Source: https://docs.jmrp.io/phonometry/devices/emission/sound-power-reverberation/ # Sound Power in the Reverberation Room (ISO 3741) 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, the accredited-style test fiche and, in section 3, the sound energy level $L_J$ that the same two methods give a single event. Which route fits which job, and the pressure and intensity alternatives, are weighed in [Sound Power](https://docs.jmrp.io/phonometry/devices/emission/sound-power/). ## 1. 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 $L_p(\text{ST})$, the Sabine absorption area $A = (55.26/c)\cdot(V/T_{60})$ and a chain of small corrections (ISO 3741 Eq. 20): $$ L_W = \bar{L}_p + 10 \log_{10}\frac{A}{A_0} + 4.34\ \frac{A}{S} + 10 \log_{10}\left( 1 + \frac{S c}{8 V f} \right) + C_1 + C_2 - 6 . $$ 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. $C_1$ (reference-quantity) and $C_2$ (radiation-impedance) carry the result to the reference meteorological conditions of 23 °C and 101.325 kPa, $$ C_1 = -10 \log_{10}\frac{p_\mathrm{s}}{p_{\mathrm{s}0}} + 5 \log_{10}\frac{273.15 + \theta}{314}, \qquad C_2 = -10 \log_{10}\frac{p_\mathrm{s}}{p_{\mathrm{s}0}} + 15 \log_{10}\frac{273.15 + \theta}{296}, $$ with the speed of sound $c = 20.05\sqrt{273 + \theta}$. The **comparison method** replaces the absorption-area, Waterhouse and $C_1$ terms by a reference sound source of known power $L_W(\text{RSS})$ measured in the same room, so the room need not be characterised: $L_W = L_W(\text{RSS}) + (L_p(\text{ST}) - L_p(\text{RSS}) + C_2)$. The right-hand panel of the clip below is this method. 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. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry-assets/main/images/anim_power_two_rooms.webm) ```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_c=20.0, static_pressure_kpa=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_c=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 $L_{W\mathrm{A}}$ in the title.*

Show the code for this figure

```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_c=20.0, static_pressure_kpa=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. Those coarse advisory criteria are the only checks made: the reverberation-room qualification itself (Annex C/D, eigenfrequency counting or a reference-source comparison) is assumed, not performed. The sound energy level $L_J$ of clause 9.2, the single-event counterpart of $L_W$ defined in the same clauses, is section 3. ### `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 $L_{W\mathrm{A}}$ | | `background_levels` | 1D or 2D array | dB | matches `levels` | $K_{1i}$ per microphone position (Eq. 14/15, before the Eq. 16 average; frequency-dependent criterion) | | `temperature_c` | float | °C | default `23.0` | Sets $c$, $C_1$, $C_2$ | | `static_pressure_kpa` | float | kPa | default `101.325` | Sets $C_1$, $C_2$ | `sound_power_comparison(levels, levels_ref, lw_ref, *, frequencies=None, background_levels=…, background_levels_ref=…, temperature_c=23.0, static_pressure_kpa=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 or `ReferenceSourceCalibration` | dB | per band | Known sound power $L_W(\text{RSS})$ of the reference source; a [`ReferenceSourceCalibration`](https://docs.jmrp.io/phonometry/devices/emission/free-field-qualification/) is read in the one-third octave bands at `frequencies` | | `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). 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). ## 2. 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](https://docs.jmrp.io/phonometry/devices/emission/sound-power-pressure/#3-the-measurement-report-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 $L_{W\mathrm{A}}$ 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 $C_\mathrm{w}$; 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. ```python 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_c=20.0, static_pressure_kpa=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), a summation not implemented here (so it has to be done by hand), not by feeding octave levels into Eq. 20, whose Waterhouse and absorption terms 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, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso3741_reverberation_power_example.pdf) *Reverberation-room sound power fiche (`ReverberationSoundPowerResult.report`), an ISO 3741 precision-grade direct-method determination with the Waterhouse and $C_1$/$C_2$ corrections and the boxed $L_{W\mathrm{A}}$.* ## 3. Sound energy level of a single event (clause 9.2) Both methods above report a rate of energy flow, so a machine that fires once and falls silent has no $L_W$ to determine. Clause 9.2 gives it the **sound energy level** $L_J = 10\log_{10}(J/J_0)$, $J_0 = 1$ pJ (clause 3.18), by the same two routes with the single event time-integrated sound pressure level $L_E$ (clause 3.4) in place of $L_p$: Eq. 30 is Eq. 20 with $\overline{L_E(\text{ST})}$ where $\overline{L_p(\text{ST})}$ stood, and Eq. 31 is Eq. 21 likewise, $$ L_J = \overline{L_E(\text{ST})} + \left[10\log_{10}\frac{A}{A_0} + 4.34\ \frac{A}{S} + 10\log_{10}\left(1 + \frac{S c}{8 V f}\right) + C_1 + C_2 - 6\right], \qquad L_J = L_W(\text{RSS}) + \left(\overline{L_E(\text{ST})} - \overline{L_p(\text{RSS})}\right) + C_2 . $$ The measurement is clause 8.5: at least five events at each position, one at a time or as one reading encompassing them all, through an interval that contains the whole event including its decay, and no moving microphone for a non-repetitive impulse. Clause 9.2.1 reduces both readings to the level of one event (Eq. 22, the energy mean of $N_\mathrm{e}$ readings, or Eq. 23, one reading less $10\log_{10} N_\mathrm{e}$), clause 9.2.2 corrects each position for its own background by the frequency-dependent criterion of 9.1.2 (Eq. 25/26), and clause 9.2.3 energy-averages the corrected positions (Eq. 27). The background is the time-averaged level measured over the same integration time $T$ as the events, compared as its exposure over that window, $L_{pi(\mathrm{B})} + 10\log_{10}(T/T_0)$, so that the energies Eq. 25 subtracts share one reference; `integration_time` is therefore required with `background_levels` (the [errata registry](https://docs.jmrp.io/phonometry/reference/errata/) records the reading). Annex F then forms the octave-band levels (Eq. F.4, the energy sum of the three thirds of Table F.1 per octave, `octave_band_levels`) and the A-weighted total (Eq. F.5, `sound_energy_level_a`). ```python import numpy as np from phonometry import emission # Five door slams recorded at the six positions of the room of section 1, one at # a time: (events, positions, bands), the 21 one-third octaves 100 Hz - 10 kHz. thirds = 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) t60_thirds = np.full(thirds.size, 2.0) slam = np.linspace(88.0, 76.0, thirds.size) # single event level, dB re E0 rng = np.random.default_rng(3741) slams = slam + rng.normal(0.0, 0.6, size=(5, 6, thirds.size)) slam_lj = emission.sound_energy_reverberation( slams, t60_thirds, volume=200.0, surface_area=220.0, frequencies=thirds, background_levels=np.full(thirds.size, 45.0), # time-averaged over the same 4 s integration_time=4.0, temperature_c=20.0, static_pressure_kpa=101.0, ) print(slam_lj.events, slam_lj.method) # 5 direct print(np.round(slam_lj.sound_energy_level[:3], 1)) # [96. 94.9 94.2] dB re 1 pJ print(round(slam_lj.sound_energy_level_a, 1)) # LJA = 99.0 dB(A) # Comparison method: the reference source of section 1, run steadily. slam_cmp = emission.sound_energy_comparison( slams, lp_rss, lw_rss, frequencies=thirds, temperature_c=20.0, ) print(round(float(slam_cmp.sound_energy_level[0]), 1), slam_cmp.method) # 95.0 comparison # Annex F octave bands, and the identity with the steady chain (LJ = LW + 10 lg(T/T0)). octaves, lj_octave = emission.octave_band_levels(slam_lj.sound_energy_level, thirds) print(octaves) # [ 125. 250. 500. 1000. 2000. 4000. 8000.] lp_thirds = np.linspace(80.0, 70.0, thirds.size) # the mean room SPL of section 1 steady = emission.sound_power_reverberation( lp_thirds, t60_thirds, volume=200.0, surface_area=220.0, frequencies=thirds, temperature_c=20.0, static_pressure_kpa=101.0, ) ten_seconds = emission.sound_energy_reverberation( lp_thirds + 10.0, t60_thirds, volume=200.0, surface_area=220.0, frequencies=thirds, temperature_c=20.0, static_pressure_kpa=101.0, ) print(np.round(ten_seconds.sound_energy_level - steady.sound_power_level, 6)[:3]) # 10.0 slam_lj.plot() # LJ per one-third octave, LJA in the title (needs matplotlib) ``` `sound_energy_reverberation` takes the room arguments of `sound_power_reverberation` and `sound_energy_comparison` the reference-source arguments of `sound_power_comparison`, both plus `events`, `background_levels` with `integration_time`; they return a `ReverberationSoundEnergyResult` (`sound_energy_level`, `mean_event_level`, `absorption_area`, `waterhouse_correction`, `background_correction`, `c1`, `c2`, `speed_of_sound`, `sound_energy_level_a`, `method`, `events`, `integration_time`) with `.plot()`, the same `NaN` fields for the comparison method as its sound power twin, and the same qualification warnings. The single event results carry no `.report()` fiche, and the averaging over several source positions of Eq. 24 is left to the caller. ## See also - [Sound Power](https://docs.jmrp.io/phonometry/devices/emission/sound-power/): choosing among the seven determination routes, what the accuracy grades promise, and the ISO 4871 noise-emission declaration a measured $L_{W\mathrm{A}}$ feeds. - [Sound Power by Pressure Methods (ISO 3744 / ISO 3746 / ISO 3745)](https://docs.jmrp.io/phonometry/devices/emission/sound-power-pressure/): the in-situ enveloping surface and the precision anechoic array. - [Sound Power by Intensity Scanning (ISO 9614)](https://docs.jmrp.io/phonometry/devices/emission/sound-power-intensity/): the routes that tolerate steady background noise. - [Room Acoustics](https://docs.jmrp.io/phonometry/buildings/rooms/room-acoustics/): measuring the reverberation time $T_{60}$ that sets the Sabine absorption area. - [Levels](https://docs.jmrp.io/phonometry/signals/levels/levels/): energy averaging and the A-weighting behind $L_{W\mathrm{A}}$. - [Theory](https://docs.jmrp.io/phonometry/reference/theory/environment-transport/): the Waterhouse and $C_1$/$C_2$ derivations. - API reference: [`emission.sound_power_reverberation`](https://docs.jmrp.io/phonometry/reference/api/power/sound-power-reverberation/). ## References - 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 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). [iso.org catalogue](https://www.iso.org/standard/52053.html). The direct (Eq. 20) and comparison (Eq. 21) methods of this guide, with the Waterhouse and meteorological corrections and the Table 1 qualification criteria. ## Standards ISO 3741:2010, *Acoustics — Determination of sound power levels and sound energy levels of noise sources using sound pressure — Precision methods for reverberation test rooms*: the direct (Eq. 20) and comparison methods with the Waterhouse and meteorological corrections, the Table 1 qualification criteria and the Annex F A-weighting.