---
title: "materials.impedance_tube"
description: "Public API of phonometry.materials.impedance_tube (auto-generated)."
sidebar:
  label: "impedance_tube"
---

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

Impedance-tube material characterisation.

Three complementary standardised methods are implemented, each kept in its
own sign convention (they are **not** interchangeable):

* **BS EN ISO 10534-2:2001** - two-microphone transfer-function method. The
  complex reflection factor `r` at the sample surface is obtained from the
  measured transfer function `H12` between two microphones, and from it the
  surface impedance and the normal-incidence absorption coefficient
  (Clause 7, Eqs. (17)-(20)). Time convention `e^{+j w t}`; the incident
  wave carries `e^{+j k0 x}` and the reflected wave `e^{-j k0 x}` (Annex D,
  Eqs. (D.1)-(D.8)). The complex wavenumber is `k0 = k0' - j k0''` with the
  attenuation constant `k0''` (Clause 2.6, Annex A). Air properties from
  Clause 7.2, Eqs. (5)/(7), use temperature in **kelvin**.

* **BS EN ISO 10534-1:2001** - standing-wave-ratio method. The reflection
  magnitude, phase, absorption coefficient and normalised impedance follow
  from the measured standing-wave ratio and the position of the first pressure
  minimum (Clause 5, Eqs. (12)-(26)).

* **ASTM E2611-19** - four-microphone transfer-matrix method. The wave field
  is decomposed into forward/backward amplitudes on each side of the specimen
  (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)) follow. Time convention `e^{+j w 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**.

The two standards adopt different sign ansaetze and different temperature
units on purpose; the helpers are named per standard so the two are never
mixed.

## absorption_from_reflection

```python
absorption_from_reflection(reflection: ArrayLike) -> Real
```

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

`alpha = 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_astm

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

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

`rho = 1,290 * (P / 101,325) * (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_density_iso

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

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

`rho = rho0 * (pa * T0) / (p0 * T)` with `rho0 = 1,186 kg/m3`,
`T0 = 293 K` and `p0 = 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.

## air_layer_transfer_matrix

```python
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 rho c sin(k d)], [j sin(k d) / (rho 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, `T11 = T22`).

**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`](/phonometry/reference/api/materials/impedance-tube/#transfermatrix).

## apply_mic_calibration

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

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

`H12 = H12_uncorrected / Hc`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `h12_uncorrected` | Uncorrected measured transfer function. |
| `calibration_factor` | Calibration factor `Hc` from [`mic_calibration_factor`](/phonometry/reference/api/materials/impedance-tube/#mic_calibration_factor). |

**Returns:** Corrected transfer function `H12`.

## characteristic_impedance

```python
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.

## face_quantities

```python
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)).

`p0 = A + B`, `pd = C e^{-j k d} + D e^{+j k d}`,
`u0 = (A - B) / (rho c)`, `ud = (C e^{-j k d} - D e^{+j k d}) / (rho 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.

## ImpedanceTubeResult

```python
ImpedanceTubeResult(
    frequency: Real,
    reflection: Complex,
    surface_impedance: Complex,
    normalized_impedance: Complex,
    absorption: Real,
)
```

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 / (rho c0)` (Eq. (19))
and `absorption` the normal-incidence coefficient `alpha = 1 - |r|^2`
(Eq. (18)).

### ImpedanceTubeResult.plot()

```python
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.report()

```python
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 / (rho c0)`) 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`](/phonometry/reference/api/building/insulation/#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`, `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]`). |

## ImpedanceTubeWarning

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

## mic_calibration_factor

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

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

`Hc = sqrt(H12^I / H12^II)` 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

```python
normalized_surface_admittance(reflection: ArrayLike) -> Complex
```

Normalised surface admittance `G rho c0` (ISO 10534-2, Eq. (20)).

`G rho c0 = (rho c0) / Z = (1 - r) / (1 + r)`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `reflection` | Complex reflection factor `r`. |

**Returns:** Normalised surface admittance (complex).

## normalized_surface_impedance

```python
normalized_surface_impedance(reflection: ArrayLike) -> Complex
```

Normalised surface impedance `Z / (rho c0)` (ISO 10534-2, Eq. (19)).

`Z / (rho c0) = (1 + r) / (1 - r)`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `reflection` | Complex reflection factor `r`. |

**Returns:** Normalised surface impedance `Z / (rho c0)` (complex).

## plane_wave_frequency_range

```python
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 c0` (Eq. (4)) and, when the tube `diameter` is given, the
cut-on bound `f_u d < 0,58 c0` for a circular tube (Eq. (2)) or
`< 0,50 c0` 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 = c0 / (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) `d`, in metres; `None` applies only the spacing bound. |
| `shape` | `"circular"` or `"rectangular"`. |

