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# Sound Power

Sound *pressure* depends on where you stand and on the room you stand in;
sound **power** does not. The sound power level `LW` is the total acoustic
energy per second a source radiates, referenced to `P0 = 1 pW`, and it is
the device-independent **emission** descriptor that goes on a datasheet,
feeds a room prediction (EN 12354) or is checked against a noise-emission
limit. This page covers the three routes phonometry implements to obtain it
and when to reach for each: an enveloping *pressure* surface in the field
(ISO 3744/3746), the diffuse field of a *reverberation room* (ISO 3741),
*intensity* scanning over a surface (ISO 9614-2), and, for the highest
accuracy, the precision grades in an *anechoic room* (ISO 3745) and by
precision *intensity* scanning (ISO 9614-3).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_power_two_rooms_dark.gif"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_power_two_rooms.gif" alt="Animation: the same source in an anechoic room and in a reverberation room produces different microphone pressures, and the free-field and diffuse-field formulas converge to the same sound power level L_W" width="640" height="360" loading="lazy"></picture>

[Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_power_two_rooms.webm)

## Choosing a method

All deliver the same quantity, a per-band `LW` and an A-weighted total
`LWA`, but under different environments, accuracy grades and practical
constraints.

| Method | Standard | Measured quantity | Environment | Accuracy grade | Use when |
| :--- | :--- | :--- | :--- | :--- | :--- |
| Enveloping surface | **ISO 3744** (engineering) / **ISO 3746** (survey) | Sound pressure on a hemisphere or box | Essentially free field over one or more reflecting planes | Grade 2 (`σR0 ≈ 1.5 dB`) / grade 3 (`≈ 3.0 dB`) | In situ or a large room; no special test facility available |
| Reverberation room | **ISO 3741** | Sound pressure in the diffuse field | Qualified hard-walled reverberation room | Grade 1 (precision) | Highest accuracy for steady, broadband sources in a lab |
| Intensity scanning | **ISO 9614-2** | Normal sound intensity scanned over a surface | Almost any, tolerant of steady extraneous noise | Grade 2 / 3 (from per-band field indicators) | On-site with background noise, or one machine among many |
| Anechoic room | **ISO 3745** | Sound pressure on a fixed microphone array | Qualified anechoic or hemi-anechoic room | Grade 1 (precision) | Reference-grade emission in a free-field laboratory |
| Precision intensity scanning | **ISO 9614-3** | Scanned normal intensity, tighter criteria | Almost any, tolerant of steady extraneous noise | Grade 1 (precision) | Precision on-site, with the ISO 9614-3 field-indicator checks |

The pressure methods correct the surface level for the room (`K2`) and for
background noise (`K1`); the reverberation method needs a *qualified* room
but reaches precision grade; intensity rejects steady background energy at
the cost of a two-microphone probe and a per-band validity check. The rest
of the page walks each in turn.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_methods_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_methods.svg" alt="The three sound power routes side by side: an enveloping pressure surface over a reflecting plane (ISO 3744/3746), a source in a reverberation room sampled by microphones (ISO 3741) and an intensity probe scanning a surface around the source (ISO 9614-2)" width="92%"></picture>

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import emission

# One steady source (octave-band LW below) determined by three routes.
freqs = np.array([125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0])
lw_true = np.array([85.0, 88.0, 90.0, 89.0, 86.0, 82.0])

# ISO 3744: SPL at 10 positions on a hemisphere, r = 2 m (10 lg(S/S0) = 14 dB).
pres = emission.sound_power_pressure(np.tile(lw_true - 14.0, (10, 1)),
                                     "hemisphere", radius=2.0,
                                     frequencies=freqs)
# ISO 9614-2: uniform normal intensity scanned over six 0.5 m2 segments.
i_n = np.tile(10.0 ** (lw_true / 10.0) * 1e-12 / 3.0, (6, 1))
inten = emission.sound_power_intensity(i_n, np.full(6, 0.5),
                                       frequencies=freqs, band_type="octave")
# ISO 3741: comparison against a reference source of known LW = 84 dB per band.
comp = emission.sound_power_comparison(lw_true - 20.0, np.full(6, 64.0),
                                       np.full(6, 84.0), frequencies=freqs)

