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

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

Sound transmission through slits, holes and apertures (Hopkins 2007, Sound
Insulation, Section 4.3.10; Gomperts 1964; Wilson & Soroka 1965).

Air paths are the real limit on the sound insulation of an otherwise heavy
construction: a small slit or hole caps the achievable sound reduction index no
matter how massive the wall. This module predicts the transmission coefficient
`tau` of the two canonical apertures and combines them with the surrounding
wall into a composite sound reduction index, the practical answer to "why do I
never reach the catalogue `Rw`".

**Straight-edged slit (Hopkins Eq. 4.99, Gomperts).** With the acoustic
wavenumber `k = 2 pi f / c0`, `K = k w` (`w` slit width), `X = d / w`
(`d` slit depth) and the end correction `e = (1/pi)(ln(8/K) - 0.57722)`:

```text
tau = m K cos^2(Ke)
      / ( 2 n^2 [ sin^2(KX + 2Ke)/cos^2(Ke)
                  + (K^2 / 2 n^2)(1 + cos(KX) cos(KX + 2Ke)) ] )
```

where `m = 8` (diffuse field) or `4` (normal incidence), and `n = 1` (slit
in the middle of a plate) or `0.5` (slit along an edge). The model assumes an
inviscid air path; maxima in `tau` (dips in `R`) occur at the resonances
`d + 2e = z lambda/2` (Eq. 4.101, `z = 1, 2, 3, ...`).

**Circular aperture (Hopkins Eq. 4.102, Wilson & Soroka).** With the piston
radiation resistance `R0(2ka) = 1 - 2 J1(2ka)/(2ka)` and reactance
`X0(2ka) = 2 H1(2ka)/(2ka)` (`J1` Bessel, `H1` Struve; radius `a`,
depth `d`):

```text
tau = 4 R0 / ( 4 R0^2 [cos(kd) - X0 sin(kd)]^2
               + [(R0^2 - X0^2 + 1) sin(kd) + 2 X0 cos(kd)]^2 )
```

**Composite (Hopkins Eq. 4.92).** For elements of area `S_n` and sound
reduction index `R_n` the resultant is the area-weighted energy sum:

```text
R = -10 lg( (1 / sum S_n) sum S_n 10^(-R_n/10) )
```

so a bare opening (`R = 0`, `tau = 1`) of relative area `S_a/S` caps the
composite at `10 lg(S / S_a)`. This is the same energetic combination used by
the EN 12354-3/-4 facade model of [`phonometry.building.facade_prediction`](/phonometry/reference/api/building/facade-prediction/).

## ApertureTransmissionResult

```python
ApertureTransmissionResult(
    frequencies: np.ndarray,
    transmission_coefficient: np.ndarray,
    kind: str,
)
```

Transmission through a slit or circular aperture (Hopkins 4.3.10).

**Attributes**

| Name | Description |
| :--- | :--- |
| `frequencies` | Band centre frequencies, in hertz. |
| `transmission_coefficient` | Transmission coefficient `tau` per band. |
| `kind` | `"slit"` or `"circular"`. |

### ApertureTransmissionResult.plot()

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

Plot the aperture sound reduction index `R(f)`.

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

### ApertureTransmissionResult.transmission_loss

*property*

Aperture sound reduction index `R = -10 lg(tau)` per band, dB.

## circular_aperture_transmission_coefficient

```python
circular_aperture_transmission_coefficient(
    frequency: ArrayLike,
    radius: float,
    depth: float,
    *,
    speed_of_sound: float = 343.0,
) -> ApertureTransmissionResult
```

Transmission coefficient of a circular aperture (Hopkins Eq. 4.102).

