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# Building Acoustics & Sound Insulation

This guide continues from the [Room Acoustics guide](room-acoustics.md):
the same impulse response, measured either side of a partition, yields its sound
insulation. Where room acoustics describes the sound field inside a single space,
building acoustics describes how much of that field passes *between* spaces. This
page follows the insulation chain — field airborne, impact and façade insulation
with single-number ratings (ISO 16283-1/2/3, ISO 717-1/2), the laboratory
characterisation of a building element (ISO 10140), the prediction of in-situ
performance from flanking transmission (EN 12354-1/2), and the measurement
uncertainty that qualifies every rating (ISO 12999-1).

## 1. Field insulation and single-number ratings (ISO 16283-1, ISO 717-1)

To rate a wall or floor, measure the energy-average level in the **source**
room ($L_1$) and the **receiving** room ($L_2$) per one-third-octave band
and form the level difference $D = L_1 - L_2$. Two normalisations make it
comparable between rooms. The **standardized level difference** references
the receiving-room reverberation time $T$ to $T_0 = 0.5$ s (so with
$T = 0.5$ s, $D_{nT} = D$ exactly), and the **apparent sound reduction
index** normalises by the partition area $S$ and the Sabine absorption area
$A$:

$$
D_{nT} = D + 10 \log_{10} \frac{T}{T_0}, \qquad
R' = D + 10 \log_{10} \frac{S}{A}, \qquad A = \frac{0.16\ V}{T}.
$$

Positions are energy-averaged with
$L = 10 \log_{10}\left( \frac{1}{n} \sum_i 10^{L_i/10} \right)$.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_insulation_setup_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_insulation_setup.svg" alt="Field airborne insulation setup: a loudspeaker in the source room, microphones energy-averaged in source and receiving rooms across the common partition" width="92%"></picture>

The band spectrum is collapsed to one number by the **reference-curve
method** of ISO 717-1: a fixed reference curve is shifted in 1 dB steps
toward the measured curve until the sum of *unfavourable* deviations
(where the measurement falls below the reference) is as large as possible
but not more than 32.0 dB (16 one-third-octave bands) or 10.0 dB (5 octave
bands). The rating (`Rw`, `R'w`, `DnT,w` …) is the shifted reference read at
500 Hz. The **spectrum adaptation terms** $C$ (pink noise) and $C_{tr}$
(urban traffic) add the low-frequency penalty of a real source.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insulation_rating_dark.png"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insulation_rating.png" alt="Measured one-third-octave sound reduction index with the shifted ISO 717-1 reference curve and the resulting weighted rating at 500 Hz" width="80%"></picture>

```python
import numpy as np
from phonometry import airborne_insulation, weighted_rating, energy_average_level

# Energy-average several microphone positions in one room (dB)
print(round(float(energy_average_level([60.0, 66.0])), 1))   # 64.0

# Field insulation per band; area S and volume V add R'
l1 = np.full(16, 80.0)                                # source-room levels
l2 = np.full(16, 40.0)                                # receiving-room levels
t2 = np.full(16, 0.5)                                 # receiving-room T (s)
ins = airborne_insulation(l1, l2, t2, area=10.0, volume=50.0)
print(round(float(ins.dnt[0]), 1))                   # 40.0  (= D since T = T0)
print(round(float(ins.r_prime[0]), 1))               # 38.0

# Single-number rating from a measured 16-band R spectrum (ISO 717-1 Annex C)
R = [20.4, 16.3, 17.7, 22.6, 22.4, 22.7, 24.8, 26.6,
     28.0, 30.5, 31.8, 32.5, 33.4, 33.0, 31.0, 25.5]
w = weighted_rating(R)
print(w.rating, w.c, w.ctr)                          # 30 -2 -3  ->  Rw(C;Ctr) = 30(-2;-3)

w.plot()   # measured R' vs shifted ISO 717-1 reference, deviations shaded (needs matplotlib)
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# One line — measured curve vs the shifted ISO 717-1 reference, deviations shaded:
w.plot()
plt.show()

# By hand, from the band curve the result now carries:
fig, ax = plt.subplots()
ax.semilogx(w.band_centers, w.measured, "o-", label="Measured R'")
ax.semilogx(w.band_centers, w.shifted_reference, "s--", label="Shifted reference")
ax.fill_between(w.band_centers, w.measured, w.shifted_reference,
                where=w.measured < w.shifted_reference, interpolate=True,
                alpha=0.3, label="Unfavourable deviations")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound reduction index [dB]")
ax.set_title(f"Rw = {w.rating} dB  (C={w.c:+d}; Ctr={w.ctr:+d})")
ax.legend()
plt.show()
```

</details>

Compute `l1`, `l2` and `t2` on the same 16 one-third-octave bands from
100 Hz to 3150 Hz — obtain `t2` from
`room_parameters(ir, fs, limits=(100, 3150), fraction=3).t30`, for example — and
pass them to `airborne_insulation`. Feed that function's `dnt` (or `r_prime`)
spectrum to `weighted_rating`, so every band aligns index-by-index with the
ISO 717-1 reference curve.

