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 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).

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.

Download the animation (WebM)

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.

Download the animation (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.

Show the code for this figure

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()

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:

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

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,

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

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.

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)

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.

Show the code for this figure

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()

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):

$$L_W={\bar{L}}_p+10{\log }_{10}\frac{A}{A_0}+4.34\frac{A}{S}+10{\log }_{10}\left(1+\frac{Sc}{8Vf}\right)+C_1+C_2-6.$$

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,

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

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).

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)

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.

Show the code for this figure

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()

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	Per-band K1 (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) is swept continuously over each of N segments of a surface enclosing the source, reporting the segment-averaged signed normal intensity <In,i>.

The partial powers sum to the total:

$$P_i=⟨I_{n,i}⟩S_i,P=\sum _i{P}_i,L_W=10{\log }_{10}\frac{P}{P_0},P_0=1\text{pW}.$$

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.

Download the animation (WebM)

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.

Download the animation (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:

$$F_{pI}=[L_p]-L_W+10{\log }_{10}\frac{S}{S_0},F_{+/-}=10{\log }_{10}\frac{{\sum }_i|P_i|}{|{\sum }_i{P}_i|}.$$

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.

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 over determinable 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)

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.

Show the code for this figure

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()

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.

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.

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).

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.

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

The MeteorologicalCorrection is a pair of scalars (plus the per-band C3 when the attenuation coefficient is supplied per band) rather than a plottable spectrum: the corrections fold into the PrecisionSoundPowerResult as its c1/c2/c3 fields, and the .report() fiche prints them on its measurement-basis strip.

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

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)

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.

Show the code for this figure

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()

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.

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.

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), flags those bands not_applicable, and draws them with the one-line result.plot() of the figure below:

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)

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.

Show the code for this figure

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()

6. The measurement report (.report())

A sound power determination ends as a document. Every result of this page stays plottable while it is being worked on — res.plot() draws the same LW spectrum interactively that the fiche typesets — and the report step wraps it into the deliverable. Both the enveloping-surface result (SoundPowerResult, ISO 3744/3746) and the precision result (PrecisionSoundPowerResult, ISO 3745) expose a .report() method that writes a one-page PDF fiche laid out like a sound-power test sheet: the standard-basis line naming the applied method and accuracy grade, an optional metadata header (client, noise source, test environment, instrumentation, climate, date), a per-band table (nominal octave/one-third-octave frequency, the surface sound-pressure level $L_p$ and the band sound-power level $L_W$), the sound-power spectrum $L_W(f)$ with a nominal band axis, and a boxed A-weighted sound power level $L_{WA}$ (dB re 1 pW) with the total $L_W$, the expanded uncertainty $U$ and the measurement surface area $S$ alongside.

The metadata is supplied through a ReportMetadata, whose applicable fields here are the source description (specimen), the test environment (test_room), the client, the instrumentation, the temperature, relative humidity and ambient pressure, the date of test (test_date) and the footer identity (laboratory, operator, report_id, notes); the measurement surface area $S$ comes from the result itself and is printed in the result box and the basis strip, together with the applied corrections (the background $K_1$ and environmental $K_2$ for the ISO 3744/3746 surface method, or the meteorological $C_1$/$C_2$/$C_3$ for the ISO 3745 precision method). Supplying requirement adds a PASS/FAIL verdict against a declared A-weighted sound-power limit (a sound-power emission is a quantity where less is better, so the source passes at or below the limit). verbose=True adds the energy-averaged level $L_p^{\prime }$ to the table, and for the ISO 3744/3746 surface result it also adds the $K_1$/$K_2$ correction columns (the ISO 3745 precision result carries no $K_1$/$K_2$; its $C_1$/$C_2$/$C_3$ appear in the basis strip). language="es" renders the Spanish fiche with comma decimals.

