Spatial Sound Decay in Workrooms

A reverberation time says what a room does to a decay in time. It says very little about what a factory hall does to a worker standing 20 m from a press. The question there is a question about distance: how much quieter does it get as you walk away, and how much louder is it than open air would be?

ISO 14257 answers both with one measurement. A calibrated source stands where a machine would, a microphone walks away from it along a straight path, and the curve of level against distance is the room’s own signature.

The curve, what it yields range by range, and the correction that takes the source’s own directivity back out. The data is the shipyard hall of Annex C.

A level on its own cannot be compared between two rooms, because it depends on how loud the source was. ISO 14257 therefore works with the sound distribution value, Equation (1): the level at a position, less the sound power level of the source that produced it.

In a free field that quantity is known exactly. A point source over a reflecting plane gives Equation (2), which is the line every measured curve is read against:

with $r_0=1$ m. The 11 dB is $10\lg (4\pi r_0^2)$, rounded.

from phonometry import room

distances = [2.0, 3.0, 4.0, 5.0, 6.0, 8.0, 12.0, 16.0, 24.0, 32.0, 48.0]
levels = [98.9, 95.1, 93.0, 92.0, 91.0, 87.9, 85.8, 83.5, 81.5, 77.0, 75.6]

d = room.sound_distribution_value(levels, 110.8)
print(d[:3].round(1))                                       # [-11.9 -15.7 -17.8]
print(room.reference_distribution_value([2.0, 4.0]).round(2))  # [-17.02 -23.04]

At 2 m the measured curve is already 5 dB above the free field, and the gap grows with distance: that is the room, holding the level up long after the direct sound has fallen away.

Clause 6 reads two numbers off the curve, over a stated distance range.

DL2, the rate of spatial decay per distance doubling, Equation (5), is the slope: the least-squares fit of $D$ against $\lg r$, converted from a decade to a doubling and signed so that a decay is positive. A free field gives 6 dB. Any room gives less, and how much less is the useful number.

DLf, the excess over a free field, Equations (6) to (8), is the height: how far above open air the room sits. It is what tells a designer that a hall is 10 dB noisier than it needs to be at the far workstations.

res = room.spatial_decay_curve(d, distances, region="middle", far_limit_m=24.0)
print(round(res.decay_rate_db, 2))    # 4.73 dB per doubling
print(round(res.mean_excess_db, 2))   # 7.33 dB
print(res.region)                     # middle

The two answer different design questions and they do not substitute for each other. A room can decay steeply and still be loud, and a room with a healthy absorbed ceiling can be quiet at the near workstations and decay hardly at all past them.

The distance ranges are 6.2: near from 1 m to $d_1$, middle from $d_1$ to $d_2$, far beyond $d_2$, with 5 m and 16 m the typical pair. The middle region is the one occupational work reads, and the clause asks for it to reach 24 m whenever the room is big enough, which is what far_limit_m above does.

print(room.distance_region(12.0))         # middle
print(room.typical_decay_range("middle")) # (2.0, 5.0)
print(room.typical_excess_range("middle"))# (2.0, 10.0)

Those last two ranges are ISO 11690-3, 4.3: what a workroom usually gives. A middle-range DL2 of 4,7 dB per doubling and an excess of 7,3 dB put this hall inside both.

The path is 5.3.1: parallel to the floor, beginning on the source, at 1,55 m for a standing workplace or 1,2 m for a seated one, with no obstacle on the floor below it and nothing large within 1,5 m of either side. The points are one of the two distributions 5.3.2 recommends, and the one used here is the logarithmic one, which is what makes a least-squares fit against $\lg r$ sensible in the first place.

Put the source where a machine would stand, with its acoustical centre either on the floor or at least 0,5 m above it, and at least 3 m from any wall or other reflecting object. Run it and check the margin over the background at every position and in every one of the six octave bands from 125 Hz to 4 kHz: 10 dB or more needs no correction, between 6 dB and 10 dB takes the ISO 3744 correction, and under 6 dB there is no measurement to make. Walk the path from the source outwards, taking a band spectrum at each of the recommended distances, measured from the acoustical centre rather than from the cabinet. Stop the last point at least 1,5 m short of any wall. Then, if the room allows it, walk a second path at right angles to the first: one path describes a direction, and two describe a room. Record the path height, the source position, and any obstacle you could not avoid, because the report is what makes the curve reproducible.

The margin is a judgement the library will make for you, one octave band at a time, and it says which of the three cases each position falls in rather than correcting anything: the correction 5.1.4 asks for between 6 dB and 10 dB is the one ISO 3744 prints, and applying it here would move a measured level behind your back.

check = room.check_background_margin([80.0, 70.0, 66.0], 60.0)
print(check.margins_db)        # [20. 10.  6.]
print(check.unusable)          # [False False  True]
print(check.satisfied)         # False

Two other conditions are read where they can be seen rather than asked about. A region that holds exactly the two positions a regression needs gives a rate that is the line through them, with nothing left over to check either position against, and 5.3.2 calls the count of its recommended distributions a minimum. And level_excess_at reading the fitted line outside the distances it was fitted over is an extrapolation, which is what the 30 m of 6.4.3 asks of a path that stopped at 24 m. Both emit SpatialDecayWarning, so a result that came out of one of them says so.