### `airborne_insulation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `l1` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Source-room levels (2D is energy-averaged) |
| `l2` | 1D or 2D array | dB | same band count | Receiving-room levels |
| `t2` | 1D array | s | > 0, one per band | Receiving-room reverberation time |
| `area` | float, optional | m² | > 0, with `volume` | Partition area `S` (enables `R'`) |
| `volume` | float, optional | m³ | > 0, with `area` | Receiving-room volume `V` |
| `t0` | float | s | default `0.5` | Reference reverberation time `T0` |

### `weighted_rating()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `values_by_band` | 1D array | dB | 16 (thirds) or 5 (octaves) | Measured `R`, `R'`, `DnT` … per band |
| `bands` | str or `None` | — | `'third-octave'` / `'octave'` / `None` | `None` infers from the count |

`airborne_insulation()` returns an `AirborneInsulationResult` (`d`, `dnt`,
`r_prime` or `None`); `weighted_rating()` returns a `WeightedRatingResult`
(`rating`, `c`, `ctr`, `unfavourable_sum`, all integers except the sum).

### Impact sound (ISO 16283-2, ISO 717-2)

Footstep noise is rated the other way round. Instead of how much a floor
*blocks*, impact insulation measures how much a standardized **tapping
machine** on the floor above puts into the room below — so a *higher* number
is *worse*. The energy-average impact sound pressure level $L_i$ in the
receiving room is normalised like the airborne case, but with a sign flip on
the reverberation term:

$$
L'_{nT} = L_i - 10 \log_{10} \frac{T}{T_0}, \qquad
L'_n = L_i + 10 \log_{10} \frac{A}{A_0}, \quad
A_0 = 10\ \text{m}^2,\ A = \frac{0.16\ V}{T}.
$$

The **standardized** impact level $L'_{nT}$ ($T_0 = 0.5$ s for dwellings)
needs only the receiving-room $T$, so with $T = 0.5$ s it equals $L_i$; the
**normalized** level $L'_n$ (referenced to a 10 m² absorption area) also needs
the receiving-room volume. Note the **minus** sign — more reverberation
*lowers* $L'_{nT}$, opposite to the airborne $D_{nT}$.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_impact_setup_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_impact_setup.svg" alt="Field impact insulation setup: a standardized tapping machine on the floor of the source room above, microphones energy-averaged in the receiving room below, and the receiving-room reverberation time" width="92%"></picture>

The single-number rating (ISO 717-2) shifts the same style of reference curve,
but an **unfavourable deviation now occurs where the measurement *exceeds* the
reference** (impact noise is worse when higher) — the sign opposite to
ISO 717-1. The rating (`Ln,w`, `L'n,w`, `L'nT,w`) is the shifted reference read
at 500 Hz; for octave bands it is then reduced by 5 dB. The spectrum
adaptation term $C_I = L_{n,\text{sum}} - 15 - L_{n,w}$ uses the energetic sum
over 100–2500 Hz (16-band thirds excluding 3150 Hz) or 125–2000 Hz (octaves).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/impact_rating_dark.png"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/impact_rating.png" alt="Measured one-third-octave normalized impact sound pressure level with the shifted ISO 717-2 reference curve and the resulting weighted rating read at 500 Hz" width="80%"></picture>

```python
import numpy as np
from phonometry import impact_insulation, weighted_impact_rating

# 16 one-third-octave impact levels Li (100 Hz - 3150 Hz), dB, from the
# ISO 717-2 Annex C worked example, and the receiving-room T per band.
li = np.array([62.1, 63.2, 63.5, 66.2, 68.5, 70.0, 71.7, 73.1,
               73.8, 73.5, 73.8, 73.3, 73.1, 73.0, 72.4, 71.2])
t2 = np.full(16, 0.5)

imp = impact_insulation(li, t2, volume=50.0)
print(round(float(imp.l_n_t[0]), 1))          # 62.1  (= Li since T = T0)
print(round(float(imp.l_n[0]), 1))            # 64.1  normalized to A0 = 10 m^2

# Weighted impact rating + spectrum adaptation term CI (ISO 717-2)
res_imp = weighted_impact_rating(imp.l_n_t)
print(res_imp.rating, res_imp.ci, res_imp.unfavourable_sum)   # 79 -11 28.0  ->  L'nT,w(CI)=79(-11)

# Octave-band data carry the extra -5 dB reduction (Clause 4.3.2)
octave = np.array([65.3, 64.5, 58.0, 55.8, 43.0])
print(weighted_impact_rating(octave).rating)  # 54

res_imp.plot()   # measured L'nT vs shifted ISO 717-2 reference, measured-above shaded (needs matplotlib)
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# One line — measured L'nT vs the shifted ISO 717-2 reference (measured-above shaded):
res_imp.plot()
plt.show()

# By hand, from the band curve the result now carries (note the opposite sign:
# an unfavourable deviation is where the MEASURED level exceeds the reference).
# Here the input was l_n_t, so the rated quantity is the field level L'nT,w:
fig, ax = plt.subplots()
ax.semilogx(res_imp.band_centers, res_imp.measured, "o-", label="Measured L'nT")
ax.semilogx(res_imp.band_centers, res_imp.shifted_reference, "s--", label="Shifted reference")
ax.fill_between(res_imp.band_centers, res_imp.shifted_reference, res_imp.measured,
                where=res_imp.measured > res_imp.shifted_reference, interpolate=True,
                alpha=0.3, label="Unfavourable deviations")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Impact sound pressure level [dB]")
ax.set_title(f"L'nT,w = {res_imp.rating} dB  (CI={res_imp.ci:+d})")
ax.legend()
plt.show()
```