fig, ax = plt.subplots()
for res, style, ms, label in ((pres, "-o", 11, "pressure (ISO 3744)"),
                              (inten, "--s", 8, "intensity (ISO 9614-2)"),
                              (comp, ":^", 5, "reference source (ISO 3741)")):
    ax.semilogx(freqs, res.sound_power_level, style, markersize=ms,
                label=f"{label}: LWA = {res.sound_power_level_a:.1f} dB")
ax.set(xlabel="Frequency [Hz]", ylabel="Sound power level LW [dB]")
ax.legend()
plt.show()
```

</details>

### A decision path

The table compresses into a short sequence of questions (ISO 3740 dedicates
its Table 3 and Annex D to exactly this decision). Work through them in
order; the first match names the standard.

1. **What is the number for?** A datasheet declaration or a limit check
   normally asks for engineering grade (grade 2, the preferred grade for
   noise declarations); a reference source, a product ranking or a dispute
   calls for precision (grade 1); a first walk-through of a noisy plant
   tolerates survey grade (grade 3). Grade 1 exists only in a qualified
   laboratory room (ISO 3741, ISO 3745) or via the precision intensity
   methods (ISO 9614-1 at discrete points, ISO 9614-3 by scanning).
2. **Can the source travel to a laboratory?** ISO 3741 wants the source
   small next to the room (volume no more than about 2 % of the room volume)
   and its noise steady; ISO 3745 wants it inside a qualified anechoic or
   hemi-anechoic room with a characteristic dimension below half the
   measurement radius, and it is the route that also yields directivity. A
   machine bolted to its foundation rules both out and leaves the in-situ
   methods.
3. **How quiet and how dry is the site?** ISO 3744 needs the background at
   least 6 dB below the source (preferably more than 15 dB) and
   `K2 ≤ 4 dB`. If only a
   3 dB margin or `K2 ≤ 7 dB` can be met, the same microphones and formulae
   degrade gracefully to ISO 3746 at survey grade.
4. **Is the background the problem?** When neighbouring machines cannot be
   switched off, or the margin is outright negative, the pressure methods
   are out. Intensity scanning (ISO 9614-2, or ISO 9614-3 for grade 1)
   tolerates steady extraneous noise even some 10 dB *above* the source,
   because only the net energy flux through the surface counts; the
   per-band field indicators then decide the grade actually achieved.

### What the accuracy grades mean

The grade is a claim about **reproducibility**: `σR0` is the standard
deviation you would see if different laboratories measured the same source,
each following the standard correctly. Typical A-weighted values are
`σR0 ≈ 0.5 dB` for grade 1 (ISO 3741), `1.5 dB` for grade 2 (ISO 3744,
ISO 9614-2) and `3 dB` or more for grade 3 (larger still when `K2` is
large or the spectrum is tonal). Per-band values are larger at the
spectrum edges. The `uncertainty` field of the pressure-method results
(enveloping surface and anechoic) is the expanded uncertainty
`U = 2·σtot` (95 % coverage), where
`σtot = √(σR0² + σomc²)` also folds in the operating/mounting instability
`σomc` that you estimate and pass in; the grade only bounds the method's
share of the budget.

In practice: a grade-2 `LWA` of 92.4 dB carries `U ≈ 3 dB`, so two grade-2
results 2 dB apart are statistically indistinguishable, and checking that
same source against a 93 dB limit is a coin flip. Choose the grade from the
decision the number has to support, not from the facility that happens to
be free.

## 1. Enveloping surface, sound pressure (ISO 3744 / ISO 3746)

Place the source on a reflecting plane and imagine a **measurement surface**
of area `S` wrapping it: a hemisphere for a compact source, a box (right
parallelepiped) for a large or elongated one. Sample the sound pressure
level at an array of microphone positions on that surface, energy-average
them, and the sound power follows because a diffuse-enough surface captures
all the radiated energy:

$$
\bar{L}_p = 10 \log_{10}\left( \frac{1}{N_M} \sum_i 10^{L_{pi}/10} \right), \qquad
L_W = \bar{L}_p - K_1 - K_2 + 10 \log_{10}\frac{S}{S_0},\quad S_0 = 1\ \text{m}^2 .
$$

Two corrections clean up the surface level. The **background-noise
correction** removes the energy that would have been there with the source
switched off, from the margin `ΔLp` between source-on and background levels,

$$
K_1 = -10 \log_{10}\left( 1 - 10^{-\Delta L_p/10} \right),
$$

and the **environmental correction** removes the reverberant build-up of the
test room from its equivalent absorption area `A`,

$$
K_2 = 10 \log_{10}\left( 1 + \frac{4 S}{A} \right).
$$

The surface area is a closed form of the geometry: a hemisphere is
`S = 2πr²` over one reflecting plane (halved and quartered for two and three
planes), and a one-plane box is `S = 4(ab + bc + ca)` with
`a = 0.5·l1 + d`, `b = 0.5·l2 + d`, `c = l3 + d` for measurement distance
`d`. ISO 3746 (survey) shares every formula but is coarser: fewer
microphone positions, a 3 dB background criterion instead of 6 dB, and
validity up to `K2 ≤ 7 dB` instead of 4 dB.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_sound_power_surfaces_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_sound_power_surfaces.svg" alt="Measurement surfaces of ISO 3744: a hemisphere of radius r enveloping a compact source on a reflecting plane, and a right parallelepiped (box) at measurement distance d around a large source, both with microphone positions marked" width="88%"></picture>

```python
import numpy as np
from phonometry import emission

# Octave-band SPL (dB) at the 10 hemisphere positions of ISO 3744 (Annex B),
# with the source running, plus the background spectrum with it switched off.
freqs = np.array([63, 125, 250, 500, 1000, 2000, 4000, 8000])
base = np.array([70.0, 74.0, 78.0, 80.0, 79.0, 76.0, 72.0, 66.0])
rng = np.random.default_rng(0)
levels = base + rng.normal(0.0, 0.5, size=(10, 8))     # (positions, bands)
background = np.full((10, 8), 55.0)

# ISO 3744 Annex B microphone coordinates on a radius-1.5 m hemisphere.
mic_xyz = emission.measurement_positions("hemisphere", radius=1.5, reflecting_planes=1)
print(mic_xyz.shape)                                    # (10, 3)

res = emission.sound_power_pressure(
    levels, "hemisphere", radius=1.5, reflecting_planes=1,
    background_levels=background, frequencies=freqs,
    reverberation_time=0.6, volume=300.0,          # room data -> K2
)
print(round(res.surface_area, 2))                       # 14.14 m^2 (= 2*pi*1.5^2)
print(round(float(res.environmental_correction[0]), 2)) # K2 = 2.32 dB
print(round(res.sound_power_level_a, 1))                # LWA = 92.4 dB
print(round(res.uncertainty, 1))                        # U = 3.0 dB (2*sigma_R0)
print(np.round(res.sound_power_level, 1))               # per-band LW

res.plot()   # sound power level bars per band, LWA in the title (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_pressure_result_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_pressure_result.svg" alt="The enveloping-surface sound power level spectrum of the ISO 3744 hemisphere example, one bar per octave band from 63 Hz to 8 kHz peaking near 500 Hz, with the A-weighted total of 92.4 dB(A) in the title" width="88%"></picture>

*One bar per band: the energy-averaged surface pressure minus the background
(`K1`) and environmental (`K2`) corrections plus the surface term
`10 lg(S/S0)` gives `LW(f)`, and the A-weighted energy sum across bands gives
the single-number `LWA` in the title.*

