materials.absorbers.four_microphone

Four-microphone transfer-matrix method for the transmission of a specimen.

ASTM E2611-19, the two-tube method: the specimen sits between an upstream and a downstream tube section with two microphones on each side, and the wave field is decomposed into forward/backward amplitudes on each side (Eqs. (17)-(20)). The face pressures and particle velocities are formed (Eq. (21)) and the transfer matrix [[T11, T12], [T21, T22]] is solved from a two-load (Eq. (22)) or a symmetric one-load (Eq. (24)) measurement. Transmission loss (Eq. (26)), hard-backed reflection/absorption (Eqs. (27)/(28)) and the material wavenumber/characteristic impedance (Eqs. (29)/(30)) all read out of those four poles, which is what makes the standard one subject: everything here exists to fill the matrix or to interpret it.

Time convention $e^{+ j ω t}$ with the forward wave carried by $e^{- j k x}$ (Eq. (21)); air properties from Clause 8.2/8.3, Eqs. (4)/(5), use temperature in degrees Celsius. Both differ from the ISO 10534-2 ansatz of impedance_tube, whose wavenumber is $k_{0} = k_{0}^{'} - j k_{0}^{''}$ and whose air properties take kelvin; the two are not interchangeable, so the air-property and working-range helpers are named per standard and each stays with the method that prescribes it. What the two transfer methods genuinely share - the tube cross-section and the plane-wave working-range arithmetic - is imported from that module.

Auto-generated from the source docstrings by scripts/generate_api_docs.py (make api-docs). Do not edit by hand.

air_density_astm

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

Air density (ASTM E2611-19, Eq. (5)).

$ρ = 1.290 \frac{P}{101.325} \frac{273.15}{273.15 + T}$ .

Parameters

Name	Description
`temperature`	Room temperature `T`, in degrees Celsius.
`atmospheric_pressure`	Atmospheric pressure `P`, in kilopascals (default 101,325 kPa).

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

air_layer_transfer_matrix

air_layer_transfer_matrix(
    wavenumber: ArrayLike,
    thickness: float,
    characteristic_impedance: float,
) -> TransferMatrix

Analytic transfer matrix of a pure air layer of thickness d.

T = [[cos (k d), j ρ c sin (k d)], [j sin (k d) / (ρ c), cos (k d)]]

the classical loss-free layer used to validate the ASTM E2611-19 reduction (it is reciprocal, $det (T) = 1$ , and symmetric, $T_{11} = T_{22}$ ).

Parameters

Name	Description
`wavenumber`	Air wavenumber `k`.
`thickness`	Layer thickness `d`, in metres.
`characteristic_impedance`	Characteristic impedance `rho c`, in rayls.

Returns: The air-layer TransferMatrix.

face_quantities

face_quantities(
    a: ArrayLike,
    b: ArrayLike,
    c: ArrayLike,
    d: ArrayLike,
    *,
    wavenumber: ArrayLike,
    thickness: float,
    characteristic_impedance: float,
) -> tuple[Complex, Complex, Complex, Complex]

Face pressures and particle velocities (ASTM E2611-19, Eq. (21)).

p_{0} = A + B, p_{d} = C e^{- j k d} + D e^{+ j k d}

u_{0} = \frac{A - B}{ρ c}, u_{d} = \frac{C e ^{- j k d} - D e ^{+ j k d}}{ρ c}

Parameters

Name	Description
`a`	Upstream forward amplitude `A`.
`b`	Upstream backward amplitude `B`.
`c`	Downstream forward amplitude `C`.
`d`	Downstream backward amplitude `D`.
`wavenumber`	Air wavenumber `k`.
`thickness`	Specimen thickness `d`, in metres.
`characteristic_impedance`	Characteristic impedance `rho c`, in rayls.

Returns: Tuple (p0, pd, u0, ud) of face pressures and velocities.

plane_wave_frequency_range_astm

plane_wave_frequency_range_astm(
    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) (ASTM E2611-19).

The upper limit is the smaller of the microphone-spacing bound $s \leq 0.8 c / (2 f_{u})$ , i.e. $f_{u} s < 0.40 c$ (6.5.4), and, when the tube diameter is given, the cut-on bound $f_{u} < K c / d$ with $K = 0.586$ for a circular tube (6.2.4.1, Eq. (2)) or $K = 0.500$ for a rectangular tube with d the largest section dimension (6.2.5). The lower limit follows 6.2.3: the spacing shall be greater than 1 % of the wavelength, i.e. $f_{l} = c / (100 s)$ .