The source itself has to qualify first. Annex A asks for more than ISO 6926: its directivity index must stay inside ± 2 dB up to 630 Hz, rising to ± 8 dB at 1 kHz and holding there to 5 kHz, its spectrum may not step more than 8 dB between adjacent one-third octaves, and its output must be stable to 0,5 dB over the measurement.

print(room.omnidirectionality_tolerance_db(630.0))   # 2.0
print(room.omnidirectionality_tolerance_db(800.0))   # 5.0
print(room.omnidirectionality_tolerance_db(1000.0))  # 8.0

The 800 Hz value is a reading rather than a quotation: the clause says the tolerance rises “linearly” between 630 Hz and 1 kHz without saying linear in what, and the band index is the only reading on which both endpoints land on printed bands.

Close to the source, a good part of what the microphone hears is not the room: it is the source’s own directivity and the reflection off the floor under it. Annex B removes both. It measures the same source over a reflecting plane outdoors or in a large room, then swaps that measured curve for the theoretical one of Equations (B.2) to (B.4):

which for a source on the floor becomes the flat 3 dB of a half space. Equation (B.1) then does the swap in energy.

free_field = [98.8, 94.7, 92.3, 90.3, 88.7, 86.1, 82.6, 79.8, 75.7, 73.7, 67.8]
reference = room.sound_distribution_value(free_field, 110.8)
corrected = room.corrected_distribution_value(d, reference, distances)

fixed = room.spatial_decay_curve(corrected, distances, region="middle", far_limit_m=24.0)
print(round(fixed.decay_rate_db, 2))    # 4.39 dB per doubling
print(round(fixed.mean_excess_db, 2))   # 6.73 dB

The correction moves the slope by 0,3 dB per doubling and the level by 0,6 dB, which is the general shape of it: it tilts a curve a little and shifts it a lot, because it is largest where the measurement starts.

That difference is also how the library found an error in the annex. Annex C prints 4,4 dB per doubling for this band and range, which is the corrected figure, and 7,3 dB of excess, which is the uncorrected one. Applying the correction to one descriptor and not to the other cannot be right, and 28 of the 36 printed results in the annex leave their own rounding if the two are swapped. The registry entry is in Errata.

Six octave bands are six numbers, and a report often wants one. Equation (3) weights the curve by the spectrum of the machine that will actually stand there, and Equation (4) fixes that spectrum as A-weighted pink noise, Table 1, so that two rooms can be compared without a machine in mind.

by_band = [-11.9, -13.6, -12.5, -11.9, -12.4, -13.6]   # 125 Hz to 4 kHz
print(round(room.normalized_distribution_value(by_band), 2))   # -12.58

Clause 4.2.3 says why the fixed spectrum is a normalisation and not an average machine: the spectra met in industry are too varied for an average of them to mean anything.

ISO 11690-3 is the other half of the subject: predicting, before the hall is built, the curve ISO 14257 would measure in it. It sorts the ways of doing that into four categories, one diffuse-field and three geometrical, and pairs each with the level of detail it has to be fed.

print(room.prediction_method("2b").family)          # geometrical
print(room.ROOM_DETAIL_LEVELS[1])
# the volume and the mean absorption coefficient of the surfaces

print(room.detail_is_sufficient("1", room_detail=3,
                                fitting_detail=3, source_detail=1).satisfied)   # False
print(room.detail_is_sufficient("2b", room_detail=3,
                                fitting_detail=3, source_detail=1).satisfied)   # True

Table E.1 reads in both directions, which is the useful part. A reader with nothing but a volume and one mean absorption coefficient is limited to the diffuse-field method, and a reader who wants to trace rays over the real shape of the room has to go and gather a description at that level of detail first.

The fittings are part of the room, and 6.2.2 measures them by density, $q=S/4V$, the surface of everything standing in the hall over four times its volume. Annex C then answers the narrower question of what the room adds at the workstation of the machine itself.

print(round(room.fitting_density(480.0, 1200.0), 2))    # 0.1
print(round(room.workstation_level(sound_power_level_db=95.0,
                                   emission_level_db=85.0,
                                   absorption_area_m2=200.0), 2))   # 85.79
print(round(room.total_workstation_level([86.0, 83.0, 80.0]), 2))   # 88.44

The increase is $10\lg (1+4S/A)$, with $S$ the area the machine’s own emission implies. It is small in a well-absorbed hall and large in a bare one, and it is the term that turns a catalogue of quiet machines into a noisy workshop.

• Covered

  • ISO 14257:2001: the sound distribution value of Equation (1) and the free-field reference of Equation (2); the single numbers of Equations (3) and (4) with the Table 1 weightings; the rate of spatial decay of Equation (5); the excess of Equations (6), (7) and (8) with the evaluation distances of 6.4.3; the distance ranges of 6.2; the source requirements of Annex A; the correction of Annex B, Equations (B.1) to (B.4); and the conditions of clause 5 as constants.
  • ISO 11690-3:1998: the detail levels of Tables 1, 2 and 3; the four categories of Table 4 with the pairing of Table E.1; the typical ranges of 4.3; the fitting density of 6.2.2; and the workstation increase of Annex C.
  • The worked example of Annex C of ISO 14257 is reproduced band by band in the conformance evidence.

• Not covered

  • Predicting the curve itself. ISO 11690-3 describes what a prediction method must model and how much detail it needs; it does not print one, and neither ray tracing nor a radiosity solver is implemented here.
  • The reverberation-room calibration of the source, which is ISO 6926 and ISO 3741, and the directivity survey of Annex A, which needs a turntable rather than a formula.
  • The full reporting checklist of clause 7, which is a list of what a report says rather than a quantity to compute.