Sound Power in the Reverberation Room

A reverberation room turns its main defect as a listening space into a measuring instrument: because the field is diffuse, a handful of microphone positions sample the whole radiated energy, and the sound power follows from the mean room level and the room’s absorption with no enveloping surface to build. That is why ISO 3741 is a grade-1 (precision) method, and why it is the laboratory route of choice for steady, broadband sources small enough to travel to a qualified room. This guide covers the direct method through the Sabine absorption area, the Waterhouse and meteorological corrections, the comparison method against a reference sound source, the qualification warnings and the accredited-style test fiche. Which route fits which job, and the pressure and intensity alternatives, are weighed in Sound Power.

1. Reverberation room, precision grade (ISO 3741)

Both methods are defined over the one-third-octave bands from 100 Hz to 10 kHz (clause 8.1). Annex E extends that range down to 50 Hz informatively; nothing extends it upwards. The lower bound is set by the room’s mode density, which is why Table 1 pairs the lowest band of interest with a minimum volume — 200 m³ at 100 Hz, 150 m³ at 125 Hz, 100 m³ at 160 Hz, 70 m³ from 200 Hz up — and the upper bound is set by air absorption. The source under test must be no larger than about 2 % of the room volume (clause 1.2; up to 5 % only where a noise test code says what that costs in uncertainty), so that it does not load the field it is being measured in, and it must be steady over the measurement interval: Eq. 20 energy-averages and reports a rate of energy flow, so it characterises the machine only if the machine is time-invariant while it is running. A single event — one press stroke, one door slam — is described by the sound energy level $L_J$ of clause 9.2 instead, which is not implemented here.

The background must lie at least 6 dB below the source level in the one-third-octave bands at and below 200 Hz and at and above 6,3 kHz, and at least 10 dB below between 250 Hz and 5 kHz (clause 5.4.1.1) — a tighter pair of criteria than the flat 6 dB of the enveloping-surface methods, which is what “the ISO 3741 criterion is frequency-dependent” in the parameter table below means. Below those margins sound_power_reverberation clamps $K_1$ to the value the criterion itself produces (1.26 dB and 0.46 dB respectively), warns, and the band becomes an upper bound rather than a determination. A band whose corrected A-weighted contribution is at least 15 dB below the largest may instead be dropped from the range of interest altogether (clause 5.4.1.2).

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 $L_p(\text{ST})$, the Sabine absorption area $A=(55.26/c)\cdot (V/T_{60})$ 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. $C_1$ (reference-quantity) and $C_2$ (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 $c=20.05\sqrt{273+\theta }$.

Two of the five terms usually go unexplained. The −6 dB is the diffuse-field relation itself: in a diffuse field the power a source feeds into the room equals ${\bar{p}}^{2}A/(4\rho c)$, and expressing that against the two reference quantities leaves a constant, which for the references of these standards is −6 dB. The 4.34 A/S term is the first-order correction from the Sabine absorption area towards the Eyring value; 4.34 is $10\lg e$ and $A/S$ is the room’s mean absorption coefficient, so the term is small in a hard room and grows as the room is damped. In the example below, $A=16.1$ m² against $S=220$ m², so it contributes 0.32 dB in every band — negligible here, and the first term to watch when the reverberation time sits near the qualification floor. The Waterhouse term runs the other way, 1.68 dB at 100 Hz and 0.02 dB at 10 kHz on the same data: the low bands are ruled by the boundary energy the interior microphones miss, the high bands by the absorption area.

The right-hand panel of the clip below is the reverberation room. The same source runs in both rooms; in the anechoic room on the left the microphones see only what the source sends their way, so the level falls with distance and the free-field route has to integrate over a measurement surface, while in the reverberation room on the right the reflected energy fills the space and the level stops depending on where a microphone is. That is the whole reason Eq. 20 can replace a surface integral with a handful of positions and a room constant — and the reason the room, not the array, is what has to be qualified. Both routes end on the same $L_W$, because sound power is a property of the source and not of the room it is measured in.

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)

Where the microphones and the source go

Everything Eq. 20 does rests on the array being in the reverberant part of the field, and the standard fixes that with one distance. The source sits on the floor, at least 1,5 m from any wall, and in a rectangular room asymmetrically (clause 7.3), so that it does not couple preferentially into one mode family. The nearest microphone must then stand off by

$$d_{\min }=D_1\sqrt{\frac{V}{T_{60}}},D_1=0.08,$$

with $D_1=0.16$ strongly recommended below 5 kHz (clause 8.3, Eq. 8). For the room of the example below, $V=200$ m³ and $T_{60}=2.0$ s, that is $d_{\min }=0.8$ m, or 1.6 m at the recommended $D_1$ — so a microphone 1 m from the source is already inside the near field this method is not measuring.

