Source: https://jmrplens.github.io/phonometry/devices/emission/sound-power-in-situ/ # Sound Power in Situ by Comparison (ISO 3747) Some machines never travel to a test room: a compressor bolted to the floor of the plant, a press line, a packaging machine wired into the rest of the line. ISO 3747 determines their sound power where they stand, by comparison. A reference sound source (RSS) of calibrated band sound power is set beside the machine, the same three or four microphones listen to each source in turn from the reverberant part of the room, and because both sources see the same room the room cancels: the sound power of the source under test is the calibrated power of the reference source carried across by the difference of the two mean levels. The method reaches engineering grade 2 where the field is reverberant enough and the source not too directional, and survey grade 3 otherwise; an impulsive source gets a sound energy level instead of a sound power level by the same route. Which route fits which job is weighed in [Sound Power](https://jmrplens.github.io/phonometry/devices/emission/sound-power/). ## 1. The comparison in situ (ISO 3747) The microphone positions must lie where the field is **reverberant**, which the standard measures as the excess of sound pressure level over the free field, $\Delta L_f \ge 7$ dB (clause 4.1, Annex A). Three or four positions are used, at least 2 m apart, none closer than 0,5 m to any boundary, distributed as evenly as possible round the machine (clause 7.4.1); the reference source stands alongside the machine, never closer than 0,5 m to its reference box, and where the machine is long it is run at several locations along the sides (clause 7.3). Both sources and the background are read at the same positions, the reference source for 30 s (clause 7.5), in **octave bands from 125 Hz to 8 kHz** (clause 3.11); Table D.1 tabulates a 63 Hz row besides, for use only where the environment and the instrumentation are satisfactory there (footnote a), which is why the API accepts that band too. The background is corrected **position by position** before anything is averaged (clause 8.1, Eq. 7): $$ K_{1i} = -10 \log_{10}\left(1 - 10^{-0.1\,\Delta L_{pi}}\right), \qquad \Delta L_{pi} = L'_{pi(\mathrm{ST})} - L_{pi(\mathrm{B})}, $$ with three rules round it: a margin above 15 dB needs no correction, a margin between 6 dB and 15 dB takes Eq. 7, and a margin **below 6 dB** caps the correction at **1,3 dB** and turns the band into an upper bound that the report must flag as not meeting the background requirement. `sound_power_in_situ` applies all three, returns the per-position $K_{1i}$ and flags every band where some margin fell below 6 dB in `background_requirement_met`. The corrected levels are energy-averaged over the positions (Eq. 8, 9) and the sound power level in each octave band is the comparison itself (clause 8.3.1, Eq. 11): $$ L_W = L_{W(\mathrm{RSS})} - \overline{L_{p(\mathrm{RSS})}} + \overline{L_{p(\mathrm{ST})}} . $$ With the reference source at $m$ locations, its calibrated powers and its per-location means are each energy-averaged over the locations first (clause 8.3.2, Eq. 12), and the correction of the reference source is evaluated per location and per position. Table 1 of the standard, which sorts the microphone positions by line of sight to each source, is guidance for placing them and is not evaluated by the library. ```python import numpy as np from phonometry import emission # A floor-standing screw compressor, 2.2 m x 1.4 m x 1.8 m, that cannot # leave the plant: four microphone positions 1.5 m from its reference box, # one reference sound source location alongside, octave bands 125 Hz - 8 kHz. freqs = np.array([125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0, 8000.0]) lw_rss = np.array([84.5, 88.0, 91.0, 92.5, 92.0, 90.5, 87.0]) # calibrated st = np.array([ # L'pi(ST): one row per microphone position, dB [82.4, 84.9, 86.1, 85.3, 82.6, 78.4, 72.1], [80.9, 84.1, 85.4, 84.7, 81.8, 77.5, 71.3], [83.1, 85.6, 86.8, 85.9, 83.2, 79.0, 72.8], [81.7, 84.5, 85.8, 85.0, 82.3, 78.0, 71.7], ]) rss = np.array([ # L'pi(RSS): the reference source at the same positions [78.2, 81.8, 84.6, 86.0, 85.6, 84.1, 80.4], [77.5, 81.1, 