building.prediction.aperture_transmission

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 $k=2\pi f/c_0$, $K=kw$ (w slit width), $X=d/w$ (d slit depth) and the end correction $e=(1/\pi )(\ln (8/K)-0.57722)$:

$$\tau =\frac{mK{\cos }^2(Ke)}{2n^2\left[\frac{{\sin }^2(KX+2Ke)}{{\cos }^2(Ke)}+\frac{K^2}{2n^2}\left(1+\cos (KX)\cos (KX+2Ke)\right)\right]}$$

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, $z=1,2,3,\dots$).

Circular aperture (Hopkins Eq. 4.102, Wilson & Soroka). With the piston radiation resistance $R_0(2ka)=1-2J_1(2ka)/(2ka)$ and reactance $X_0(2ka)=2H_1(2ka)/(2ka)$ ($J_1$ Bessel, $H_1$ Struve; radius a, depth d):

$$\tau =\frac{4R_0}{4R_0^2{\left[\cos (kd)-X_0\sin (kd)\right]}^2+{\left[\left(R_0^2-X_0^2+1\right)\sin (kd)+2X_0\cos (kd)\right]}^2}$$

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

$$R=-10{\log }_{10}\left[\frac{1}{\sum S_n}\sum S_{n}10^{-R_n/10}\right]$$

so a bare opening ($R=0$, $\tau =1$) of relative area $S_{\mathrm{a}}/S$ caps the composite at $10{\log }_{10}(S/S_{\mathrm{a}})$. This is the same energetic combination used by the EN 12354-3/-4 facade model of phonometry.building.prediction.facade.

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ApertureTransmissionResult

ApertureTransmissionResult(
    frequencies: np.ndarray,
    transmission_coefficient: np.ndarray,
    kind: str,
    width: float | None = None,
    radius: float | None = None,
    depth: float | None = None,
)

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".
width	Slit width, in metres, retained (with depth) so plot_geometry can draw the section; appended after the original fields and None for circular apertures or hand-built results.
radius	Circular-aperture radius, in metres, or None.
depth	Wall thickness, in metres, or None.

ApertureTransmissionResult.plot()

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.plot_geometry()

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

Draw the wall-aperture cross-section to scale.

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

Raises

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

ApertureTransmissionResult.transmission_loss

property

Aperture sound reduction index $R=-10{\log }_{10}(\tau )$ per band, dB.

circular_aperture_transmission_coefficient

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 (kind "circular").

Raises

Exception	When
ValueError	for a non-positive input.

composite_transmission_loss

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

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

The area-weighted energy combination of N elements (wall, window, slit, open aperture …) sharing a partition:

$$R=-10{\log }_{10}\left[\frac{1}{\sum S_n}\sum S_{n}10^{-R_n/10}\right]$$

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.

plot_aperture_geometry

plot_aperture_geometry(
    depth: float,
    ax: Axes | None = None,
    *,
    width: float | None = None,
    radius: float | None = None,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the section through a wall aperture to scale.

Wall of thickness depth with a slit of the given width (or the diametral section of a circular hole of the given radius), incident sound on the left and the transmitted wavefronts sketched on the right. Give exactly one of width/radius, matching slit_transmission_coefficient / circular_aperture_transmission_coefficient.

Parameters

Name	Description
depth	Wall thickness d, in metres.
ax	Existing axes, or None to create a figure.
width	Slit width w, in metres.
radius	Circular-hole radius, in metres.
language	Label language, "en" (default) or "es".
kwargs	Forwarded to the wall rectangles.

Returns: The axes.

slit_resonance_frequencies

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

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 (kind "slit").

Raises

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

transmission_loss_from_coefficient

transmission_loss_from_coefficient(tau: ArrayLike) -> np.ndarray

Sound reduction index $R=-10{\log }_{10}(\tau )$ 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.