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import emission

# Octave-band SPL (dB) at the 10 hemisphere positions of ISO 3744 (Annex B),
# with the source running, plus the background spectrum with it switched off.
freqs = np.array([63, 125, 250, 500, 1000, 2000, 4000, 8000])
base = np.array([70.0, 74.0, 78.0, 80.0, 79.0, 76.0, 72.0, 66.0])
rng = np.random.default_rng(0)
levels = base + rng.normal(0.0, 0.5, size=(10, 8))     # (positions, bands)
background = np.full((10, 8), 55.0)
res = emission.sound_power_pressure(
    levels, "hemisphere", radius=1.5, reflecting_planes=1,
    background_levels=background, frequencies=freqs,
    reverberation_time=0.6, volume=300.0,          # room data -> K2
)

# res is the SoundPowerResult computed above. One line:
res.plot()
plt.show()

# By hand: a bar spectrum of LW with the A-weighted total in the title.
freqs = res.frequencies
positions = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(positions, res.sound_power_level, width=0.7, color="#1f77b4")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Enveloping-surface sound power (ISO 3744)  "
    f"LWA = {res.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

The A-weighted total `LWA` is combined from the band powers with the ISO 3744
Annex E A-weighting corrections, so it needs `frequencies`. Passing the room
data (`reverberation_time` + `volume`, or `absorption_area`, or
`mean_absorption_coefficient` + `room_surface`) enables `K2`; omit it and the
field is treated as free (`K2 = 0`). If the background margin drops below the
grade criterion or `K2` exceeds the validity limit, a `SoundPowerWarning`
flags that the levels are upper bounds; the determination still returns.

### K1 and K2 pitfalls

Both corrections subtract energy from the surface level, so overestimating
either one understates the emission. That is why the standards cap them, and
why most disputes over an enveloping-surface result trace back to one of
these habits:

- **K1 has a cliff, not a slope.** At a 15 dB margin the correction is a
  negligible 0.14 dB; at the 6 dB engineering criterion it is already
  1.26 dB, the largest value the grade accepts. Below the criterion the
  standard does not let the formula run on: `K1` is capped and the result is
  reported as an upper bound. Never extrapolate the subtraction into a
  smaller margin; raise the margin (quieter site, closer surface) or switch
  to the intensity method.
- **K1 assumes a stationary background.** The source-off reading must be
  taken at the same positions with the room in the same state, and the
  background energy must be the same during both readings. A ventilation
  system that cycles or a vehicle passing during either reading invalidates
  the pair; the energy subtraction also assumes source and background are
  incoherent, which holds for unrelated noise but not for the source's own
  reflections.
- **K2 removes the average room build-up, not discrete reflections.** A
  nearby wall, a trolley or another machine just outside the surface adds a
  specular contribution concentrated at a few microphones. That imbalance
  shows up in the apparent directivity index `DIi*`, and no room-average
  correction can remove it: move the surface, remove the reflector or treat
  it with absorption.
- **K2 is only as good as `A`.** With `A` from Sabine (`0.16·V/T`), errors
  in the reverberation time or the volume propagate directly. At the
  `K2 = 4 dB` validity limit about 60 % of the measured energy is room, not
  source, and a 20 % error in `A` still moves `LW` by about 0.5 dB. Prefer a
  measured `T60` over a guessed absorption coefficient, and keep the
  measurement distance small enough that `K2` stays well under the limit.

### `sound_power_pressure()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `levels_positions` | 2D array | dB | `(NM, NB)` | One row per position, one column per band (or a single A-weighted column) |
| `surface` | str | — | `'hemisphere'` / `'box'` | Measurement-surface shape |
| `radius` | float | m | > 0 (hemisphere) | Hemisphere radius `r` |
| `dimensions` | (float, float, float) | m | > 0 (box) | Reference-box `(l1, l2, l3)` |
| `distance` | float | m | > 0 (box) | Measurement distance `d` |
| `reflecting_planes` | int | — | `1` / `2` / `3`, default `1` | Halves/quarters the hemisphere area |
| `background_levels` | 2D array or spectrum | dB | `(NM, NB)`, or `(NB,)` / `(1, NB)` | Enables `K1`; a single spectrum broadcasts to every position |
| `frequencies` | 1D array | Hz | nominal band centres | Enables `LWA` (Annex E) |
| `absorption_area` | float or 1D array | m² | > 0 | `A` for `K2` (direct); per-band array → per-band `K2` |
| `reverberation_time`, `volume` | float/array, float | s, m³ | > 0 | `A = 0.16 V/T` for `K2`; per-band `T` → per-band `K2` |
| `mean_absorption_coefficient`, `room_surface` | float/array, float | —, m² | `(0,1]`, > 0 | `A = α·Sv` (Eq. A.7); per-band `α` → per-band `K2` |
| `grade` | str | — | `'engineering'` (default) / `'survey'` | ISO 3744 vs ISO 3746 |
| `omc_uncertainty` | float | dB | default `0.0` | `σomc`, operating/mounting instability, folded into `U` |

Returns a `SoundPowerResult`: `sound_power_level` (per-band `LW`),
`surface_pressure_level` (`Lp` after K1/K2), `mean_pressure_level`,
`background_correction`/`environmental_correction` (`K1`/`K2`),
`directivity_index` (apparent `DIi*` per microphone position **and** frequency
band, shape `(NM, NB)`; ISO 3744 clause 8.6), `surface_area`,
`sound_power_level_a` (`LWA`), `uncertainty` (expanded, 95 %) and `grade`.
`measurement_positions('hemisphere', radius=…, reflecting_planes=…, tones=…,
grade=…)` returns the normative `(N, 3)` microphone coordinates (Table B.1 for
tonal sources, B.2 for broadband).