import numpy as np
from phonometry import ReportMetadata, emission

freqs = np.array([63, 125, 250, 500, 1000, 2000, 4000, 8000], float)
# Ten identical position spectra over a hemisphere of radius 4 m; background a
# uniform 10 dB below and an equivalent absorption area A = 1500 m^2 (so K1, K2
# are meaningful and within the engineering validity limit).
surface = np.array([72.0, 76, 80, 82, 81, 78, 73, 66])
res = emission.sound_power_pressure(
    np.tile(surface, (10, 1)), "hemisphere", radius=4.0,
    background_levels=np.tile(surface - 10.0, (10, 1)),
    frequencies=freqs, absorption_area=1500.0, grade="engineering",
)

res.report(
    "sound_power.pdf",
    metadata=ReportMetadata(
        client="Example manufacturing plant",
        specimen="Hydraulic power pack (floor-standing)",
        test_room="Hemi-anechoic room over a reflecting floor",
        instrumentation="Class 1 sound level meter (IEC 61672-1), s/n 0042",
        laboratory="Phonometry reference example",
        report_id="EXAMPLE-3744",
        requirement=105.0,
    ),
)   # LWA = 103.7 dB(A) re 1 pW -> declared limit 105 dB(A): PASS

The example fiche is regenerated with make reports and kept rendered in the repository; click the preview to open the PDF.

One-page ISO 3744 sound-power determination fiche: a header with the client, the noise source, the hemi-anechoic test environment and the instrumentation and climate, the octave-band table (63 Hz to 8 kHz) of surface sound-pressure levels Lp and band sound-power levels LW, the sound-power spectrum LW(f) with a nominal band axis, the boxed A-weighted sound power level LWA = 103.7 dB(A) re 1 pW with the total LW = 105.8 dB, the expanded uncertainty U = 3.0 dB and the measurement surface S = 100.53 m2, and a PASS verdict against the declared 105 dB(A) limit, closed by a basis strip stating the applied K1 = 0.5 dB and K2 = 1.0 dB corrections.

Download the report (PDF)

Sound power determination fiche (SoundPowerResult.report), an ISO 3744 engineering-grade hemisphere measurement with the K1/K2 corrections and the boxed LWA.

The intensity determination (ISO 9614-2)

The intensity-scanning result (SoundPowerIntensityResult, ISO 9614-2) writes the same one-page fiche through its own .report(). The standard-basis line names ISO 9614-2:1996 and the measurement grade, the per-band table lists the intensity-derived band sound-power level $L_W$, and the boxed $L_{WA}$ carries the total $L_W$, the measurement surface $S$ and the determination grade (the intensity result has no expanded uncertainty $U$). verbose=True adds the field indicators $F_{pI}$ (surface pressure-intensity) and $F_{+/-}$ (negative partial power) and the per-band achieved grade; the basis strip states the partial-power model (the segment partial powers $P_i=I_{n,i}{S}_i$ summing to $P$) and the Annex B qualification criteria. A band whose net power is non-positive is not determinable (clause 9.2) and prints an em dash.

import numpy as np
from phonometry import ReportMetadata, emission

freqs = np.array([125, 250, 500, 1000, 2000, 4000], float)
# Six equal 0.5 m^2 segments (S = 3.0 m^2); one uniform normal-intensity
# spectrum scanned twice, with the surface SPL and the instrument residual
# index that qualify every band at engineering grade.
intensity = np.array([0.6e-4, 1.0e-4, 1.5e-4, 1.4e-4, 0.9e-4, 0.5e-4])
scan = np.tile(intensity, (6, 1))
res = emission.sound_power_intensity(
    scan, np.full(6, 0.5), normal_intensity_2=scan.copy(),
    pressure_levels=np.full((6, 6), 80.0), pressure_residual_index=15.0,
    frequencies=freqs, band_type="octave", grade="engineering",
)

res.report(
    "sound_power_intensity.pdf",
    metadata=ReportMetadata(
        client="Example manufacturing plant",
        specimen="Hydraulic power pack (floor-standing)",
        test_room="Machine hall with steady background noise",
        instrumentation="Class 1 p-p intensity probe (IEC 61043), s/n 0042",
        laboratory="Phonometry reference example",
        report_id="EXAMPLE-9614",
        requirement=93.0,
    ),
)   # LWA = 90.9 dB(A) re 1 pW -> declared limit 93 dB(A): PASS

The example fiche is regenerated with make reports and kept rendered in the repository; click the preview to open the PDF.