If the room has not been qualified for discrete tones per Annex D, the array is six discrete positions, each more than 1,0 m from any room surface, more than $d_{\min }$ from the source, and separated from one another by at least half a wavelength at the lowest band of interest (1,7 m at 100 Hz). A continuous traverse may replace them for the levels themselves, provided every point on it keeps $d_{\min }$ from the source and 1,0 m from any room surface, stays 0,5 m clear of any diffuser, does not lie in a plane within 10° of a room surface, and runs for at least $3\lambda$ or 10,3 m, whichever is the smaller. The six discrete positions are still needed to estimate the standard deviations below.

Each level is time-averaged over at least 30 s in the bands centred at and below 160 Hz and at least 10 s from 200 Hz up (clause 8.4.1) — the low bands need the longer interval because a diffuse field is a modal average, and there are few modes down there to average over. A traversing microphone must cover at least two full traverses. The background reading is taken immediately before or immediately after the source reading, at the same positions and over the same interval; that is what background_levels is, and a spectrum measured on another day is not it.

The SoundPowerWarning at an inter-position spread of 1.5 dB is the entry point to the escalation, not a defect in itself. Clause 8.4.2.2 computes the standard deviation $s_M$ of the six initial levels band by band (Eq. 10); above 1,5 dB in any band the source is radiating significant discrete components for this room, and the answer is more sampling, not a different formula: either qualify the room per Annex D, or read the required number of positions from Table 4 — six stays six up to 160 Hz, but 400 Hz to 630 Hz goes to 12 positions for $1,5<s_M\le 3$ dB and 24 above 3 dB, and 800 Hz upwards to 15 and 30 — and add source positions per Eq. 12 and Table 5. Averaging longer at six positions buys nothing here, because the scatter is spatial. The number of positions, the traverse geometry and the measurement time interval all belong in the test report.

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 $L_W(f)$, and the A-weighted energy sum across the 21 bands gives the $L_{WA}$ 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.

The five terms of Eq. 20 discussed above are all fields of that same rev, so the balance between them can simply be drawn:

The whole of Eq. 20, band by band, on the example below. The absorption area carries the determination; the Waterhouse term is the only one with real frequency dependence and it lives entirely in the low bands; and the meteorological pair is a fraction of a decibel at ordinary conditions — which is precisely why it is worth stating that the term everyone omits, 4.34 A/S, grows with the room’s absorption rather than with frequency.

Show the code for this figure

import matplotlib.pyplot as plt

# Every term is a field of the result the first snippet of this page returns.
absorption = 10.0 * np.log10(rev.absorption_area)
eyring = 4.34 * rev.absorption_area / 220.0
terms = [("10 lg(A/A0)", absorption), ("4.34 A/S", eyring),
         ("Waterhouse", rev.waterhouse_correction),
         ("C1 + C2", np.full(freqs.size, rev.c1 + rev.c2)),
         ("-6 dB", np.full(freqs.size, -6.0))]

x = np.arange(freqs.size)
up = np.zeros(freqs.size)
down = np.zeros(freqs.size)
fig, ax = plt.subplots(figsize=(11, 5.6))
for label, values in terms:
    ax.bar(x, values, bottom=np.where(values >= 0.0, up, down), width=0.72,
           label=label)
    up = up + np.clip(values, 0.0, None)
    down = down + np.clip(values, None, 0.0)
ax.plot(x, rev.sound_power_level - lp, "o-", label="LW - Lp")
ax.set_xticks(x, [f"{f:g}" for f in freqs], rotation=60)
ax.set(xlabel="Frequency [Hz]", ylabel="Contribution to LW - Lp [dB]")
ax.legend()
plt.show()

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 $L_{WA}$
background_levels	1D or 2D array	dB	matches levels	Per-band $K_1$ (frequency-dependent criterion)
temperature	float	°C	default 23.0	Sets $c$, $C_1$, $C_2$
static_pressure	float	kPa	default 101.325	Sets $C_1$, $C_2$

Both this function and the comparison method of section 2 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).

2. The comparison method (Eq. 21)

The comparison method replaces the absorption-area, Waterhouse and $C_1$ terms by a reference sound source of known power $L_W(\text{RSS})$ run in the same room, so the room itself never has to be characterised:

$$L_W=L_W(\text{RSS})+\left(L_p(\text{ST})-L_p(\text{RSS})+C_2\right).$$

A reference sound source is a small aerodynamic broadband source of stable output whose band-wise sound power level has been calibrated to ISO 6926, and it is that calibration — not the room — that the determination now rests on. The calibrated $L_W(\text{RSS})$ is stated for reference meteorological conditions, which is why $C_2$ survives in Eq. 21 while $C_1$ and the room terms cancel.

The procedure is two measurements and two backgrounds. The test source is measured first, at the six positions or over the traverse of section 1. The RSS is then run in the same room, at the same microphone positions or the same traverse, with the source under test left standing where it was, and with its own background reading (that is what background_levels_ref is for). The RSS stands on the floor more than 1,5 m from the walls; the preferred position is the one the source under test occupies, and where the source cannot be moved the RSS goes as close to it as practicable but never closer than 1,5 m (clause 8.6.1). One RSS position suffices even when the source under test needs several. Both measurements must be made at the same temperature, pressure and humidity (clause 8.6.2).