</details>

Feed `impact_insulation`'s `l_n_t` (or `l_n`) straight into
`weighted_impact_rating`; the rating and `CI` reproduce the ISO 717-2 Annex C
values (thirds `L'nT,w = 79`, `CI = −11`; octave `54`, `CI = 0`).

#### `impact_insulation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `li` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Energy-average impact SPL (2D is averaged over positions) |
| `t2` | 1D array | s | > 0, one per band | Receiving-room reverberation time |
| `volume` | float, optional | m³ | > 0 | Receiving-room `V` (enables `L'n`) |
| `t0` | float | s | default `0.5` | Reference reverberation time `T0` |

#### `weighted_impact_rating()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `values_by_band` | 1D array | dB | 16 (thirds) or 5 (octaves) | Measured `Ln`, `L'n` or `L'nT` per band |
| `bands` | str or `None` | — | `'third-octave'` / `'octave'` / `None` | `None` infers from the count |

`impact_insulation()` returns an `ImpactInsulationResult` (`l_n_t`, `l_n` or
`None`); `weighted_impact_rating()` returns an `ImpactRatingResult` (`rating`,
`ci` integers, `unfavourable_sum` in dB).

### Field façade insulation (ISO 16283-3)

The same source/receiver logic reaches the building **façade**, but now the
source is *outdoors* — a loudspeaker at 45° or the road traffic itself. Rather
than a level difference across an internal partition, ISO 16283-3 references the
receiving-room level $L_2$ to the level **2 m in front of the façade**
$L_{1,2m}$, giving the level difference $D_{2m}$ and, exactly as in the airborne
case, its standardized and normalized forms:

$$
D_{2m} = L_{1,2m} - L_2, \quad
D_{2m,nT} = D_{2m} + 10 \log_{10}\frac{T}{T_0}, \quad
D_{2m,n} = D_{2m} - 10 \log_{10}\frac{A}{A_0},
$$

with $T_0 = 0.5$ s, $A_0 = 10$ m² and $A = 0.16\ V/T$ (dwellings). When the
microphone sits **on the test element** (surface level $L_{1,s}$) the *element*
method also yields an apparent sound reduction index, carrying a fixed
angle-of-incidence correction — $-1.5$ dB for the 45° loudspeaker method,
$-3$ dB for the all-angle road-traffic method:

$$
R'_{45°} = L_{1,s} - L_2 + 10 \log_{10}\frac{S}{A} - 1.5, \qquad
R'_{tr,s} = L_{1,s} - L_2 + 10 \log_{10}\frac{S}{A} - 3.
$$