One-page ISO 9614-2 sound-power-by-intensity determination fiche: a header with the client, the noise source, the machine-hall test environment and the intensity probe and climate, the octave-band table (125 Hz to 4 kHz) of intensity-derived band sound-power levels LW, the sound-power spectrum LW(f) with a nominal band axis, the boxed A-weighted sound power level LWA = 90.9 dB(A) re 1 pW with the total LW = 92.5 dB, the measurement surface S = 3.00 m2 and the engineering grade, and a PASS verdict against the declared 93 dB(A) limit, closed by a basis strip stating the partial-power model, the field indicators FpI and F+/- and the Annex B qualification criteria.

Download the report (PDF)

Sound power by intensity fiche (SoundPowerIntensityResult.report), an ISO 9614-2 engineering-grade scan with the field indicators and the boxed LWA.

The reverberation-room determination (ISO 3741)

The reverberation-room result (ReverberationSoundPowerResult, ISO 3741) writes the same one-page fiche through its own .report(). The standard-basis line names ISO 3741:2010 and the precision accuracy grade (grade 1) and states which method was used, the direct method using the room equivalent absorption area (Eq. 20) or the comparison method using a reference sound source (Eq. 21). The per-band table lists the mean room sound-pressure level $L_p$ and the band sound-power level $L_W$, and the boxed $L_{WA}$ carries the total $L_W$ and the determination method (the reverberation result has no expanded uncertainty $U$). verbose=True adds the background correction $K_1$ and, for the direct method, the equivalent absorption area $A$ and the Waterhouse boundary correction $C_w$; the basis strip states the correction model (Eq. 20 or Eq. 21), the applied meteorological corrections $C_1$/$C_2$ and the speed of sound, and cites the Annex F A-weighting.

import numpy as np
from phonometry import ReportMetadata, emission

freqs = np.array([125, 250, 500, 1000, 2000, 4000, 8000], float)
# Octave-band mean room sound-pressure levels in a qualified
# reverberation room of V = 200 m3, S = 240 m2, with a uniform T60 = 2.0 s.
lp = np.array([80.0, 83.0, 85.0, 84.0, 80.0, 75.0, 68.0])
res = emission.sound_power_reverberation(
    lp, 2.0, volume=200.0, surface_area=240.0, frequencies=freqs,
    temperature=20.0, static_pressure=101.325,
)

res.report(
    "sound_power_reverberation.pdf",
    metadata=ReportMetadata(
        client="Example manufacturing plant",
        specimen="Hydraulic power pack (floor-standing)",
        test_room="Qualified reverberation room, V = 200 m3, T60 = 2.0 s",
        instrumentation="Class 1 sound level meter (IEC 61672-1), s/n 0042",
        laboratory="Phonometry reference example",
        report_id="EXAMPLE-3741",
        requirement=96.0,
    ),
)   # LWA = 94.3 dB(A) re 1 pW -> declared limit 96 dB(A): PASS

The example fiche is regenerated with make reports and kept rendered in the repository; click the preview to open the PDF.

One-page ISO 3741 reverberation-room sound-power determination fiche: a header with the client, the noise source, the qualified reverberation test room and the instrumentation and climate, the octave-band table (125 Hz to 8 kHz) of mean room sound-pressure levels Lp and band sound-power levels LW, the sound-power spectrum LW(f) with a nominal band axis, the boxed A-weighted sound power level LWA = 94.3 dB(A) re 1 pW with the total LW = 96.7 dB and the direct determination method, and a PASS verdict against the declared 96 dB(A) limit, closed by a basis strip stating the Eq. 20 correction model with the Sabine absorption area, the Waterhouse boundary term and the meteorological corrections C1 and C2, and the Annex F A-weighting.

Download the report (PDF)

Reverberation-room sound power fiche (ReverberationSoundPowerResult.report), an ISO 3741 precision-grade direct-method determination with the Waterhouse and C1/C2 corrections and the boxed LWA.

7. Declaring the noise emission (ISO 4871)

A measured sound power level is not yet a declaration. ISO 4871:1996 is the standard for the noise-emission declaration a manufacturer prints in technical documents: which quantities are stated, in which form, and how a declared value is verified. The preferred quantity is the A-weighted sound power level L_WA, optionally accompanied by the A-weighted emission sound pressure level L_pA at a work station.