84.0, 85.4, 85.0, 83.4, 79.8], [78.9, 82.4, 85.1, 86.5, 86.1, 84.6, 80.9], [78.0, 81.6, 84.4, 85.8, 85.4, 83.9, 80.2], ]) background = np.array([ # Lpi(B): the factory floor, loud at the low end [75.0, 74.0, 72.0, 70.0, 67.0, 62.0, 57.0], [76.5, 75.0, 73.0, 71.0, 68.0, 63.0, 58.0], [74.5, 73.5, 71.5, 69.5, 66.5, 61.5, 56.5], [75.5, 74.5, 72.5, 70.5, 67.5, 62.5, 57.5], ]) res = emission.sound_power_in_situ( st, rss, lw_rss, freqs, background_levels=background, conditions=emission.GradeConditions( excess_levels=[8.2, 7.6, 8.9, 8.0], # dLfA at each position (Annex A) directivity_range=4.0, # +/-4 dB round the machine (7.2) ), sigma_omc=0.5, # a steady source (9.2 NOTE) ) print(np.round(res.sound_power_level, 1)) # [88.8 91.5 92.6 91.8 88.9 84.7 78.6] print(round(res.sound_power_level_a, 1)) # 96.1 dB(A) print(res.background_requirement_met) # [False True True True True True True] print(np.round(res.background_correction[1], 2)) # position 2: [1.3 0.57 0.26 0.19 0.18 0.16 0.21] print(res.grade, res.sigma_r0, round(res.expanded_uncertainty, 1)) # engineering 1.5 3.2 ``` The 125 Hz band is the one to read twice: at one of the four positions the margin over the floor is below 6 dB for the machine, and it is below 6 dB at every position for the quieter reference source, so $K_1$ takes its 1,3 dB cap and the 88,8 dB the band reports is an **upper bound**, which is what the `False` says. *The comparison position by position: both sources at the four microphones after the Eq. 7 correction above, with the hollow markers showing where each level was before it, and Eq. 11 band by band below, with the calibrated $L_{W(\mathrm{RSS})}$ beside it and the 125 Hz upper bound hatched.*

Show the code for this figure

```python import matplotlib.pyplot as plt # res is the InSituSoundPowerResult computed above. One line: res.plot() # LW bars, LWA in the title, the upper-bound band hatched plt.show() # By hand: both sources per position above, Eq. 11 below. corrected_st = st - res.background_correction corrected_rss = rss - res.background_correction_ref[0] x = np.arange(freqs.size) fig, (axt, axb) = plt.subplots(2, 1, figsize=(10, 8.4), sharex=True) for i, offset in enumerate(np.linspace(-0.27, 0.27, st.shape[0])): axt.plot(x + offset, corrected_st[i], "o", color="#1f77b4") axt.plot(x + offset, corrected_rss[i], "s", color="#2ca02c") axt.plot(x + offset, background[i], "x", color="#9e9e9e") axt.plot(x, res.mean_source_level, "-", color="#1f77b4", label="mean, source under test") axt.plot(x, res.mean_reference_level, "-", color="#2ca02c", label="mean, reference source") axt.set_ylabel("Sound pressure level [dB]") axt.legend() bars = axb.bar(x, res.sound_power_level, width=0.62, color="#1f77b4") for bar, met in zip(bars, res.background_requirement_met, strict=True): if not met: bar.set_hatch("//") axb.plot(x, res.reference_power_level, "D--", color="#d62728", label="calibrated LW(RSS)") axb.set_xticks(x, [f"{f:g}" for f in freqs]) axb.set_xlabel("Frequency [Hz]") axb.set_ylabel("Sound power level LW [dB re 1 pW]") axb.legend() plt.show() ```

A machine that is long relative to the measurement distance gets the reference source at several locations along its sides (clause 7.3.3), and the levels arrive as one grid per location: ```python # The same compressor with the reference source run at two locations, one # along each long side; the second sits nearer a wall and reads 1.2 dB lower, # and was calibrated in a similar position 0.4 dB lower. two = emission.sound_power_in_situ( st, np.stack([rss, rss - 1.2]), np.stack([lw_rss, lw_rss - 0.4]), freqs, background_levels=background, ) print(two.reference_levels.shape) # (2, 7): mean level per location (Eq. 10) print(np.round(two.sound_power_level, 1)) # [89.1 92. 93. 