The façade quantity is airborne, so its single-number rating uses the
**ISO 717-1** reference curve through `weighted_rating` unchanged (Annex F).

```python
import numpy as np
from phonometry import facade_insulation, weighted_rating

# Outdoor level 2 m in front of the façade, receiving-room level and T per
# one-third-octave band; surface_level is the microphone on the test element.
l1_2m = np.full(16, 75.0)                              # L1,2m outdoors
l2 = np.full(16, 33.0)                                 # receiving-room L2
t2 = np.full(16, 0.5)                                  # receiving-room T (s)

fac = facade_insulation(l1_2m, l2, t2, volume=50.0, area=11.5,
                        surface_level=np.full(16, 78.0), method="loudspeaker")
print(round(float(fac.d_2m[0]), 1))                    # 42.0  D2m = L1,2m - L2
print(round(float(fac.d_2m_nt[0]), 1))                 # 42.0  (= D2m since T = T0)
print(round(float(fac.d_2m_n[0]), 1))                  # 40.0  normalized to A0 = 10 m^2
print(round(float(fac.r_prime[0]), 1))                 # 42.1  R'45deg (loudspeaker, -1.5 dB)

# The road-traffic element method carries the -3 dB all-angle correction instead
tr = facade_insulation(l1_2m, l2, t2, volume=50.0, area=11.5,
                       surface_level=np.full(16, 78.0), method="road_traffic")
print(round(float(tr.r_prime[0]), 1))                  # 40.6  R'tr,s (traffic, -3 dB)

# The façade quantity is airborne: rate D2m,nT with the ISO 717-1 engine
print(weighted_rating(fac.d_2m_nt).rating)             # 42  Dls,2m,nT,w

fac.plot()   # per-band D2m,nT with D2m, D2m,n and R' overlaid (needs matplotlib)
```

`surface_level`, `area` and `volume` are all optional: with only `l1_2m`, `l2`
and `t2` the function returns `d_2m` and `d_2m_nt`; add `volume` for `d_2m_n`;
add `surface_level` **and** `area` **and** `volume` for `r_prime`. Positions are
energy-averaged with the surface-level formula (Clause 9.5.1); band levels are
assumed already corrected for background noise.

#### `facade_insulation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `l1_2m` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Level 2 m in front of the façade `L1,2m` |
| `l2` | 1D or 2D array | dB | same band count | Receiving-room levels |
| `t2` | 1D array | s | > 0, one per band | Receiving-room reverberation time |
| `area` | float, optional | m² | > 0, with `surface_level`, `volume` | Test-element area `S` (enables `R'`) |
| `volume` | float, optional | m³ | > 0 | Receiving-room `V` (enables `D2m,n`; required for `R'`) |
| `surface_level` | 1D/2D array, optional | dB | same band count | Surface level `L1,s` on the element (enables `R'`) |
| `method` | str | — | `'loudspeaker'` (−1.5 dB) / `'road_traffic'` (−3 dB) | Angle-of-incidence correction of `R'` |
| `t0` | float | s | default `0.5` | Reference reverberation time `T0` |
| `frequencies` | 1D array, optional | Hz | — | Band centres carried on the result for plotting |

`facade_insulation()` returns a `FacadeInsulationResult` (`d_2m`, `d_2m_nt`,
`d_2m_n` or `None`, `r_prime` or `None`, `frequencies`); feed any 16-band façade
quantity to `weighted_rating` for its ISO 717-1 single number.

## 2. Laboratory measurement (ISO 10140)

Everything above is a **field** measurement (the primed quantities $R'$, $L'_n$):
the number a real building achieves, flanking transmission and all. To rate an
element on its own — a wall type, a floating floor, a window — you take it to a
qualified **laboratory** (ISO 10140), where suppressed flanking makes the
*direct* transmission the whole story. The formulas lose their primes: the
**sound reduction index** $R$ (not $R'$) and the **normalized impact level**
$L_n$ (not $L'_n$), with the receiving room's absorption area $A = 0.16\ V/T$
now a known property of the facility:

$$
R = L_1 - L_2 + 10 \log_{10}\frac{S}{A}, \qquad
L_n = L_i + 10 \log_{10}\frac{A}{A_0}, \quad A_0 = 10\ \text{m}^2.
$$

| | Field (ISO 16283) | Laboratory (ISO 10140) |
| :--- | :--- | :--- |
| Airborne | $R'$ apparent (with flanking) | $R$ direct (flanking suppressed) |
| Impact | $L'_n$ apparent | $L_n$ direct |
| Absorption area | measured in the room | property of the facility |