## 2. Reverberation room, precision grade (ISO 3741)

In a qualified hard-walled **reverberation room** the field is diffuse, so a
handful of microphones sample the whole radiated energy and the method
reaches grade 1. The sound power comes from the mean room level `Lp(ST)`,
the Sabine absorption area `A = (55.26/c)·(V/T60)` and a chain of small
corrections (ISO 3741 Eq. 20):

$$
L_W = \bar{L}_p + 10 \log_{10}\frac{A}{A_0} + 4.34\ \frac{A}{S}
      + 10 \log_{10}\left( 1 + \frac{S c}{8 V f} \right) + C_1 + C_2 - 6 .
$$

The bracketed term is the **Waterhouse correction**: near the room
boundaries the sound energy density is higher than in the interior, and
this term (which vanishes as frequency grows) restores the energy the
interior microphones miss. `C1` (reference-quantity) and `C2`
(radiation-impedance) carry the result to the reference meteorological
conditions of 23 °C and 101.325 kPa,

$$
C_1 = -10 \log_{10}\frac{p_s}{p_{s0}} + 5 \log_{10}\frac{273.15 + \theta}{314}, \qquad
C_2 = -10 \log_{10}\frac{p_s}{p_{s0}} + 15 \log_{10}\frac{273.15 + \theta}{296},
$$

with the speed of sound `c = 20.05·√(273 + θ)`. The **comparison method**
replaces the absorption-area, Waterhouse and `C1` terms by a reference sound
source of known power `LW(RSS)` measured in the same room, so the room need
not be characterised: `LW = LW(RSS) + (Lp(ST) − Lp(RSS) + C2)`.

```python
import numpy as np
from phonometry import emission

# One-third-octave mean room SPL (dB), 100 Hz - 10 kHz, and the room's T60.
freqs = np.array([100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000,
                  1250, 1600, 2000, 2500, 3150, 4000, 5000, 6300, 8000, 10000],
                 dtype=float)
lp = np.linspace(80.0, 70.0, freqs.size)
t60 = np.full(freqs.size, 2.0)

rev = emission.sound_power_reverberation(
    lp, t60, volume=200.0, surface_area=220.0, frequencies=freqs,
    temperature=20.0, static_pressure=101.0,
)
print(round(rev.speed_of_sound, 1))                     # c = 343.2 m/s
print(round(float(rev.absorption_area[0]), 1))          # A = 16.1 m^2 at 100 Hz
print(round(float(rev.waterhouse_correction[0]), 2))    # 1.68 dB at 100 Hz
print(round(float(rev.sound_power_level[0]), 1))        # LW = 87.9 dB
print(round(rev.sound_power_level_a, 1))                # LWA = 92.1 dB

# Comparison method: a reference source of known LW measured at the same spots.
lw_rss = np.full(freqs.size, 85.0)
lp_rss = np.linspace(78.0, 69.0, freqs.size)
cmp = emission.sound_power_comparison(lp, lp_rss, lw_rss, frequencies=freqs, temperature=20.0)
print(round(float(cmp.sound_power_level[0]), 1), cmp.method)   # 86.9 comparison

rev.plot()   # reverberation-room LW spectrum, LWA in the title (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_reverberation_result_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_reverberation_result.svg" alt="The reverberation-room sound power level spectrum of the ISO 3741 example, one bar per one-third-octave band from 100 Hz to 10 kHz falling gently with frequency, with the A-weighted total of 92.1 dB(A) in the title" width="88%"></picture>

*The mean room level carried through the absorption-area, Waterhouse and
meteorological terms of Eq. 20 gives the one-third-octave `LW(f)`, and the
A-weighted energy sum across the 21 bands gives the `LWA` in the title.*

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import emission

# One-third-octave mean room SPL (dB), 100 Hz - 10 kHz, and the room's T60.
freqs = np.array([100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000,
                  1250, 1600, 2000, 2500, 3150, 4000, 5000, 6300, 8000, 10000],
                 dtype=float)
lp = np.linspace(80.0, 70.0, freqs.size)
t60 = np.full(freqs.size, 2.0)
rev = emission.sound_power_reverberation(
    lp, t60, volume=200.0, surface_area=220.0, frequencies=freqs,
    temperature=20.0, static_pressure=101.0,
)

# rev is the ReverberationSoundPowerResult computed above. One line:
rev.plot()
plt.show()

# By hand: a bar spectrum of LW with the A-weighted total in the title.
freqs = rev.frequencies
positions = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(positions, rev.sound_power_level, width=0.7, color="#1f77b4")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Reverberation-room sound power (ISO 3741)  "
    f"LWA = {rev.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

`levels` may be a 1D mean spectrum or a 2D `(NM, NB)` array averaged over
positions. When the room volume, its reverberation time or the microphone
count fail an ISO 3741 qualification criterion (Table 1 minimum volume, the
`V/S` reverberation floor, fewer than 6 positions, or an inter-position
spread above 1.5 dB), an advisory `SoundPowerWarning` is emitted and the
result still returns.

### `sound_power_reverberation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `levels` | 1D or 2D array | dB | per band, or `(NM, NB)` | Mean room SPL; 2D is energy-averaged over positions |
| `t60` | float or 1D array | s | > 0 | Room reverberation time (scalar broadcasts) |
| `volume` | float | m³ | > 0 | Room volume `V` |
| `surface_area` | float | m² | > 0 | Total room surface `S` (Waterhouse, `A/S`) |
| `frequencies` | 1D array | Hz | one per band | Required (Waterhouse needs `f`); enables `LWA` |
| `background_levels` | 1D or 2D array | dB | matches `levels` | `K1i` per microphone position (Eq. 14/15, before the Eq. 16 average; frequency-dependent criterion) |
| `temperature` | float | °C | default `23.0` | Sets `c`, `C1`, `C2` |
| `static_pressure` | float | kPa | default `101.325` | Sets `C1`, `C2` |

`sound_power_comparison(levels, levels_ref, lw_ref, *, frequencies=None,
background_levels=…, background_levels_ref=…, temperature=23.0,
static_pressure=101.325)` takes the same room levels plus the reference
source's levels and known power.

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `levels_ref` | 1D or 2D array | dB | matches `levels` | Mean room SPL with the reference source (RSS) running |
| `lw_ref` | 1D array | dB | per band | Known sound power `LW(RSS)` of the reference source |
| `background_levels` | 1D or 2D array | dB | matches `levels` | Background for the test source; per-band `K1` on `Lp(ST)` |
| `background_levels_ref` | 1D or 2D array | dB | matches `levels_ref` | Background for the **reference** source; per-band `K1` on `Lp(RSS)` |

`background_levels_ref` background-corrects the reference-source room level
`Lp(RSS)` exactly as `background_levels` does for the test source; both need
`frequencies` (the ISO 3741 criterion is frequency-dependent). Both return a
`ReverberationSoundPowerResult`
(`sound_power_level`, `mean_pressure_level`, `absorption_area`,
`waterhouse_correction`, `background_correction`, `c1`, `c2`,
`speed_of_sound`, `sound_power_level_a`, `method`; the absorption/Waterhouse/`c1`
fields are `NaN` for the comparison method).

## 3. Intensity scanning (ISO 9614-2)

Sound **intensity** is the net energy flux, so it distinguishes energy
*leaving* the source from steady energy merely passing through the surface,
which is why the intensity method tolerates background noise that would
defeat the pressure methods. A p-p probe (see the
[Sound Intensity guide](intensity.md)) is swept continuously over each of
`N` segments of a surface enclosing the source, reporting the segment-averaged
signed normal intensity `<In,i>`.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_pp_probe_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_pp_probe.svg" alt="A two-microphone p-p sound intensity probe: two pressure microphones separated by a spacer, from which the pressure gradient and hence the normal intensity are estimated" width="70%"></picture>

The partial powers sum to the total:

$$
P_i = \langle I_{n,i} \rangle\ S_i, \qquad P = \sum_i P_i, \qquad
L_W = 10 \log_{10}\frac{P}{P_0},\quad P_0 = 1\ \text{pW} .
$$

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_intensity_scan_power_dark.gif"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_intensity_scan_power.gif" alt="Animation: a p-p probe traces the serpentine scan over the top face of the measurement box while the normal-intensity arrows appear behind it, and the partial powers of the five faces accumulate into the sound power level L_W" width="640" height="360" loading="lazy"></picture>

[Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_intensity_scan_power.webm)

A band in which `P < 0` (net inflow, from a stronger source outside the
surface) is **not determinable** and reported as `NaN`. Two normative field
indicators qualify each band. The **surface pressure-intensity indicator**
`FpI` measures how reactive the field is, and the **negative-partial-power
indicator** `F+/-` measures how much energy circulates in and out:

$$
F_{pI} = [L_p] - L_W + 10 \log_{10}\frac{S}{S_0}, \qquad
F_{+/-} = 10 \log_{10}\frac{\sum_i \lvert P_i \rvert}{\lvert \sum_i P_i \rvert} .
$$

The probe's **dynamic capability** `Ld = δpI0 − K` (pressure-residual
intensity index minus the bias factor `K`, 10 dB for grade 2 and 7 dB for
grade 3) must exceed `FpI` (criterion 1); `F+/- ≤ 3 dB` is criterion 2
(mandatory for grade 2); and the two repeated sweeps must agree within the
Table 2 limit `s` per segment (criterion 3). A band is **engineering** grade
when criteria 1, 2 and 3 hold, **survey** when 1 and 3 hold, else `none`.
An A-weighted total additionally omits the bands failing criteria 1 and/or 2
(clause 10.6 b); the result flags them in `a_weighting_omitted_bands`.