With two different spacings s1/s2, call with the larger one for the upper bound and the smaller one for the lower bound (each bound is binding for every microphone pair).

Parameters

Name	Description
`spacing`	Microphone spacing `s`, in metres.
`speed_of_sound`	Speed of sound `c`, in metres per second.
`diameter`	Tube diameter (circular) or largest section dimension (rectangular/square) `d`, in metres; `None` applies only the spacing bound.
`shape`	`"circular"`, `"rectangular"` or `"square"`.

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

speed_of_sound_astm

speed_of_sound_astm(temperature: ArrayLike) -> Real

Speed of sound in air (ASTM E2611-19, Eq. (4)).

$c = 20.047 273.15 + T$ .

Parameters

Name	Description
`temperature`	Room temperature `T`, in degrees Celsius.

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

transfer_matrix_one_load

transfer_matrix_one_load(
    load: tuple[ArrayLike, ArrayLike, ArrayLike, ArrayLike],
    *,
    l1: float,
    s1: float,
    l2: float,
    s2: float,
    thickness: float,
    wavenumber: ArrayLike,
    characteristic_impedance: float,
    frequency: ArrayLike | None = None,
    diameter: float | None = None,
    shape: str = 'circular',
) -> TransferMatrix

One-load transfer matrix, symmetric specimen (ASTM E2611-19, Eqs. (23)-(24)).

Valid only for a reciprocal and symmetric specimen ( $T_{11} = T_{22}$ and $T_{11} T_{22} - T_{12} T_{21} = 1$ , Eq. (23)). A single termination suffices:

DEN T_{11} = T_{22} T_{12} T_{21} = p_{0} u_{d} + p_{d} u_{0} = (p_{d} u_{d} + p_{0} u_{0}) / DEN = (p_{0}^{2} - p_{d}^{2}) / DEN = (u_{0}^{2} - u_{d}^{2}) / DEN

Parameters

Name	Description
`load`	Microphone transfer functions `(H1, H2, H3, H4)`.
`l1`	Upstream reference distance `l1`, in metres.
`s1`	Upstream microphone spacing `s1`, in metres.
`l2`	Downstream reference distance `l2`, in metres.
`s2`	Downstream microphone spacing `s2`, in metres.
`thickness`	Specimen thickness `d`, in metres.
`wavenumber`	Air wavenumber `k`.
`characteristic_impedance`	Characteristic impedance `rho c`.
`frequency`	Optional frequency vector `f`, in hertz, retained on the result so `TransferMatrix.plot` needs no arguments.
`diameter`	Optional tube diameter (circular) or largest section dimension (rectangular/square), in metres, that activates the plane-wave working-range check (6.2.3-6.2.5, 6.5.4).
`shape`	Tube cross-section, `"circular"`, `"rectangular"` or `"square"`.

Returns: The specimen TransferMatrix (measurement context retained on the result).

transfer_matrix_two_load

transfer_matrix_two_load(
    load_a: tuple[ArrayLike, ArrayLike, ArrayLike, ArrayLike],
    load_b: tuple[ArrayLike, ArrayLike, ArrayLike, ArrayLike],
    *,
    l1: float,
    s1: float,
    l2: float,
    s2: float,
    thickness: float,
    wavenumber: ArrayLike,
    characteristic_impedance: float,
    frequency: ArrayLike | None = None,
    diameter: float | None = None,
    shape: str = 'circular',
) -> TransferMatrix

Two-load transfer matrix (ASTM E2611-19, Eqs. (17)-(22)).

Each load is the tuple (H1, H2, H3, H4) of the four microphone transfer functions measured with a different downstream termination. The two loads give four equations for the four unknowns (Eq. (22)):

DEN T_{11} T_{12} T_{21} T_{22} = p_{d a} u_{d b} - p_{d b} u_{d a} = (p_{0 a} u_{d b} - p_{0 b} u_{d a}) / DEN = (p_{0 b} p_{d a} - p_{0 a} p_{d b}) / DEN = (u_{0 a} u_{d b} - u_{0 b} u_{d a}) / DEN = (p_{d a} u_{0 b} - p_{d b} u_{0 a}) / DEN

Parameters

Name	Description
`load_a`	Microphone transfer functions `(H1, H2, H3, H4)` for load a.
`load_b`	Microphone transfer functions `(H1, H2, H3, H4)` for load b.
`l1`	Upstream reference distance `l1`, in metres.
`s1`	Upstream microphone spacing `s1`, in metres.
`l2`	Downstream reference distance `l2`, in metres.
`s2`	Downstream microphone spacing `s2`, in metres.
`thickness`	Specimen thickness `d`, in metres.
`wavenumber`	Air wavenumber `k`.
`characteristic_impedance`	Characteristic impedance `rho c`.
`frequency`	Optional frequency vector `f`, in hertz, retained on the result so `TransferMatrix.plot` needs no arguments.
`diameter`	Optional tube diameter (circular) or largest section dimension (rectangular/square), in metres, that activates the plane-wave working-range check (6.2.3-6.2.5, 6.5.4).
`shape`	Tube cross-section, `"circular"`, `"rectangular"` or `"square"`.