The single-number ratings reuse the very same ISO 717-1/2 engines
(`weighted_rating`, `weighted_impact_rating`) — an $R$ spectrum rates to $R_w$
exactly as an $R'$ spectrum rated to $R'_w$. Before forming the index the
receiving-room levels must be **corrected for background noise** (Clause 4.3):
the energy subtraction $10 \log_{10}(10^{L_{sb}/10} - 10^{L_b/10})$ applies for a
6–15 dB signal-to-background margin, a fixed 1.3 dB correction (the *limit of
measurement*) at or below 6 dB, and no correction at or above 15 dB.

```python
import numpy as np
from phonometry import (lab_airborne_insulation, lab_impact_insulation,
                        background_correction)

# Source/receiving levels and receiving-room T over the 16 one-third-octave
# bands; S is the free test-opening area, V the receiving-room volume.
l1 = np.full(16, 80.0)
l2 = np.full(16, 40.0)
t2 = np.full(16, 0.5)
lab = lab_airborne_insulation(l1, l2, t2, area=10.0, volume=50.0)
print(round(float(lab.r[0]), 1))              # 38.0  R = L1 - L2 + 10 lg(S/A)
print(round(float(lab.absorption[0]), 1))     # 16.0  A = 0.16 V / T (m^2)
print(lab.rating.rating, lab.rating.c, lab.rating.ctr)   # 38 0 0  ->  Rw(C;Ctr)

# Impact: the tapping-machine level Li normalized to A0 = 10 m^2 gives Ln
li = np.array([62.1, 63.2, 63.5, 66.2, 68.5, 70.0, 71.7, 73.1,
               73.8, 73.5, 73.8, 73.3, 73.1, 73.0, 72.4, 71.2])
imp = lab_impact_insulation(li, t2, volume=50.0)
print(round(float(imp.l_n[0]), 1))            # 64.1  Ln = Li + 10 lg(A/A0)
print(imp.rating.rating, imp.rating.ci)       # 81 -11  ->  Ln,w(CI)

# Background correction: margins 6 / 1 / 20 dB -> capped / capped / unchanged
corrected = background_correction([30.0, 33.0, 50.0], [24.0, 32.0, 30.0])
print(np.round(corrected, 1))                 # [28.7 31.7 50.0]  (1.3 dB cap twice)

lab.rating.plot()   # measured R vs shifted ISO 717-1 reference (needs matplotlib)
```

A margin at or below 6 dB emits a `LabInsulationWarning` and flags the band as
the limit of measurement; catch it with `warnings.simplefilter("error",
LabInsulationWarning)`. The automatic rating is formed only when exactly 16
one-third-octave or 5 octave values are supplied (`rating` is `None` otherwise).

### `lab_airborne_insulation()` / `lab_impact_insulation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `l1` / `l2` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Source / receiving levels (airborne) |
| `li` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Impact SPL from the tapping machine (impact) |
| `t2` | 1D array | s | > 0, one per band | Receiving-room reverberation time |
| `area` | float | m² | > 0 | Free test-opening area `S` (airborne only) |
| `volume` | float | m³ | > 0 | Receiving-room volume `V` |

`lab_airborne_insulation()` returns a `LabAirborneInsulationResult` (`r`,
`absorption`, `rating`); `lab_impact_insulation()` a
`LabImpactInsulationResult` (`l_n`, `absorption`, `rating`);
`background_correction(signal_and_background, background)` returns the corrected
levels directly.

## 3. Predicting performance (EN 12354)

A laboratory rating describes an element in isolation, yet the sound a building
actually transmits also travels *around* the partition — along the floor, up the
façade, through the flanking walls — re-radiating into the receiving room. This
**flanking transmission** is the whole difference between the laboratory $R$ and
the field $R'$. EN 12354 predicts the in-situ apparent rating from the
laboratory ratings of the elements plus the vibration transmission of their
junctions.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_flanking_paths_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_flanking_paths.svg" alt="The direct path Dd through the separating element and the three flanking paths Ff, Df and Fd across each junction between a flanking element and the separating element" width="92%"></picture>

Each junction between a flanking element and the separating element carries
three paths — $Ff$ (flanking→flanking), $Df$ (direct→flanking) and $Fd$
(flanking→direct) — alongside the single direct path $Dd$. The **simplified
single-number model** combines them energetically (Formula 26):

$$
R'_w = -10 \log_{10}\Big[ 10^{-R_{Dd,w}/10}
      + \sum 10^{-R_{Ff,w}/10} + \sum 10^{-R_{Df,w}/10}
      + \sum 10^{-R_{Fd,w}/10} \Big],
$$

with the direct path $R_{Dd,w} = R_{s,w} + \Delta R_{Dd,w}$ (Formula 27) and each
flanking path (Formula 28a)

$$
R_{ij,w} = \tfrac{R_{i,w} + R_{j,w}}{2} + \Delta R_{ij,w} + K_{ij}
         + 10 \log_{10}\frac{S_s}{l_0\ l_f},
$$