```python
import numpy as np
from phonometry import emission

# 6 surface segments x 6 octave bands: signed normal intensity (W/m^2) from two
# repeated sweeps, the segment areas, and the per-segment surface SPL (dB).
freqs = np.array([125, 250, 500, 1000, 2000, 4000], dtype=float)
areas = np.full(6, 0.5)                                 # 0.5 m^2 per segment
rng = np.random.default_rng(0)
scan1 = np.abs(rng.normal(1e-4, 2e-5, size=(6, 6)))     # (segments, bands)
scan2 = scan1 * (1.0 + rng.normal(0.0, 0.02, size=(6, 6)))
pressure = np.full((6, 6), 80.0)

res = emission.sound_power_intensity(
    scan1, areas, normal_intensity_2=scan2, pressure_levels=pressure,
    pressure_residual_index=12.0, frequencies=freqs,
    band_type="octave", grade="engineering",
)
print(np.round(res.sound_power_level, 1))               # per-band LW
print(round(res.sound_power_level_a, 1))                # LWA, determinable + qualified bands
print(round(float(res.dynamic_capability_index[0]), 1)) # Ld = 12 - 10 = 2.0 dB
print(round(float(res.surface_pressure_intensity_index[0]), 2))   # FpI
print(list(res.achieved_grade))                         # per-band grade

res.plot()   # LW spectrum; non-positive (undeterminable) bands hatched (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_intensity_result_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_intensity_result.svg" alt="The intensity-scanning sound power level spectrum of the ISO 9614-2 example, one bar per octave band from 125 Hz to 4 kHz all near 85 dB, with the A-weighted total of 90.9 dB(A) in the title" width="88%"></picture>

*The partial powers `<In,i>·Si` of the six segments sum to each band's `LW`;
every band here nets positive power and passes the field-indicator criteria at
engineering grade, so all six bars stand, and the A-weighted total of
90.9 dB(A) heads the title.*

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import emission

# 6 surface segments x 6 octave bands: signed normal intensity (W/m^2) from two
# repeated sweeps, the segment areas, and the per-segment surface SPL (dB).
freqs = np.array([125, 250, 500, 1000, 2000, 4000], dtype=float)
areas = np.full(6, 0.5)                                 # 0.5 m^2 per segment
rng = np.random.default_rng(0)
scan1 = np.abs(rng.normal(1e-4, 2e-5, size=(6, 6)))     # (segments, bands)
scan2 = scan1 * (1.0 + rng.normal(0.0, 0.02, size=(6, 6)))
pressure = np.full((6, 6), 80.0)
res = emission.sound_power_intensity(
    scan1, areas, normal_intensity_2=scan2, pressure_levels=pressure,
    pressure_residual_index=12.0, frequencies=freqs,
    band_type="octave", grade="engineering",
)