Returns: The specimen TransferMatrix (measurement context retained on the result).

TransferMatrix

TransferMatrix(
    t11: Complex,
    t12: Complex,
    t21: Complex,
    t22: Complex,
    l1: float | None = None,
    s1: float | None = None,
    l2: float | None = None,
    s2: float | None = None,
    thickness: float | None = None,
    diameter: float | None = None,
    shape: str | None = None,
    frequency: Real | None = None,
    air_characteristic_impedance: float | None = None,
)

Acoustic transfer matrix [[T11, T12], [T21, T22]] (ASTM E2611-19).

Relates the pressure and normal particle velocity across a specimen, $[p; u]_{x = 0} = T [p; u]_{x = d}$ (Eq. (16)). Each entry is complex and may be scalar or a per-frequency array of matching shape.

The trailing fields retain the measurement context when the matrix comes out of transfer_matrix_two_load / transfer_matrix_one_load (tube geometry l1/s1/l2/s2, specimen thickness, tube diameter and canonical cross-section shape, the frequency vector when supplied to the solver, and the air air_characteristic_impedance rho c); all default to None so a hand-built matrix (for example air_layer_transfer_matrix) is unchanged.

TransferMatrix.absorption_hard_backed()

TransferMatrix.absorption_hard_backed(
    characteristic_impedance: float,
) -> Real

Hard-backed absorption coefficient (ASTM E2611-19, Eq. (28)).

$α = 1 - ∣ R ∣^{2}$ .

Parameters

Name	Description
`characteristic_impedance`	Characteristic impedance `rho c`.

Returns: Absorption coefficient alpha.

TransferMatrix.characteristic_impedance_material()

TransferMatrix.characteristic_impedance_material() -> Complex

Characteristic impedance of the material (ASTM E2611-19, Eq. (30)).

$Z = T_{12} / T_{21}$ .

Returns: Complex characteristic impedance Z, in rayls.

TransferMatrix.determinant()

TransferMatrix.determinant() -> Complex

Determinant $T_{11} T_{22} - T_{12} T_{21}$ (unity for a reciprocal specimen).

TransferMatrix.material_wavenumber()

TransferMatrix.material_wavenumber(thickness: float) -> Complex

Propagation wavenumber inside the material (ASTM E2611-19, Eq. (29)).

$k^{'} = arccos (T_{11}) / d$ (complex arccos).

Parameters

Name	Description
`thickness`	Specimen thickness `d`, in metres.

Returns: Complex material wavenumber k', in reciprocal metres.

TransferMatrix.plot()

TransferMatrix.plot(
    frequency: ArrayLike | None = None,
    characteristic_impedance: float | None = None,
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot the transmission loss with the hard-backed absorption overlaid.

Reads the four-pole entries out as the two ASTM E2611-19 spectra a laboratory quotes: the normal-incidence transmission loss TLn(f) (Eq. (26), the primary curve, left axis) and the hard-backed absorption coefficient alpha(f) (Eq. (28), a muted companion on a 0..1 right axis). The four-pole entries carry no frequency axis of their own, so the plot needs the measurement’s frequency vector (matching the shape of the entries) and the air characteristic impedance rho c. A matrix built by the solvers retains both (self.frequency / self.air_characteristic_impedance), so plot() takes no arguments there; only a hand-built matrix (for example air_layer_transfer_matrix) must supply them.

Requires matplotlib (pip install phonometry[plot]); returns the Axes of the transmission-loss curve.

Parameters

Name	Description
`frequency`	Frequency vector `f`, in hertz, matching the shape of the matrix entries; `None` uses the stored `frequency`.
`characteristic_impedance`	Characteristic impedance `rho c` of the air in the tube, in rayls; `None` uses the stored `air_characteristic_impedance`.
`ax`	Existing axes, or `None` to create a figure.
`language`	Plot language: `"en"` (default) or `"es"`.
`kwargs`	Forwarded to the transmission-loss `plot` call.

Returns: The axes.