where $l_0 = 1$ m is the reference coupling length, $l_f$ the junction coupling
length and $K_{ij}$ the junction's **vibration reduction index** (Annex E,
empirical in the mass ratio $M = \log_{10}(m'_{\perp,i}/m'_i)$).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/prediction_flanking_demo_dark.png"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/prediction_flanking_demo.png" alt="Per-path sound reduction indices for the EN 12354-1 Annex H.3 example and each path's share of the transmitted energy, showing the direct path dominating at R'w = 52 dB" width="80%"></picture>

```python
import numpy as np
from phonometry import (junction_vibration_reduction, flanking_element,
                        predicted_airborne_insulation)

# EN 12354-1 Annex H.3: a separating wall Rs,w = 57 dB, area Ss = 11.5 m², with
# four flanking elements. The simplified model reads each junction's Kij at
# 500 Hz from the mass ratio m'perp / m' (Annex E) — here the floor's rigid
# cross-junction (the mass ratio is itself rounded, hence 12.5 vs Annex 12.4):
print(round(junction_vibration_reduction("rigid_cross", "through", 1.61), 1))  # 12.5  KFf
print(round(junction_vibration_reduction("rigid_cross", "corner",  1.61), 1))  #  8.9  KFd = KDf

# Build each element's three flanking paths (Ff, Df, Fd) from the Annex H
# tabulated Kij, then combine the direct path Dd energetically (Formula 26).
elements = [   # (name, Rw, KFf, KFd = KDf, coupling length lf)
    ("floor",    49, 12.4,  8.9, 4.50),
    ("ceiling",  46, 14.4,  9.2, 4.50),
    ("facade",   42, 12.6,  6.7, 2.55),
    ("int-wall", 33, 33.5, 15.7, 2.55),
]
paths = []
for name, rw, k_ff, k_fd, lf in elements:
    paths += flanking_element(label=name, r_flanking=rw, r_separating=57,
                              k_ff=k_ff, k_fd=k_fd, k_df=k_fd,
                              separating_area=11.5, coupling_length=lf)

res = predicted_airborne_insulation(r_direct=57.0, flanking_paths=paths)
print(round(res.r_prime_w, 1))                          # 52.2  ->  R'w = 52 dB
print(res.dominant.label, round(res.dominant.fraction, 2))   # Dd 0.33 (direct dominates)
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# Uses `paths` and `res` from the snippet above.
# Per-path sound reduction index and each path's share of the transmitted
# energy for the Annex H.3 result computed above.
labels = [p.label for p in res.paths]
r_w = [p.r_w for p in res.paths]
frac = [100.0 * p.fraction for p in res.paths]

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(9, 6), sharex=True)
ax1.bar(labels, r_w, color="tab:blue")
ax1.axhline(res.r_prime_w, ls="--", color="k", label=f"R'w = {res.r_prime_w:.1f} dB")
ax1.set_ylabel("Path Rij,w [dB]"); ax1.legend()
ax2.bar(labels, frac, color="tab:orange")
ax2.set_ylabel("Energy share [%]"); ax2.set_xlabel("Transmission path")
for ax in (ax1, ax2):
    ax.tick_params(axis="x", rotation=45)
fig.suptitle("EN 12354-1 Annex H.3 — flanking transmission")
fig.tight_layout()
plt.show()
```

</details>

Every added flanking path strictly lowers $R'_w$ below the direct $R_{Dd,w} = 57$;
`res.paths` exposes each path's share of the transmitted energy so the dominant
path is visible. `flanking_element` is a convenience that builds one junction's
three paths at once; the single-path constructor behind it, `flanking_path`,
builds one `Ff`, `Df` or `Fd` path at a time (Formula 28a). Clause 4.4.2 also
enforces a floor $K_{ij} \ge K_{ij,\min}$ from the junction geometry — compute
it with `junction_min_vibration_reduction` and pass it to
`flanking_path(..., kij_min=...)`, which raises a below-floor $K_{ij}$ to the
minimum:

```python
from phonometry import junction_min_vibration_reduction
# Kij,min = 10 lg[lf·l0·(1/Si + 1/Sj)]; large elements give a low (here negative)
# floor, so a realistic tabulated Kij is rarely clamped.
print(round(junction_min_vibration_reduction(coupling_length=4.5,
                                             s_i=11.5, s_j=11.5), 1))     # -1.1
```

The impact counterpart (EN 12354-2, Formula 21) is a direct subtraction:
$L'_{n,w} = L_{n,w,eq} - \Delta L_w + K$, with the bare-floor equivalent level
$L_{n,w,eq} = 164 - 35 \log_{10}(m'/m'_0)$ (Annex B), the covering improvement
$\Delta L_w$ (ISO 717-2) and the flanking correction $K$ from Table 1.

```python
from phonometry import (equivalent_impact_level, impact_flanking_correction,
                        predicted_impact_insulation, standardized_impact_level)

# EN 12354-2 Annex E.3: a 0.14 m concrete floor (m' = 322 kg/m²) with a floating
# floor (ΔLw = 33 dB), rooms one above the other, mean flanking mass 145 kg/m².
ln_eq = equivalent_impact_level(322.0)                   # 164 - 35 lg(m')
k = impact_flanking_correction(322.0, 145.0)             # Table 1 (sep 322, flk 145)
imp = predicted_impact_insulation(ln_w_eq=ln_eq, delta_l_w=33.0, k_correction=k)
print(round(ln_eq, 1), k, round(imp.l_prime_n_w, 1))     # 76.2 2 45.2  ->  L'n,w = 45 dB
print(round(standardized_impact_level(imp.l_prime_n_w, 50.0), 1))   # 43.0  L'nT,w
```