Because the RSS supplies the room constant, the minimum source-to-microphone distance may also be taken from the RSS reading itself (clause 8.3, Eq. 9), $d_{\min }=D_2\cdot 10^{0.05(L_{Wr}-L_{pr})}$ with $D_2=0.4$ and 0,8 recommended below 5 kHz. On the example below, $L_{Wr}-L_{pr}=7$ dB in the 100 Hz band, so $d_{\min }=0.90$ m, or 1.8 m at the recommended $D_2$ — close enough to the 0.8/1.6 m of Eq. 8 to be a useful cross-check.

What it buys and what it costs: no $T_{60}$, no volume, no surface area and no Waterhouse term, so a room that has never been surveyed can still return a determination — and, correspondingly, absorption_area, waterhouse_correction and c1 come back NaN, because the method never computes them. What it costs is that the result inherits the RSS calibration whole, so a source whose ISO 6926 certificate has expired quietly biases every band, and the RSS position has to sample the room the way the test source does or the difference $L_p(\text{ST})-L_p(\text{RSS})$ is measuring the two positions rather than the two sources.

sound_power_comparison() parameters

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 $L_W(\text{RSS})$ of the reference source, from its ISO 6926 calibration
background_levels	1D or 2D array	dB	matches levels	Background for the test source; per-band $K_1$ on $L_p(\text{ST})$
background_levels_ref	1D or 2D array	dB	matches levels_ref	Background for the reference source; per-band $K_1$ on $L_p(\text{RSS})$

background_levels_ref background-corrects the reference-source room level $L_p(\text{RSS})$ exactly as background_levels does for the test source; both need frequencies (the ISO 3741 criterion is frequency-dependent).

3. The measurement report (.report())

The reverberation-room result (ReverberationSoundPowerResult, ISO 3741) writes a one-page PDF fiche laid out like a sound-power test sheet through its own .report(), sharing the layout and the ReportMetadata container of the pressure-method fiche. 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

One caveat on that example: it runs Eq. 20 on octave bands for compactness, and that is not a conforming ISO 3741 determination. Both methods are defined on one-third-octave bands, and octave-band results are formed afterwards by summing the one-third-octave band powers per Annex F (Eq. F.1), not by feeding octave levels into Eq. 20 — the Waterhouse and absorption terms are frequency-dependent and would then be evaluated at the octave mid-frequency instead of at each third. Section 1’s example, 100 Hz to 10 kHz in thirds, is the conforming shape.

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.

What this guide covers

• Covered

  The ISO 3741 reverberation-room determination (sound_power_reverberation): the direct method (Eq. 20) through the Sabine absorption area, the Waterhouse boundary correction and the $C_1$/$C_2$ meteorological corrections, and the comparison method against a reference sound source (sound_power_comparison, Eq. 21), both with per-band background correction and the Annex F A-weighted total, rendered as the accredited-style sound-power fiche through .report().

• Not covered

  The sound energy level $L_J$ of clause 9.2, the single-event counterpart of $L_W$ defined in the same clauses, is not implemented, so an impulsive or single-event source falls outside this guide. Neither is the Annex F octave-band summation, so octave results have to be summed from the one-third-octave band powers by hand. The reverberation-room qualification itself (Annex C/D, eigenfrequency counting or a reference-source comparison) is assumed, not performed; the library only warns on the coarse advisory criteria ISO 3741 states explicitly (the Table 1 minimum volume, the $V/S$ reverberation floor, fewer than 6 positions, an inter-position spread above 1.5 dB). Choosing among the six routes, and the ISO 4871 declaration a result feeds, live in Sound Power.

See also

• Sound Power: choosing among the five determination routes, the accuracy grades and the ISO 4871 noise-emission declaration.
• Sound Power by Pressure Methods (ISO 3744 / ISO 3746 / ISO 3745): the in-situ enveloping surface and the precision anechoic array.
• Sound Power by Intensity Scanning (ISO 9614): the routes that tolerate steady background noise.
• Room Acoustics: measuring the reverberation time $T_{60}$ that sets the Sabine absorption area.
• Levels: energy averaging and the A-weighting behind $L_{WA}$.
• Theory: the Waterhouse and $C_1$/$C_2$ derivations.
• API reference: emission.sound_power_reverberation.
• Theory: Sound power determination: the ISO 3741 comparison and direct methods derived from the diffuse-field relation.

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 diffuse-field relation between pressure and power that the reverberation-room method rests on.
• 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 direct (Eq. 20) and comparison (Eq. 21) methods of this guide, with the Waterhouse and meteorological corrections and the Table 1 qualification criteria.