Raises

Exception	When
ValueError	If `frequency` or `characteristic_impedance` is neither supplied nor stored on the matrix.

TransferMatrix.plot_geometry()

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

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

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

Raises

Exception	When
ValueError	If the matrix does not retain its tube geometry (`l1`/`s1`/`l2`/`s2`/`thickness`).

TransferMatrix.reflection_hard_backed()

TransferMatrix.reflection_hard_backed(
    characteristic_impedance: float,
) -> Complex

Hard-backed reflection coefficient (ASTM E2611-19, Eq. (27)).

$R = (T_{11} - ρ c T_{21}) / (T_{11} + ρ c T_{21})$ .

Parameters

Name	Description
`characteristic_impedance`	Characteristic impedance `rho c`.

Returns: Complex reflection coefficient R.

TransferMatrix.transmission_loss()

TransferMatrix.transmission_loss(characteristic_impedance: float) -> Real

Normal-incidence transmission loss in dB (ASTM E2611-19, Eq. (26)).

With

t = \frac{2 e ^{j k d}}{T _{11} + T _{12} / ( ρ c ) + ρ c T _{21} + T _{22}} (Eq. 25)

T L = 20 lo g_{10} \frac{1}{t} = 20 lo g_{10} \frac{∣ T _{11} + T _{12} / ( ρ c ) + ρ c T _{21} + T _{22} ∣}{2} (Eq. 26)

(the $e^{j k d}$ factor has unit magnitude for a real wavenumber).

Parameters

Name	Description
`characteristic_impedance`	Characteristic impedance `rho c`.

Returns: Transmission loss TLn, in decibels.

wave_decomposition

wave_decomposition(
    h1: ArrayLike,
    h2: ArrayLike,
    h3: ArrayLike,
    h4: ArrayLike,
    *,
    l1: float,
    s1: float,
    l2: float,
    s2: float,
    wavenumber: ArrayLike,
    diameter: float | None = None,
    shape: str = 'circular',
) -> tuple[Complex, Complex, Complex, Complex]

Decompose the wave field into (A, B, C, D) (ASTM E2611-19, Eqs. (17)-(20)).

The exponents are implemented exactly as printed:

A = \frac{j ( H _{1} e ^{- j k l_{1}} - H _{2} e ^{- j k (l_{1} + s_{1})} )}{2 sin ( k s _{1} )}

B = \frac{j ( H _{2} e ^{+ j k (l_{1} + s_{1})} - H _{1} e ^{+ j k l_{1}} )}{2 sin ( k s _{1} )}

C = \frac{j ( H _{3} e ^{+ j k (l_{2} + s_{2})} - H _{4} e ^{+ j k l_{2}} )}{2 sin ( k s _{2} )}

D = \frac{j ( H _{4} e ^{- j k l_{2}} - H _{3} e ^{- j k (l_{2} + s_{2})} )}{2 sin ( k s _{2} )}

A/B are the forward/backward complex amplitudes on the upstream (source) side and C/D those on the downstream side, all referenced to the front face $x = 0$ . With the $e^{+ j ω t}$ / forward- $e^{- j k x}$ convention these exponents correspond to the microphone whose transfer function is H2 sitting nearest the front face at distance l1 (and H1 at $l_{1} + s_{1}$ ), and to H3 nearest the downstream side at l2 (and H4 at $l_{2} + s_{2}$ ), with l1, l2 measured from the front reference plane. The convention was locked down against the analytic air-layer transfer matrix (see air_layer_transfer_matrix).

Parameters

Name	Description
`h1`	Transfer function `H1,ref` (upstream, farther microphone).
`h2`	Transfer function `H2,ref` (upstream, nearer microphone).
`h3`	Transfer function `H3,ref` (downstream, nearer microphone).
`h4`	Transfer function `H4,ref` (downstream, farther microphone).
`l1`	Distance `l1` from the front reference plane, in metres.
`s1`	Upstream microphone spacing `s1`, in metres.
`l2`	Distance `l2` from the front reference plane, in metres.
`s2`	Downstream microphone spacing `s2`, in metres.
`wavenumber`	Air wavenumber `k` (real or complex), scalar or per band.
`diameter`	Optional tube diameter (circular) or largest section dimension (rectangular/square), in metres, that activates the plane-wave working-range check (6.2.3-6.2.5, 6.5.4).
`shape`	Tube cross-section, `"circular"`, `"rectangular"` or `"square"`.

Returns: Tuple (A, B, C, D) of complex amplitudes.

Created and maintained by José Manuel Requena Plens ORCID All projects