### `junction_vibration_reduction()` / `flanking_element()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `junction_type` | str | — | `'rigid_cross'` / `'rigid_t'` / `'flexible_t'` / `'lightweight_facade'` | Junction geometry (Annex E) |
| `path` | str | — | `'through'` (K13) / `'corner'` (K12 = K23) | Path branch |
| `mass_ratio` | float | — | > 0 | `m'⊥,i / m'i` (Formula E.2) |
| `frequency` | float | Hz | default `500` | Only `flexible_t` is frequency-dependent |
| `r_flanking` / `r_separating` | float | dB | — | Weighted indices of the flanking / separating element |
| `k_ff` / `k_fd` / `k_df` | float | dB | — | Junction `Kij` for the three paths |
| `separating_area` | float | m² | > 0 | Separating-element area `Ss` |
| `coupling_length` | float | m | > 0 | Junction coupling length `lf` |
| `delta_r_ff` / `delta_r_fd` / `delta_r_df` | float | dB | default `0` | Lining improvements per path |

### `flanking_path()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `label` | str | — | — | Display name of the path |
| `kind` | str | — | `'Ff'` / `'Df'` / `'Fd'` | Which flanking branch the path is |
| `r_source` / `r_receive` | float | dB | — | Weighted indices of the source-side / receive-side elements |
| `k_ij` | float | dB | — | Junction vibration-reduction index for this path |
| `separating_area` | float | m² | > 0 | Separating-element area `Ss` |
| `coupling_length` | float | m | > 0 | Junction coupling length `lf` |
| `delta_r` | float | dB | default `0` | Lining improvement on this path |
| `kij_min` | float | dB | default `None` | When given, `k_ij` is floored at this Formula E.4 minimum |

`predicted_airborne_insulation()` returns an `AirbornePredictionResult`
(`r_prime_w`, `r_direct_w`, `paths` of `PathContribution`, `dominant`);
`predicted_impact_insulation()` an `ImpactPredictionResult` (`l_prime_n_w`,
`ln_w_eq`, `delta_l_w`, `k_correction`). The simplified model carries a reported
standard deviation of about 2 dB (Clause 5).

## 4. Measurement uncertainty (ISO 12999-1)

A rating without an uncertainty is only half a result. ISO 12999-1 does not
re-measure anything; it tabulates the **standard uncertainty** $u$ of every
sound-insulation quantity — derived from inter-laboratory tests — and prescribes
how to expand and combine it. Which standard deviation is $u$ depends on the
**measurement situation** (Clause 5.2):

| Situation | Meaning | Standard uncertainty $u$ |
| :--- | :--- | :--- |
| **A** | laboratory characterisation (ISO 10140) | reproducibility $\sigma_R$ |
| **B** | same location, different teams | in-situ $\sigma_{situ}$ |
| **C** | same location, same operator repeated | repeatability $\sigma_r$ |

The expanded uncertainty is $U = k\ u$ (Formula 2) with the coverage factor $k$
of Table 8. A two-sided interval $Y = y \pm U$ (Formula 3, $k = 1.96$ at 95 %)
*reports* a value; the **one-sided** factor ($k = 1.65$ at 95 %) *declares
conformity* with a requirement (Formulae 4/5).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insulation_uncertainty_demo_dark.png"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insulation_uncertainty_demo.png" alt="A weighted rating reported with its two-sided 95 % expanded uncertainty in situations A, B and C, the reproducibility uncertainty widest and the repeatability uncertainty narrowest" width="80%"></picture>

```python
from phonometry import (band_uncertainty, single_number_uncertainty,
                        uncertain_value, satisfies_lower_requirement)

# Situation B (same building, different teams) -> the in-situ standard deviation.
print(single_number_uncertainty("r_w", "B"))       # 0.9  dB  (Table 3)
u = band_uncertainty("airborne", "B")              # per-band u (Table 2)
print(len(u.frequencies), u.uncertainties[10])     # 21 1.1  (the 500 Hz band)

# Report R'w = 52 dB with a two-sided 95 % interval (k = 1.96, Table 8):
uv = uncertain_value(52.0, "rprime_w", "B")        # aliases resolve to r_w
print(uv.coverage_factor, round(uv.expanded_uncertainty, 1))    # 1.96 1.8
print(round(uv.lower, 1), round(uv.upper, 1))      # 50.2 53.8  ->  52 ± 1.8 dB

# Declaring conformity uses the ONE-sided factor (k = 1.65): does R'w provably
# clear a 50 dB requirement?
uc = uncertain_value(52.0, "rprime_w", "B", one_sided=True)
print(satisfies_lower_requirement(52.0, uc.expanded_uncertainty, 50.0))   # True
```