# res is the SoundPowerIntensityResult computed above. One line:
res.plot()
plt.show()

# By hand: a bar spectrum of LW with the A-weighted total in the title.
freqs = res.frequencies
positions = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(positions, res.sound_power_level, width=0.7, color="#1f77b4")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Intensity-scanning sound power (ISO 9614-2)  "
    f"LWA = {res.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

Supplying `normal_intensity_2` (the second sweep) averages the two for the
partial powers and evaluates criterion 3; `pressure_levels` enables `FpI`;
`pressure_residual_index` (`δpI0`) plus a second sweep enables the per-band
achieved grade. The probe's finite-difference intensity has a
frequency-dependent bias handled in the [intensity guide](intensity.md).

### `sound_power_intensity()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `normal_intensity` | 2D array | W/m² | `(N_seg, N_bands)` | Signed segment-averaged normal intensity `<In,i>` (first sweep) |
| `areas` | 1D array | m² | > 0, `(N_seg,)` | Segment areas `Si` |
| `normal_intensity_2` | 2D array | W/m² | same shape | Second sweep → criterion 3 and averaging |
| `pressure_levels` | 2D array | dB | same shape | Segment SPL `Lpi` → `FpI` |
| `pressure_residual_index` | float or 1D array | dB | — | `δpI0` → `Ld` / criterion 1 |
| `frequencies` | 1D array | Hz | nominal centres | `LWA` and Table 2 limits |
| `band_type` | str | — | `'third'` (default) / `'octave'` | Table 2 lookup |
| `grade` | str | — | `'engineering'` (default) / `'survey'` | Selects `K` |
| `repeatability_limit` | float or 1D array | dB | default Table 2 | Override criterion-3 `s` |

Returns a `SoundPowerIntensityResult`: `partial_power`/`partial_power_level`
per segment and band, `sound_power`/`sound_power_level` (band total, `NaN`
where `negative_band`), `surface_pressure_intensity_index` (`FpI`),
`negative_partial_power_index` (`F+/-`), `repeatability`,
`dynamic_capability_index` (`Ld`), `achieved_grade`, `surface_area`,
`sound_power_level_a` and `grade`.

## 4. Precision grade, anechoic room (ISO 3745)

When the highest accuracy is required, ISO 3745 measures sound power in a
qualified **anechoic** or **hemi-anechoic** room, where the free field lets a
fixed array of microphones sample the radiated sound pressure directly. It is the
grade-1 counterpart to the enveloping-surface method of Section 1, with
standardized microphone coordinates, a per-position background correction and an
explicit meteorological correction.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_precision_anechoic_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_precision_anechoic.svg" alt="ISO 3745 precision sound power in an anechoic room: wedge-lined walls, the device under test at the centre and a hemispherical array of microphones at a fixed radius, with the sound power level formed from the surface-averaged pressure plus the area, background and meteorological corrections" width="92%"></picture>

**Sound power level (Clause 8).** The band sound power level is the
surface-averaged pressure level plus the surface term and the corrections:

$$
L_W = \overline{L_p} + 10\lg\frac{S}{S_0} + C_1 + C_2 + C_3,
$$

with $S = 4\pi r^2$ over the sphere or $S = 2\pi r^2$ over the hemisphere,
$S_0 = 1\ \text{m}^2$. $C_1$ and $C_2$ are the meteorological corrections
(reference and radiation-impedance terms); $C_3$ accounts for air absorption over
the measurement radius. The microphone positions are the standardized
unit-vector arrays of Tables D.1 (sphere), E.1 (hemisphere) and E.2 (hemisphere,
broadband).

```python
import numpy as np
from phonometry import emission

# The 40 standardized hemisphere positions (unit vectors scaled by the radius).
pos = emission.precision_positions("hemisphere", radius=1.0, count=40)
print(pos.shape)                      # (40, 3)

# Octave/third-octave band SPL (dB) at each of the 40 positions; here a uniform
# 74 dB in one band. The result carries S = 2*pi*r^2 and LW with C1+C2+C3.
levels = np.full((40, 1), 74.0)
res = emission.sound_power_anechoic(levels, "hemisphere", radius=1.0)
print(round(res.surface_area, 3))                 # 6.283  (2*pi*1^2)
print(np.round(res.sound_power_level, 2))         # [81.85]
```

**Background and meteorological corrections.** The $K_1$ background correction is
applied **per position** and floored where the signal-to-background difference is
small (Eq. 11); the meteorological correction is evaluated from the measured
temperature and static pressure.

```python
import numpy as np
from phonometry import emission

# K1 for a 6 dB signal-to-background difference in a <=200 Hz edge band: the
# floor is 1.26 dB (Eq. 11). Source and background levels are [positions, bands].
k1 = emission.precision_background_correction(
    np.array([[56.0]]), np.array([[50.0]]), np.array([200.0]))
print(round(float(k1[0, 0]), 4))      # 1.2563

# Meteorological corrections at the 23 C, 101.325 kPa reference (Eq. 16):
mc = emission.meteorological_corrections(23.0, 101.325)
print(round(mc.c1, 4), round(mc.c2, 4))   # -0.1282 0.0

# Expanded uncertainty (Clause 10.5 EXAMPLE): sigma_R0 = 0.5, sigma_omc = 2.0,
# k = 2 -> U = 4.1 dB.
print(round(emission.precision_uncertainty(0.5, 2.0, 2.0), 3))   # 4.123
```