A declaration takes one of two alternative forms (clause 4):

• the dual-number form (clause 3.16): the measured value L_WA and its uncertainty K_WA stated together but separately; and
• the single-number form (clause 3.15): the derived declared value L_WAd = L_WA + K_WA, an upper limit that repeated measurements are unlikely to exceed at the stated confidence level.

K_WA combines the measurement (reproducibility) and, for a batch, the production spread; for a single machine K = 1.645 sigma_R (Annex A.2.2). A NoiseEmissionDeclaration holds one or more per-operating-mode declarations and renders the ISO 4871 fiche through .report(). The quickest route is to declare() straight from a measured sound power:

import numpy as np
import phonometry as ph
from phonometry import ReportMetadata

# ... a measured LWA from ISO 3744 ...
result = ph.sound_power_pressure(levels, "hemisphere", radius=1.0,
                                 frequencies=freqs)

declaration = result.declare(
    uncertainty=2.0,                 # K_WA in dB (defaults to the expanded U)
    machine="Type 990, Model 11-TC",
    operating_conditions="50 Hz, 230 V, rated load",
    basic_standards="ISO 3744",
    verification_level=result.sound_power_level_a,  # L_1 for clause 6.2
)
declaration.report(
    "iso4871.pdf",
    metadata=ReportMetadata(measurement_standard="ISO 3744"),
)   # -> L_WAd = L_WA + K_WA, verified when L_1 <= L_WAd

Or build the declaration directly, reproducing the ISO 4871 Annex B example (two operating modes, L_WA = 88 and 95 dB with K_WA = 2 dB, giving declared L_WAd = 90 and 97 dB):

mode1 = ph.OperatingModeDeclaration(
    "Operating mode 1", sound_power_level=88.0, sound_power_uncertainty=2.0,
    emission_pressure_level=78.0, emission_pressure_uncertainty=2.0,
    verification_level=89.0,          # passes: 89 <= 90
)
mode2 = ph.OperatingModeDeclaration(
    "Operating mode 2", sound_power_level=95.0, sound_power_uncertainty=2.0,
    emission_pressure_level=86.0, emission_pressure_uncertainty=2.0,
    verification_level=98.0,          # fails: 98 > 97
)
ph.NoiseEmissionDeclaration(
    (mode1, mode2), machine="Type 990, Model 11-TC",
    basic_standards=("ISO 3744", "ISO 11202"), form="dual-number",
).report("iso4871.pdf")

The example fiche is regenerated with make reports and kept rendered in the repository; click the preview to open the PDF.

One-page ISO 4871 noise-emission declaration fiche: a header with the machine identification and operating conditions, the declared dual-number table across two operating-mode columns listing the measured A-weighted sound power level L_WA, its uncertainty K_WA, the emission sound pressure level L_pA and the derived declared value L_WAd = L_WA + K_WA (90 and 97 dB), the noise-test-code and basic-standards footnote, and a clause 6.2 verification table where mode 1 passes and mode 2 fails.

Download the report (PDF)

Noise emission declaration fiche (NoiseEmissionDeclaration.report), the ISO 4871 Annex B dual-number table with the declared L_WAd = L_WA + K_WA and the clause 6.2 verification verdict.

What this guide covers

Covered. The five routes to LW: ISO 3744/3746 enveloping-surface sound pressure (surface areas, K1/K2, the Annex B microphone positions and the Annex E A-weighting band corrections); ISO 3741 reverberation-room direct and comparison methods (Eq. 20/21, the Waterhouse and C1/C2 meteorological corrections, the Table 1 qualification criteria); ISO 9614-2 intensity scanning (partial powers, the FpI and F+/- field indicators and the clause 9.2 not-applicable flagging for P < 0 bands); ISO 3745 precision anechoic/hemi-anechoic (the Annex D/E fixed arrays, the per-position K1i background correction and C1/C2/C3); ISO 9614-3 precision intensity scanning (its own field indicators and grade-1 qualification criteria); and ISO 4871 declaration (the dual- and single-number forms, L_WAd = L_WA + K_WA and the clause 6.2 verification).