Impact quantities offer situations B/C only (Table 4, no 500 Hz band in the 2020
edition), and $\Delta L$ only situation A. Descriptors are case-insensitive with
aliases (`rprime_w`/`dnt_w`→`r_w`, `lprime_n_w`→`ln_w`); combine independent
components in quadrature with `combine_uncertainties`, and reduce by $m$
independent measurements with `reduce_by_independent_measurements` ($u/\sqrt{m}$).

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
from phonometry import uncertain_value

# The same R'w = 52 dB reported in each situation with its two-sided 95 % U.
situations = ["A", "B", "C"]
vals = [uncertain_value(52.0, "r_w", s) for s in situations]

fig, ax = plt.subplots(figsize=(7, 4))
ax.errorbar(situations, [v.value for v in vals],
            yerr=[v.expanded_uncertainty for v in vals],
            fmt="o", capsize=8, color="tab:blue")
for s, v in zip(situations, vals):
    ax.annotate(f"±{v.expanded_uncertainty:.1f}", (s, v.upper),
                textcoords="offset points", xytext=(8, 4))
ax.set_ylabel("R'w [dB]"); ax.set_xlabel("Measurement situation")
ax.set_title("R'w = 52 dB with 95 % expanded uncertainty (ISO 12999-1)")
fig.tight_layout()
plt.show()
```

</details>

### `band_uncertainty()` / `single_number_uncertainty()` / `uncertain_value()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `measurand` | str | — | `'airborne'` / `'impact'` / `'impact_reduction'` | Selects Table 2 / 4 / 6 |
| `quantity` | str | — | `'r_w'`, `'ln_w'`, `'delta_lw'` (+ aliases, `+c`/`+ctr` variants) | Single-number descriptor |
| `situation` | str | — | `'A'` / `'B'` / `'C'` | Measurement situation (Clause 5.2) |
| `value` | float | dB | — | Best estimate `y` to attach `U` to |
| `coverage` | float | — | default `0.95` | Confidence level (Table 8) |
| `one_sided` | bool | — | default `False` | One-sided factor for conformity checks |
| `upper_limit` | bool | — | default `False` | Select the σR95 upper limit (airborne, situation A) |

`band_uncertainty()` returns a `BandUncertainty` (`frequencies`,
`uncertainties`, `.to_arrays()`); `single_number_uncertainty()` a float;
`uncertain_value()` an `UncertainValue` (`value`, `standard_uncertainty`,
`coverage_factor`, `expanded_uncertainty`, `.lower`, `.upper`). The read-only
`COVERAGE_FACTORS` mapping exposes Table 8 keyed by `(confidence, one_sided)`.

---

**Standards.** ISO 16283-1:2014, ISO 16283-2 and ISO 16283-3:2016, *Acoustics —
Field measurement of sound insulation in buildings and of building elements* —
the level differences, normalisations and element methods of §1; ISO 717-1 and
ISO 717-2 — the reference-curve single-number ratings and the spectrum
adaptation terms C, Ctr and CI; ISO 10140-2:2010, ISO 10140-3:2010 and ISO 10140-4:2010 — the
laboratory R and Ln with the background-noise correction of §2; EN 12354-1:2000
and EN 12354-2:2000 — the simplified flanking-transmission predictions of §3
(Annex E junctions, worked examples H.3 and E.3); ISO 12999-1:2020 — the
standard uncertainties per measurement situation and the coverage factors
of §4.

## See also

- [Room Acoustics](room-acoustics.md) — the impulse response,
  room parameters and sound absorption that this guide's insulation chain builds on.
- [Levels](levels.md) — energy averaging and the level metrics behind
  source/receiving-room levels.
- [Filter Banks](filter-banks.md) — the IEC 61260 fractional-octave filters
  used for the insulation spectra.
- [Sound Power](sound-power.md) — the `LW` methods that share the
  absorption-area machinery of the receiving room.
- [Theory](theory.md) — the reference-curve derivation behind the
  weighted single-number ratings.