92.2 89.3 85.1 78.9] print(round(two.sound_power_level_a, 1)) # 96.5 dB(A) ``` ### `sound_power_in_situ()` parameters | Parameter | Type | Units | Range / default | Notes | | :--- | :--- | :--- | :--- | :--- | | `levels` | 2D array | dB | `(n, bands)` | Measured $L'_{pi(\mathrm{ST})}$, one row per microphone position, three or four positions (7.4.1) | | `levels_ref` | 2D or 3D array | dB | `(n, bands)` or `(m, n, bands)` | The reference source at the same positions, one grid per location, corrected per its manufacturer but not for background | | `lw_ref` | 1D or 2D array | dB | `(bands,)` or `(m, bands)` | Calibrated $L_{W(\mathrm{RSS})}$, per location when each was calibrated in its own similar position (Eq. 12) | | `frequencies` | 1D array | Hz | 63 Hz to 8 kHz octaves | Nominal octave mid-band frequencies of Table D.1; required. Clause 3.11 puts the general-purpose range at 125 Hz to 8 kHz; the 63 Hz row of Table D.1 carries footnote a, which allows it only where the test environment, the reference sound source and the instrumentation are satisfactory at that frequency | | `background_levels` | 1D or 2D array | dB | `(bands,)` or `(n, bands)` | $L_{pi(\mathrm{B})}$; one spectrum serves every position. `None` warns and leaves `background_requirement_met` `False` in every band, since 7.5 measures it at each position and 8.1 needs the margin | | `background_levels_ref` | 1D or 2D array | dB | same shapes | Background for the reference-source reading; `None` reuses `background_levels` (7.5) | | `temperature` | float | °C | default `23.0` | Air temperature at the test, for $C_2$ | | `static_pressure` | float | kPa | default `101.325` | Static pressure at the test, for $C_2$; see `static_pressure_from_altitude` | | `conditions` | `GradeConditions` | | | The two conditions Table 2 reads together: `excess_levels`, the $\Delta L_{f\mathrm{A}}$ at each position `(n,)` in decibels (Annex A), and `directivity_range`, the half-width of the A-weighted directivity survey in decibels (7.2). Either one left out leaves the determination at survey grade | | `sigma_omc` | float | dB | $\ge 0$ | Standard deviation of the operating and mounting conditions (9.2); `None` leaves the uncertainty `NaN` | | `coverage_factor` | float | | default `2.0` | $k$ of Eq. 23; 1,6 for a one-sided comparison with a limit | The function returns an `InSituSoundPowerResult`: `sound_power_level` (Eq. 11 or 12, at the conditions of the test), `mean_source_level` and `mean_reference_level` (Eq. 8, 9), `reference_levels` per location (Eq. 10), `reference_power_level`, `background_correction` per position and band, `background_correction_ref` per location, position and band, `background_requirement_met` per band, `c2`, `grade`, `sigma_r0`, `sigma_omc`, `sigma_tot`, `expanded_uncertainty`, `coverage_factor`, `sound_power_level_a` and `quantity`. `sound_power_level_ref` adds `c2` to the level (section 3). ## 2. Impulsive sources: the sound energy level (Eq. 13 to 20) A press stroke or a door slam has no steady power to report. Clause 8.4 measures the **single event level** $L_E$ at each position instead, either one event at a time, at least five of them (clause 7.6), or once over a run of $N$ successive events, and clause 8.5 turns the mean into a **sound energy level** $L_J$ in dB re 1 pJ by the same comparison. Measured one at a time, each event is corrected for background with Eq. 14 and the corrected levels are energy-averaged into the mean single event level of the position (Eq. 13, 15); measured once over $N$ events, the level is corrected and reduced by $10 \log_{10} N$ to one event (Eq. 16, 17). The per-position levels are energy-averaged (Eq. 18), and $$ L_J = L_{W(\mathrm{RSS})} - \overline{L_{p(\mathrm{RSS})}} + \overline{L_{E(\mathrm{ST})}} $$ (Eq. 19), or its several-location form (Eq. 20). `sound_energy_in_situ` takes the events as a 3D `(n, N, bands)` array or as a 2D `(n, bands)` measurement with `events=N`, and returns the same result type with `quantity='energy'`, the level in `sound_energy_level` and the Annex D total in `sound_energy_level_a`. Eq. 14 subtracts the **time-averaged** background level from a **time-integrated** event level, asking only that both be measured over the same integration time $T$; as printed the difference is a true margin for $T$ = 1 s, and for a longer $T$ the background holds $10 \log_{10}(T/T_0)$ dB more energy over the event's interval (clause 3.4, NOTE 1). ISO 3741 and ISO 3744 print the same line, so it is the family's convention and not a misprint of this part; the library applies Eq. 14 as printed by default and offers `integration_time` to carry the background to the event's interval first. ```python # The same room, a press beside the