Not covered. None of the methods implement the underlying room or facility qualification test: ISO 3741’s reverberation-room qualification (Annex C/D, eigenfrequency counting or a reference-source comparison) and ISO 3745’s free-field qualification of the anechoic or hemi-anechoic environment are both assumed, not performed; the library only warns on the coarse advisory criteria ISO 3741 states explicitly (the Table 1 minimum volume and the V/S reverberation-time floor). ISO 9614-1, the discrete fixed-point intensity method, is not one of the five routes here: only its field indicators are reused, and the discrete-point power determination itself belongs to the sound intensity guide. The C3 meteorological correction of ISO 3745 needs an air-absorption coefficient the caller supplies (air_absorption_coefficient=); this module does not compute it from ISO 9613-1 itself.

See also

• Sound Intensity (p-p): the two-microphone probe, its finite-difference bias and the ISO 9614-1 field indicators behind the scanning method.
• Room Acoustics: the reverberation time and equivalent absorption area (ISO 354) that feed K2 and the ISO 3741 absorption area.
• Predicting Sound Insulation (EN 12354): the airborne and impact insulation ratings and the EN 12354 emission predictions that consume a source LW.
• Levels: energy averaging and the A-weighting behind LWA.
• Theory: the Waterhouse, K1/K2 and C1/C2 derivations.
• API reference: emission.sound_power, emission.sound_power_reverberation and emission.sound_power_intensity.

Quick answers

What is the difference between sound power and sound pressure?

Sound pressure depends on where you stand and on the room; 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 or is checked against a noise-emission limit; ISO 3744, ISO 3741, ISO 9614-2, ISO 3745 and ISO 9614-3 all determine it.

What do the accuracy grades in sound power measurement 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. A grade-2 LWA carries U ≈ 3 dB, so two grade-2 results 2 dB apart are statistically indistinguishable.

How do I measure sound power when background noise cannot be switched off?

Use intensity scanning: ISO 9614-2 (grade 2 or 3) or ISO 9614-3 (grade 1, precision). Because sound intensity is the net energy flux through the measurement surface, steady extraneous noise even some 10 dB above the source is tolerated, whereas the ISO 3744 pressure method needs the background at least 6 dB below the source. The per-band field indicators then decide the grade actually achieved.

References

• Beranek, L. L., & Mellow, T. J. (2012). Acoustics: Sound fields and transducers. Academic Press. https://doi.org/10.1016/C2011-0-05897-0ISBN 978-0-12-391421-7. 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.
• Fahy, F. J. (1995). Sound intensity (2nd ed.). E&FN Spon. https://doi.org/10.4324/9780203475386ISBN 978-0-419-19810-9. 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.
• International Organization for Standardization. (1996). Acoustics — Declaration and verification of noise emission values of machinery and equipment (ISO 4871:1996). The declaration behind section 6: the dual/single-number forms, L_WAd = L_WA + K_WA (clause 3.15) and the clause 6.2 verification.
• International Organization for Standardization. (1996). Acoustics — Determination of sound power levels of noise sources using sound intensity — Part 2: Measurement by scanning (ISO 9614-2:1996). The scanning method of section 3: the partial powers, the FpI and F+/- field indicators and the grade criteria.
• International Organization for Standardization. (2002). Acoustics — Determination of sound power levels of noise sources using sound intensity — Part 3: Precision method for measurement by scanning (ISO 9614-3:2002). The grade-1 scanning method of section 5, its field indicators and the clause 9.2 not-applicable flagging.
• 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). The enveloping-surface method of section 1: hemisphere and box surface areas, the K1/K2 corrections, the Annex B microphone positions and the Annex E A-weighting.
• 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). The reverberation-room method of section 2: the direct (Eq. 20) and comparison methods with the Waterhouse and meteorological corrections and the Table 1 qualification criteria.
• International Organization for Standardization. (2010). Acoustics — Determination of sound power levels and sound energy levels of noise sources using sound pressure — Survey method using an enveloping measurement surface over a reflecting plane (ISO 3746:2010). The survey grade of section 1, sharing the enveloping-surface formulae with coarser criteria.
• 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). The precision anechoic-room method of section 4: the Clause 8 power level, the per-position background correction (Eq. 11), the meteorological corrections and the standardized microphone arrays.
• International Organization for Standardization. (2019). Acoustics — Determination of sound power levels of noise sources — Guidelines for the use of basic standards (ISO 3740:2019). The selection guide behind 'Choosing a method': grades, environments, source-size and background criteria for the whole family.