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

## reflection_factor

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

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

`r = ((H12 - HI) / (HR - H12)) * exp(+2 j k0 x1)` with the incident- and
reflected-wave transfer functions `HI = exp(-j k0 s)` (Eq. (D.5)) and
`HR = exp(+j k0 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`](/phonometry/reference/api/materials/impedance-tube/#apply_mic_calibration)). |
| `spacing` | Microphone spacing `s = x1 - x2`, in metres. |
| `x1` | Distance from the sample surface to the farther microphone (position 1), in metres. |
| `wavenumber` | Complex wavenumber `k0` (from [`tube_wavenumber`](/phonometry/reference/api/materials/impedance-tube/#tube_wavenumber)), scalar or per band. |

**Returns:** Complex reflection factor `r` at the reference plane.

## speed_of_sound_astm

```python
speed_of_sound_astm(temperature: ArrayLike) -> Real
```

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

`c = 20,047 * sqrt(273,15 + T)`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `temperature` | Room temperature `T`, in **degrees Celsius**. |

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

## speed_of_sound_iso

```python
speed_of_sound_iso(temperature: ArrayLike) -> Real
```

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

`c0 = 343,2 * sqrt(T / 293)`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `temperature` | Air temperature `T`, in **kelvin**. |

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

## standing_wave_absorption

```python
standing_wave_absorption(swr: ArrayLike) -> Real
```

Absorption coefficient from the standing-wave ratio (ISO 10534-1).

Combining `alpha = 1 - |r|^2` (Eq. (9)) with `|r| = (s - 1)/(s + 1)`
(Eq. (14)) gives `alpha = 4 s / (s + 1)^2`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `swr` | Standing-wave ratio `s` (>= 1). |

**Returns:** Absorption coefficient `alpha` in `[0, 1]`.

## standing_wave_normalized_impedance

```python
standing_wave_normalized_impedance(
    swr: ArrayLike,
    first_min_distance: ArrayLike,
    wavelength: ArrayLike,
) -> Complex
```

Normalised impedance from the standing wave (ISO 10534-1, Eqs. (24)-(26)).

`z = Z / Z0 = (1 + r) / (1 - r)`; the real/imaginary split is Eqs. (25)/(26).

**Parameters**

| Name | Description |
| :--- | :--- |
| `swr` | Standing-wave ratio `s` (>= 1). |
| `first_min_distance` | Distance `x_min1` to the first minimum, in metres. |
| `wavelength` | Wavelength `lambda0`, in metres. |

**Returns:** Normalised surface impedance `z` (complex).

## standing_wave_ratio_from_level

```python
standing_wave_ratio_from_level(level_difference: ArrayLike) -> Real
```

Standing-wave ratio from a level difference (ISO 10534-1, Eq. (15)).

`s = 10^(dL / 20)` with `dL = L_max - L_min` in decibels.

**Parameters**

| Name | Description |
| :--- | :--- |
| `level_difference` | Level difference `dL = L_max - L_min`, in dB. |

**Returns:** Standing-wave ratio `s` (>= 1).

## standing_wave_reflection

```python
standing_wave_reflection(
    swr: ArrayLike,
    first_min_distance: ArrayLike,
    wavelength: ArrayLike,
) -> Complex
```

Complex reflection factor from the standing wave (ISO 10534-1, Eqs. (17)-(23)).

`r = |r| e^{j phi}` with `|r| = (s - 1)/(s + 1)` (Eq. (14)) and the
phase at the first pressure minimum `phi = pi (4 x_min1 / lambda0 - 1)`
(Eq. (20)).

**Parameters**

| Name | Description |
| :--- | :--- |
| `swr` | Standing-wave ratio `s` (>= 1). |
| `first_min_distance` | Distance `x_min1` from the reference plane to the first pressure minimum (toward the source), in metres. |
| `wavelength` | Wavelength `lambda0`, in metres (Eq. (27)). |

**Returns:** Complex reflection factor `r`.

## standing_wave_reflection_magnitude

```python
standing_wave_reflection_magnitude(swr: ArrayLike) -> Real
```

Reflection magnitude from the standing-wave ratio (ISO 10534-1, Eq. (14)).

`|r| = (s - 1) / (s + 1)`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `swr` | Standing-wave ratio `s` (>= 1). |

**Returns:** Reflection magnitude `|r|` in `[0, 1]`.

## surface_impedance

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

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

`Z = rho c0 * (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).