compressor: six strokes measured one at a # time at the four positions, each integrated over 4 s, with the same # reference source reading and the same floor. strokes = np.array([ [96.3, 98.1, 99.4, 98.8, 96.0, 92.1, 86.4], [95.1, 97.4, 98.7, 98.0, 95.3, 91.4, 85.6], [97.0, 98.9, 100.1, 99.5, 96.6, 92.8, 87.0], [95.8, 97.9, 99.0, 98.4, 95.8, 91.9, 86.0], ]) events = np.stack([strokes + d for d in (-0.3, 0.2, 0.0, 0.4, -0.2, 0.1)], axis=1) energy = emission.sound_energy_in_situ( events, rss, lw_rss, freqs, # (4 positions, 6 events, 7 bands) background_levels=background, integration_time=4.0, conditions=emission.GradeConditions(excess_levels=[8.2, 7.6, 8.9, 8.0], directivity_range=4.0), sigma_omc=2.0, ) print(energy.quantity) # energy print(np.round(energy.sound_energy_level, 1)) # [103.7 105.3 106.1 105.3 102.4 98.6 93. ] print(round(energy.sound_energy_level_a, 1)) # 109.7 dB(A) re 1 pJ print(round(energy.expanded_uncertainty, 1)) # 5.0 dB: grade 2 with sigma_omc = 2 dB ``` The 5,0 dB is the standard's own example in clause 9.5, reproduced because the inputs are the same: grade 2, $\sigma_\mathrm{omc}$ = 2,0 dB, $k$ = 2. ### `sound_energy_in_situ()` parameters | Parameter | Type | Units | Range / default | Notes | | :--- | :--- | :--- | :--- | :--- | | `event_levels` | 3D or 2D array | dB | `(n, N, bands)` or `(n, bands)` | $L'_{Ei,q(\mathrm{ST})}$ per event, or $L'_{Ei,N(\mathrm{ST})}$ of one measurement over $N$ events | | `events` | int | | $\ge 1$ | $N$ for the 2D form (Eq. 17); must be `None` with the 3D form. Clause 7.6 asks for at least five events, so fewer than five warns and the determination is nonconforming, on either form | | `integration_time` | float | s | $> 0$ | $T$ of the event measurement; `None` applies Eq. 14 as printed | | `levels_ref`, `lw_ref`, `frequencies`, `background_levels`, `background_levels_ref`, `temperature`, `static_pressure`, `excess_levels`, `directivity_range`, `sigma_omc`, `coverage_factor` | | | | As in `sound_power_in_situ()` | ## 3. Grade, uncertainty and reference conditions **Grade (Table 2).** Engineering grade 2 needs the A-weighted excess $\Delta L_{f\mathrm{A}} \ge 7$ dB at every microphone position and a source directivity range within ±7 dB (clause 7.2); either indicator failing, or not determined, gives survey grade 3. The table pairs each grade with a typical upper bound of the reproducibility of the method, **1,5 dB** for grade 2 and **4,0 dB** for grade 3. The excess itself is Annex A (Eq. A.1), $$ \Delta L_f(r) = L_{p(\mathrm{RSS}),r} - L_{W(\mathrm{RSS})} + 11\ \mathrm{dB} + 20 \log_{10}\frac{r}{r_0}, \qquad r_0 = 1\ \mathrm{m}, $$ which `excess_sound_pressure_level` evaluates. **Uncertainty (clause 9).** $\sigma_\mathrm{tot} = \sqrt{\sigma_{R0}^2 + \sigma_\mathrm{omc}^2}$ (Eq. 22) and $U = k\,\sigma_\mathrm{tot}$ (Eq. 23), with $k$ = 2 for the two-sided 95 % interval or 1,6 for a one-sided comparison with a limit value; $\sigma_\mathrm{omc}$ is the user's number (9.2, E.3), near 0,5 dB for a steady source, 2 dB with material flow, 4 dB for a press under load. The standard's example (9.5), grade 2 with $\sigma_\mathrm{omc}$ = 2,0 dB and $k$ = 2, gives $U = 2\sqrt{1{,}5^2 + 2^2}$ dB = 5 dB. Without a `sigma_omc` the result leaves `sigma_tot` and `expanded_uncertainty` `NaN`. **Reference conditions (Annex C).** The radiation-impedance correction $C_2 = -10 \log_{10}(p_\mathrm{s}/p_{\mathrm{s},0}) + 15 \log_{10}((273{,}15 + \theta)/296)$ carries the level to 101,325 kPa and 23,0 °C, the same $C_2$ as ISO 3741 clause 9.1.4, and Eq. C.2 estimates the static pressure from the altitude, $p_\mathrm{s} = p_{\mathrm{s},0}\,(1 - a H_\mathrm{a})^b$ with $a$ = 2,2560 × 10⁻⁵ m⁻¹ and $b$ = 5,2553. The result exposes `c2` and the properties `sound_power_level_ref` and `sound_energy_level_ref` (Eq. C.1, C.3). $\theta_\mathrm{ref}$ is printed as 296 K beside a reference temperature of 23,0 °C (296,15 K), so at exactly the reference conditions $C_2$ is +0,003 3 dB and not zero; ISO 3741 and ISO 3744 print the same 296 K, so the library keeps the family's rounding. ```python # A plant 640 m above sea level, at 27 degC, with no barometer on site. ps = emission.static_pressure_from_altitude(640.0) print(round(ps, 2)) # 93.87 kPa (Eq. C.2) high = emission.sound_power_in_situ( st, rss, lw_rss, freqs, background_levels=background, temperature=27.0, static_pressure=ps, ) print(round(high.c2, 3)) # 0.423 dB print(round(float(high.sound_power_level[3]), 2), round(float(high.sound_power_level_ref[3]), 2)) # 91.76 at the test conditions, 92.18 under the reference conditions (1 kHz) # Annex A along a line of sight from the reference source, A-weighted. print(np.round(emission.excess_sound_pressure_level([81.0, 76.5, 73.8], 92.5, [1.5, 3.0, 6.0]), 2)) # [3.02 4.54 7.86]: reverberant enough for grade 2 only from about 6 m out ``` **A-weighted totals (Annex D).