters

Name	Description
`h12`	Measured transfer function `H12` between microphone positions 1 and 2 (Clause 7.6, Eq. (14)); complex, scalar or per band. It must already be corrected for microphone mismatch (see `apply_mic_calibration`).
`spacing`	Microphone spacing $s = x_{1} - x_{2}$ , in metres.
`x1`	Distance from the sample surface to the farther microphone (position 1), in metres.
`wavenumber`	Complex wavenumber `k0` (from `tube_wavenumber`), scalar or per band.

Returns: Complex reflection factor r at the reference plane.

speed_of_sound_iso

speed_of_sound_iso(temperature: ArrayLike) -> Real

Speed of sound in air (ISO 10534-2:2001, Eq. (5)).

$c_{0} = 343.2 T /293$ .

Parameters

Name	Description
`temperature`	Air temperature `T`, in kelvin.

Returns: Speed of sound c0, in metres per second.

surface_impedance

surface_impedance(
    reflection: ArrayLike,
    characteristic_impedance: float,
) -> Complex

Absolute surface impedance Z (ISO 10534-2, Eq. (19)).

$Z = ρ c_{0} (1 + r) / (1 - r)$ .

Parameters

Name	Description
`reflection`	Complex reflection factor `r`.
`characteristic_impedance`	Characteristic impedance of air `rho c0`, in rayls (`rho` and `c0` from the Clause 7.2 helpers).

Returns: Surface impedance Z, in rayls (complex).

tube_attenuation_constant

tube_attenuation_constant(
    frequency: ArrayLike,
    speed_of_sound: float,
    diameter: float,
) -> Real

Lower-bound tube attenuation constant k0'' (ISO 10534-2, Eq. (A.18)).

$k_{0}^{''} = 1.94 \times 1 0^{- 2} f / (c_{0} d)$ (nepers per metre). This ignores porous-wall and object losses and is therefore a lower limit (Clause A.2.1.5).

Parameters

Name	Description
`frequency`	Frequency `f`, in hertz (scalar or per band).
`speed_of_sound`	Speed of sound `c0`, in metres per second.
`diameter`	Circular-tube diameter `d`, in metres, or the hydraulic diameter `4 * area / perimeter` for a rectangular tube (see `hydraulic_diameter`).

Returns: Attenuation constant k0'', in nepers per metre.

tube_wavenumber

tube_wavenumber(
    frequency: ArrayLike,
    speed_of_sound: float,
    *,
    attenuation: ArrayLike | None = None,
) -> Complex

Complex wavenumber $k_{0} = k_{0}^{'} - j k_{0}^{''}$ (ISO 10534-2, Clause 2.6).

The real part is $k_{0}^{'} = 2 π f / c_{0}$ (Eq. (2)); the optional attenuation constant k0'' enters with a minus sign on the imaginary part (Clause 2.6 NOTE, Eq. (A.1)).

Parameters

Name	Description
`frequency`	Frequency `f`, in hertz (scalar or per band).
`speed_of_sound`	Speed of sound `c0`, in metres per second.
`attenuation`	Attenuation constant `k0''`, in nepers per metre (scalar or matching `frequency`); `None` gives the lossless real wavenumber. Obtain a lower-bound estimate from `tube_attenuation_constant`.

Returns: Complex wavenumber k0, in reciprocal metres.

two_microphone_impedance

two_microphone_impedance(
    h12: ArrayLike,
    *,
    frequency: ArrayLike,
    spacing: float,
    x1: float,
    speed_of_sound: float,
    characteristic_impedance: float,
    attenuation: ArrayLike | None = None,
    diameter: float | None = None,
    shape: str = 'circular',
) -> ImpedanceTubeResult

Full two-microphone reduction (ISO 10534-2:2001, Clause 7).

Builds the complex wavenumber (Clause 2.6), the reflection factor (Eq. (17)), the surface impedance (Eq. (19)) and the absorption coefficient (Eq. (18)) from the measured transfer function H12. When diameter is supplied, frequencies outside the plane-wave range (Eqs. (1)-(4)) raise an ImpedanceTubeWarning; the results are still returned.

Parameters

Name	Description
`h12`	Measured (mismatch-corrected) transfer function `H12`.
`frequency`	Frequency vector `f`, in hertz.
`spacing`	Microphone spacing `s`, in metres.
`x1`	Distance from the sample to the farther microphone, in metres.
`speed_of_sound`	Speed of sound `c0`, in m/s (see `speed_of_sound_iso`).
`characteristic_impedance`	Characteristic impedance `rho c0`, in rayls.
`attenuation`	Optional tube attenuation constant `k0''`, in nepers/m (see `tube_attenuation_constant`).
`diameter`	Optional tube diameter/lateral dimension, in metres, that activates the plane-wave range check.
`shape`	Tube cross-section, `"circular"`, `"rectangular"` or `"square"`.

Returns: An ImpedanceTubeResult (the tube geometry is retained on the result).

Created and maintained by José Manuel Requena Plens ORCID All projects