Over several bands `sound_power_anechoic` returns a plottable
`PrecisionSoundPowerResult` carrying the per-band `LW` and the A-weighted total:

```python
import numpy as np
from phonometry import emission

# A mid-frequency-peaked machine measured over the 40-position hemisphere array
# (Annex E). levels_positions is the (40, NB) surface pressure spectrum: a base
# spectrum peaked near 1 kHz plus a small per-position spatial spread.
freqs = np.array([125, 250, 500, 1000, 2000, 4000, 8000], float)
base = 70.0 + 8.0 * np.exp(-(np.log2(freqs / 1000.0) ** 2) / 2.0)
rng = np.random.default_rng(7)
levels = base[None, :] + rng.normal(0.0, 1.0, (40, freqs.size))

result = emission.sound_power_anechoic(levels, "hemisphere", radius=1.0, frequencies=freqs)
print(round(result.sound_power_level_a, 1))   # 89.3
result.plot()   # LW spectrum, LWA in the title (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/precision_anechoic_power_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/precision_anechoic_power.svg" alt="The precision sound power level spectrum of a mid-frequency-peaked machine measured over the ISO 3745 hemisphere array, one bar per band peaking near 1 kHz, with the A-weighted total of 89.3 dB(A) in the title" width="88%"></picture>

*One bar per band: the surface-averaged pressure plus the area, background and
meteorological corrections give `LW(f)`, and the A-weighted energy sum across
bands gives the single-number `LWA` in the title.*

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import emission

# A mid-frequency-peaked machine measured over the 40-position hemisphere array
# (Annex E). levels_positions is the (40, NB) surface pressure spectrum: a base
# spectrum peaked near 1 kHz plus a small per-position spatial spread.
freqs = np.array([125, 250, 500, 1000, 2000, 4000, 8000], float)
base = 70.0 + 8.0 * np.exp(-(np.log2(freqs / 1000.0) ** 2) / 2.0)
rng = np.random.default_rng(7)
levels = base[None, :] + rng.normal(0.0, 1.0, (40, freqs.size))
result = emission.sound_power_anechoic(levels, "hemisphere", radius=1.0, frequencies=freqs)

# result is the PrecisionSoundPowerResult computed above. One line:
result.plot()
plt.show()

# By hand: a bar spectrum of LW with the A-weighted total in the title.
freqs = result.frequencies
positions = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(positions, result.sound_power_level, width=0.7, color="#1f77b4")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Precision sound power (ISO 3745)  LWA = {result.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

## 5. Precision intensity scanning (ISO 9614-3)

ISO 9614-3 is the grade-1 scanning method: like ISO 9614-2 it integrates the
normal intensity over a surface enclosing the source, but with a continuous
scan, tighter field-indicator criteria and an explicit uncertainty budget.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_intensity_scan_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_intensity_scan.svg" alt="ISO 9614-3 precision sound intensity scanning: a source enclosed by a measurement surface divided into segments, a two-microphone intensity probe scanned along a serpentine path over each segment, and the sound power formed by summing the normal intensity times segment area, subject to the field-indicator acceptance criteria" width="92%"></picture>

**Power and level (Clause 7).** The partial power of each segment is
$P_i = I_{n,i}\,S_i$; the total $P = \sum_i P_i$ gives
$L_W = 10\lg(P/P_0)$, $P_0 = 1\ \text{pW}$. A band whose net intensity is
negative (more power flowing in than out) is flagged not-applicable rather than
logged. The field indicators (temporal variability $F_T$, the signed and
unsigned pressure–intensity indicators, and the non-uniformity $F_S$) drive the
five acceptance criteria.

```python
import numpy as np
from phonometry import emission

# A fully enclosing surface with a uniform normal intensity In = W/S recovers
# the source power exactly: LW = 10*lg(W/P0). Here W = 100 uW -> 80 dB.
areas = np.array([0.5, 1.0, 0.25, 2.0])
w = 1.0e-4
i_n = np.full(areas.shape, w / float(areas.sum()))
res = emission.sound_power_intensity_precision(i_n, areas)
print(round(float(res.sound_power[0]), 6))          # 0.0001
print(round(float(res.sound_power_level[0]), 2))    # 80.0
```

Across several bands the result carries the per-band `LW` (`NaN` where the net
power is non-positive) and flags those bands `not_applicable`:

```python
import numpy as np
from phonometry import emission

# Four partial surfaces scanned over five one-third-octave bands. Each cell of
# partial_intensity is the signed normal intensity In_i (W/m^2); areas are the
# partial-surface areas Si. The 250 Hz band has net-negative power (a locally
# reactive field), so ISO 9614-3 flags it not-applicable (clause 9.2) -> NaN.
freqs = np.array([250, 500, 1000, 2000, 4000], float)
areas = np.array([0.5, 1.0, 0.75, 0.5])
base_intensity = np.array([2.0e-6, 8.0e-6, 2.0e-5, 1.0e-5, 3.0e-6])
partial_intensity = base_intensity[None, :] * np.array([1.0, 1.1, 0.9, 1.05])[:, None]
partial_intensity[:, 0] = [2.0e-6, -3.0e-6, -4.0e-6, -1.0e-6]   # net-negative band

result = emission.sound_power_intensity_precision(partial_intensity, areas, frequencies=freqs)
print(result.not_applicable_band.tolist())   # [True, False, False, False, False]
print(round(result.sound_power_level_a, 1))   # 80.6
result.plot()   # LW spectrum; the not-applicable band is hatched (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/intensity_scan_power_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/intensity_scan_power.svg" alt="The precision intensity-scanning sound power level spectrum over five one-third-octave bands, four determinate bars and a hatched, greyed 250 Hz band flagged not-applicable because its net intensity is negative, with the A-weighted total of 80.6 dB(A) in the title" width="88%"></picture>

*The 250 Hz band nets negative (more energy flowing in than out), so ISO 9614-3
declares it not-applicable; the figure hatches and greys it while the four
determinate bands and the A-weighted total stand.*