## transfer_matrix_one_load

```python
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,
) -> TransferMatrix
```

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

Valid only for a reciprocal **and** symmetric specimen (`T11 = T22` and
`T11 T22 - T12 T21 = 1`, Eq. (23)). A single termination suffices:

```text
DEN = p0 ud + pd u0
T11 = T22 = (pd ud + p0 u0) / DEN
T12 = (p0^2 - pd^2) / DEN
T21 = (u0^2 - ud^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`. |

**Returns:** The specimen [`TransferMatrix`](/phonometry/reference/api/materials/impedance-tube/#transfermatrix).

## transfer_matrix_two_load

```python
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,
) -> 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)):

```text
DEN = p_da u_db - p_db u_da
T11 = (p0a u_db - p0b u_da) / DEN
T12 = (p0b p_da - p0a p_db) / DEN
T21 = (u0a u_db - u0b u_da) / DEN
T22 = (p_da u0b - p_db u0a) / 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`. |

**Returns:** The specimen [`TransferMatrix`](/phonometry/reference/api/materials/impedance-tube/#transfermatrix).

## TransferMatrix

```python
TransferMatrix(t11: Complex, t12: Complex, t21: Complex, t22: Complex)
```

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.

### TransferMatrix.absorption_hard_backed()

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

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

`alpha = 1 - |R|^2`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `characteristic_impedance` | Characteristic impedance `rho c`. |

**Returns:** Absorption coefficient `alpha`.

### TransferMatrix.characteristic_impedance_material()

```python
TransferMatrix.characteristic_impedance_material() -> Complex
```

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

`Z = sqrt(T12 / T21)`.

**Returns:** Complex characteristic impedance `Z`, in rayls.

### TransferMatrix.determinant()

```python
TransferMatrix.determinant() -> Complex
```

Determinant `T11 T22 - T12 T21` (unity for a reciprocal specimen).

### TransferMatrix.material_wavenumber()

```python
TransferMatrix.material_wavenumber(thickness: float) -> Complex
```

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

`k' = arccos(T11) / d` (complex `arccos`).

**Parameters**

| Name | Description |
| :--- | :--- |
| `thickness` | Specimen thickness `d`, in metres. |

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

### TransferMatrix.reflection_hard_backed()

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

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

`R = (T11 - rho c T21) / (T11 + rho c T21)`.

**Parameters**

| Name | Description |
| :--- | :--- |
| `characteristic_impedance` | Characteristic impedance `rho c`. |

**Returns:** Complex reflection coefficient `R`.

### TransferMatrix.transmission_loss()

```python
TransferMatrix.transmission_loss(characteristic_impedance: float) -> Real
```

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

With `t = 2 e^{j k d} / (T11 + T12/(rho c) + rho c T21 + T22)`
(Eq. (25)), `TL = 20 log10 |1/t| = 20 log10 |T11 + T12/(rho c) +
rho c T21 + T22| / 2` (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.

## tube_attenuation_constant

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

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

`k0'' = 1,94e-2 * sqrt(f) / (c0 * 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. |

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

## tube_wavenumber

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

Complex wavenumber `k0 = k0' - j k0''` (ISO 10534-2, Clause 2.6).

The real part is `k0' = 2 pi f / c0` (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`](/phonometry/reference/api/materials/impedance-tube/#tube_attenuation_constant). |

**Returns:** Complex wavenumber `k0`, in reciprocal metres.

## two_microphone_impedance

```python
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`](/phonometry/reference/api/materials/impedance-tube/#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`](/phonometry/reference/api/materials/impedance-tube/#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`](/phonometry/reference/api/materials/impedance-tube/#tube_attenuation_constant)). |
| `diameter` | Optional tube diameter/lateral dimension, in metres, that activates the plane-wave range check. |
| `shape` | Tube cross-section, `"circular"` or `"rectangular"`. |

**Returns:** An [`ImpedanceTubeResult`](/phonometry/reference/api/materials/impedance-tube/#impedancetuberesult).

## wave_decomposition

```python
wave_decomposition(
    h1: ArrayLike,
    h2: ArrayLike,
    h3: ArrayLike,
    h4: ArrayLike,
    *,
    l1: float,
    s1: float,
    l2: float,
    s2: float,
    wavenumber: ArrayLike,
) -> 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:

```text
A = j (H1 e^{-j k l1}       - H2 e^{-j k (l1+s1)}) / (2 sin(k s1))
B = j (H2 e^{+j k (l1+s1)}  - H1 e^{+j k l1})      / (2 sin(k s1))
C = j (H3 e^{+j k (l2+s2)}  - H4 e^{+j k l2})      / (2 sin(k s2))
D = j (H4 e^{-j k l2}       - H3 e^{-j k (l2+s2)}) / (2 sin(k s2))
```

`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 w 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 `l1 + s1`), and to `H3` nearest the downstream side at `l2`
(and `H4` at `l2 + s2`), 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`](/phonometry/reference/api/materials/impedance-tube/#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. |

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