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequency` | Band centre frequencies `f`, in hertz (array, > 0). |
| `radius` | Aperture radius `a`, in m (> 0). |
| `depth` | Aperture depth `d` (wall thickness), in m (> 0). |
| `speed_of_sound` | Speed of sound in air `c0` (Default: 343 m/s). |

**Returns:** An [`ApertureTransmissionResult`](/phonometry/reference/api/building/aperture-transmission/#aperturetransmissionresult) (kind `"circular"`).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive input. |

## composite_transmission_loss

```python
composite_transmission_loss(
    areas: ArrayLike,
    reduction_indices: ArrayLike,
) -> np.ndarray
```

Composite sound reduction index of parallel elements (Hopkins Eq. 4.92).

`R = -10 lg( (1/sum S_n) sum S_n 10^(-R_n/10) )`, the area-weighted
energy combination of `N` elements (wall, window, slit, open aperture ...)
sharing a partition. A bare opening enters with `R = 0` (`tau = 1`).

**Parameters**

| Name | Description |
| :--- | :--- |
| `areas` | Element areas `S_n`, in m^2 (1-D, length `N`, all > 0). |
| `reduction_indices` | Element sound reduction indices `R_n`, in dB. Either a 1-D array of length `N` (one value per element) or a 2-D array of shape `(N, M)` (`N` elements over `M` bands). |

**Returns:** The composite `R`: a scalar array for 1-D input, or one value per band (length `M`) for 2-D input.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive area, mismatched shapes, or an empty element set. |

## slit_resonance_frequencies

```python
slit_resonance_frequencies(
    depth: float,
    width: float,
    *,
    orders: int = 3,
    speed_of_sound: float = 343.0,
) -> np.ndarray
```

Slit resonance frequencies `d + 2e = z lambda/2` (Hopkins Eq. 4.101).

Maxima in the transmission coefficient (dips in `R`) occur where the
effective slit depth is a half-wavelength multiple. Solved iteratively
because the end correction `e` depends weakly on frequency.

**Parameters**

| Name | Description |
| :--- | :--- |
| `depth` | Slit depth `d`, in m (> 0). |
| `width` | Slit width `w`, in m (> 0). |
| `orders` | Number of resonance orders `z = 1..orders` (>= 1). |
| `speed_of_sound` | Speed of sound in air `c0` (Default: 343 m/s). |

**Returns:** The resonance frequencies (Hz), one per order.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive input, `orders < 1`, or a slit so wide relative to its depth that the effective depth `d + 2e` is non-positive (no resonance exists; `width` must be much less than the wavelength). |

## slit_transmission_coefficient

```python
slit_transmission_coefficient(
    frequency: ArrayLike,
    width: float,
    depth: float,
    *,
    field: str = 'diffuse',
    position: str = 'mid',
    speed_of_sound: float = 343.0,
) -> ApertureTransmissionResult
```

Transmission coefficient of a straight-edged slit (Hopkins Eq. 4.99).

**Parameters**

| Name | Description |
| :--- | :--- |
| `frequency` | Band centre frequencies `f`, in hertz (array, > 0). |
| `width` | Slit width `w`, in m (> 0). |
| `depth` | Slit depth `d` (wall thickness across the slit), in m (> 0). |
| `field` | `"diffuse"` (`m = 8`) or `"normal"` (`m = 4`). |
| `position` | `"mid"` (`n = 1`) or `"edge"` (`n = 0.5`). |
| `speed_of_sound` | Speed of sound in air `c0` (Default: 343 m/s). |

**Returns:** An [`ApertureTransmissionResult`](/phonometry/reference/api/building/aperture-transmission/#aperturetransmissionresult) (kind `"slit"`).

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive input or unknown field/position. |

## transmission_loss_from_coefficient

```python
transmission_loss_from_coefficient(tau: ArrayLike) -> np.ndarray
```

Sound reduction index `R = -10 lg(tau)` from a transmission coefficient.

**Parameters**

| Name | Description |
| :--- | :--- |
| `tau` | Transmission coefficient(s) `tau` (> 0). Values above 1 (a resonating aperture that transmits more than the incident intensity) give a negative `R`. |

**Returns:** The sound reduction index `R`, in dB.

**Raises**

| Exception | When |
| :--- | :--- |
| ValueError | for a non-positive coefficient. |