** $L_{W\mathrm{A}}$ and $L_{J\mathrm{A}}$ are the energy sums of the band levels plus the Table D.1 corrections $C_k$ (−26,2, −16,1, −8,6, −3,2, 0,0, 1,2, 1,0 and −1,1 dB from 63 Hz to 8 kHz, the ISO 3744 Annex E octave values digit for digit); every `frequencies` entry must be one of those eight nominal centres, distinct and in ascending order. ## What this guide covers **Covered.** The ISO 3747 comparison in situ (`sound_power_in_situ`): the per-position background correction of clause 8.1 with its 6 dB and 15 dB rules and the 1,3 dB cap, the mean corrected levels of clause 8.2, the sound power level for one (Eq. 11) or several (Eq. 12) reference-source locations, the single-event levels of clause 8.4 in both forms and the sound energy level of clause 8.5 (`sound_energy_in_situ`), the Table 2 grade with its $\sigma_{R0}$, the clause 9 uncertainty (Eq. 22, 23), the Annex A excess of sound pressure level (`excess_sound_pressure_level`), the Annex C reference conditions with Eq. C.2 (`static_pressure_from_altitude`) and the Annex D A-weighted totals. **Not covered.** The Annex E uncertainty budget (Table E.2 and its sensitivity coefficients) is documented but not modelled: $\sigma_{R0}$ is the typical upper bound of Table 2 by grade, not a machine-specific value from a round robin (9.3.2) or from the model of 9.3.3. The zoning of Table 1, the directivity survey of clause 7.2 and the placement rules of clause 7.3 are the operator's decisions, made before the levels exist. The reference source's own corrections for speed, temperature and static pressure are its manufacturer's, applied before the levels arrive. ## See also - [Sound Power](https://jmrplens.github.io/phonometry/devices/emission/sound-power/): choosing among the determination routes, the accuracy grades and the ISO 4871 noise-emission declaration. - [Sound Power in the Reverberation Room](https://jmrplens.github.io/phonometry/devices/emission/sound-power-reverberation/): the precision comparison of ISO 3741 in a qualified room, whose $C_2$ this guide shares. - [Sound Power by Pressure Methods](https://jmrplens.github.io/phonometry/devices/emission/sound-power-pressure/): the enveloping surface for a source that can be measured in its direct field. - [Sound Power by Intensity Scanning](https://jmrplens.github.io/phonometry/devices/emission/sound-power-intensity/): the routes that tolerate steady background noise without a reference source. ## References - International Organization for Standardization. (2010). *Acoustics — Determination of sound power levels and sound energy levels of noise sources using sound pressure — Engineering/survey methods for use in situ in a reverberant environment* (ISO 3747:2010). [iso.org catalogue](https://www.iso.org/standard/46426.html). The comparison method of this guide: the background correction of clause 8.1, the mean levels and the sound power and sound energy levels of clauses 8.2 to 8.5, the Table 2 grades, the clause 9 uncertainty and Annexes A, C and D. - 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 precision comparison method of a qualified room, whose Eq. 21 is Eq. 11 of this guide plus the same radiation-impedance correction $C_2$. ## Standards ISO 3747:2010, *Acoustics — Determination of sound power levels and sound energy levels of noise sources using sound pressure — Engineering/survey methods for use in situ in a reverberant environment*: the comparison with a reference sound source, the background correction of clause 8.1, the several-location forms, the single-event path, the Table 2 grades, the clause 9 uncertainty and Annexes A, C and D.