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import emission

# Four partial surfaces scanned over five one-third-octave bands. Each cell of
# partial_intensity is the signed normal intensity In_i (W/m^2); areas are the
# partial-surface areas Si. The 250 Hz band has net-negative power (a locally
# reactive field), so ISO 9614-3 flags it not-applicable (clause 9.2) -> NaN.
freqs = np.array([250, 500, 1000, 2000, 4000], float)
areas = np.array([0.5, 1.0, 0.75, 0.5])
base_intensity = np.array([2.0e-6, 8.0e-6, 2.0e-5, 1.0e-5, 3.0e-6])
partial_intensity = base_intensity[None, :] * np.array([1.0, 1.1, 0.9, 1.05])[:, None]
partial_intensity[:, 0] = [2.0e-6, -3.0e-6, -4.0e-6, -1.0e-6]   # net-negative band
result = emission.sound_power_intensity_precision(partial_intensity, areas, frequencies=freqs)

# result is the PrecisionIntensityResult computed above. One line:
result.plot()
plt.show()

# By hand: determinate bands as LW bars; a not-applicable band (its LW is NaN)
# is flagged by a full-height greyed, hatched span rather than a zero-height bar.
freqs = result.frequencies
positions = np.arange(freqs.size)
neg = result.not_applicable_band
lw = np.nan_to_num(result.sound_power_level)
fig, ax = plt.subplots()
ax.bar(positions[~neg], lw[~neg], width=0.7, color="#1f77b4")
for pos in positions[neg]:
    ax.axvspan(pos - 0.35, pos + 0.35, facecolor="#888888", alpha=0.28,
               hatch="//", edgecolor="#888888")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Precision intensity scanning (ISO 9614-3)  "
    f"LWA = {result.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

## See also

- [Sound Intensity (p-p)](intensity.md): the two-microphone probe, its
  finite-difference bias and the ISO 9614-1 field indicators behind the
  scanning method.
- [Room Acoustics](room-acoustics.md): the reverberation time
  and equivalent absorption area (ISO 354) that feed `K2` and the ISO 3741
  absorption area; impact and airborne insulation.
- [Levels](levels.md): energy averaging and the A-weighting behind `LWA`.
- [Theory](theory-environment-transport.md): the Waterhouse, K1/K2 and C1/C2 derivations.
- API reference: [`emission.sound_power`](https://jmrplens.github.io/phonometry/reference/api/power/sound-power/), [`emission.sound_power_reverberation`](https://jmrplens.github.io/phonometry/reference/api/power/sound-power-reverberation/) and [`emission.sound_power_intensity`](https://jmrplens.github.io/phonometry/reference/api/power/sound-power-intensity/).

## References

- Fahy, F. J. (1995). *Sound intensity* (2nd ed.). E&FN Spon.
  ISBN 978-0-419-19810-9.
  [doi:10.4324/9780203475386](https://doi.org/10.4324/9780203475386).
  The monograph on sound energy flux: why intensity separates the energy
  leaving the source from steady energy passing through, behind the
  scanning methods of sections 3 and 5.
- Beranek, L. L., & Mellow, T. J. (2012). *Acoustics: Sound fields and
  transducers*. Academic Press. ISBN 978-0-12-391421-7.
  [doi:10.1016/C2011-0-05897-0](https://doi.org/10.1016/C2011-0-05897-0).
  Radiation and sound fields: the free-field and diffuse-field relations
  between pressure and power that the enveloping-surface and
  reverberation-room methods rest on.
- International Organization for Standardization. (2019). *Acoustics —
  Determination of sound power levels of noise sources — Guidelines for the
  use of basic standards* (ISO 3740:2019).
  [iso.org catalogue](https://www.iso.org/standard/45107.html).
  The selection guide behind "Choosing a method": grades, environments,
  source-size and background criteria for the whole family.
- International Organization for Standardization. (2010). *Acoustics —
  Determination of sound power levels and sound energy levels of noise
  sources using sound pressure — Precision methods for reverberation test
  rooms* (ISO 3741:2010).
  [iso.org catalogue](https://www.iso.org/standard/52053.html).
  The reverberation-room method of section 2.
- International Organization for Standardization. (2010). *Acoustics —
  Determination of sound power levels and sound energy levels of noise
  sources using sound pressure — Engineering methods for an essentially free
  field over a reflecting plane* (ISO 3744:2010).
  [iso.org catalogue](https://www.iso.org/standard/52055.html).
  The enveloping-surface method of section 1.
- International Organization for Standardization. (2012). *Acoustics —
  Determination of sound power levels and sound energy levels of noise
  sources using sound pressure — Precision methods for anechoic rooms and
  hemi-anechoic rooms* (ISO 3745:2012).
  [iso.org catalogue](https://www.iso.org/standard/45362.html).
  The precision anechoic-room method of section 4.

## Standards

ISO 3744:2010, *Acoustics — Determination of sound power levels
and sound energy levels of noise sources using sound pressure — Engineering
methods for an essentially free field over a reflecting plane*: the
enveloping-surface method: hemisphere and box surface areas, the `K1`/`K2`
corrections, the Annex B microphone positions and the Annex E A-weighting.
ISO 3746:2010, *… Survey method using an enveloping measurement surface over a
reflecting plane*: the survey grade sharing the same formulae with coarser
criteria. ISO 3741:2010, *… Precision methods for reverberation test rooms*:
the direct (Eq. 20) and comparison methods with the Waterhouse and
meteorological corrections and the Table 1 qualification criteria.
ISO 3745:2012, *… Precision methods for anechoic rooms and hemi-anechoic
rooms*: the Clause 8 power level, the per-position background correction
(Eq. 11), the meteorological corrections and the standardized microphone
arrays. ISO 9614-2:1996, *Acoustics — Determination of sound power levels of
noise sources using sound intensity — Part 2: Measurement by scanning* — the
partial powers, the `FpI` and `F+/-` field indicators and the grade criteria.
ISO 9614-3:2002, *… Part 3: Precision method for measurement by scanning*:
the grade-1 scanning method, its field indicators and the clause 9.2
not-applicable flagging.
