Sound Intensity (p-p)

Standards: IEC 61043 ISO 9614Key references: Fahy 1995

Sound pressure tells you how loud a point is; sound intensity tells you where the energy is going. It is the acoustic power flux (W/m²), a signed vector quantity, which is why intensity probes can localize sources, separate them from background noise and measure sound power in situ (ISO 9614) where a pressure measurement alone cannot.

The two-microphone principle (IEC 61043)

A p-p probe holds two matched microphones a small distance $Δ r$ apart. The pressure at the probe center is their mean, and the particle velocity comes from the pressure gradient (Euler’s equation, finite-difference form):

p = \frac{p _{1} + p _{2}}{2}, u = - \frac{1}{ρ _{0} Δ r} \int (p_{2} - p_{1}) d t, I = \overline{p u}

In practice the estimator works in the frequency domain through the cross-spectrum of the two channels (the standard’s equivalent form):

I (f) = - \frac{Im { G _{12} ( f )}}{2 π f ρ _{0} Δ r}

Everything on this page is then read as a level. The sound intensity level is $L_{I} = 10 lo g_{10} (∣ I ∣ / I_{0})$ with $I_{0} = 1 pW/m^{2}$ : the level is of the magnitude, and the direction is carried separately, which is what total_direction reports as $\pm 1$ . The two references were chosen to line up: in a plane progressive wave, where $I = p^{2} / (ρ c)$ , the pressure and intensity levels agree to within $10 lo g_{10} (ρ c /400) = 0.14$ dB at ordinary air conditions (IEC 61043 clause 5). That is why the left panel of the figure below shows two curves lying on top of each other, and why the pressure-intensity index has a floor near 0.14 dB rather than 0 — everything a field measurement shows above that floor is either a genuinely reactive or noise-contaminated field, or the instrument’s own phase error.

The probe itself is small enough to draw at true scale. plot_pp_probe_geometry puts the classic 12 mm solid spacer between the two face-to-face capsules, and a computed IntensityResult that retained its spacing redraws its own probe with res.plot_geometry().

The finite difference at true scale: two half-inch capsules face to face across 12 mm of solid spacer, and that $Δ r$ is both the sensitivity of the gradient estimate and the origin of the high-frequency error.

Show the code for this figure

import matplotlib.pyplot as plt
from phonometry import emission

# The classic 12 mm solid spacer between two half-inch microphones.
emission.plot_pp_probe_geometry()
plt.show()

# A computed IntensityResult retains its spacing:
#   res = emission.sound_intensity(p1, p2, fs, spacing=0.012)
#   res.plot_geometry()

The measurement itself is two synchronous channels: p1 and p2 are the calibrated pressures of the two capsules in pascals, p1 the one nearer the source, acquired on the same analyser at the same sampling rate with the probe axis normal to the measurement surface.

import numpy as np
from phonometry import emission

fs = 48000
rng = np.random.default_rng(0)
# The two probe-microphone pressures in Pa, p1 closest to the source.
#   In a real measurement these are your two calibrated probe recordings;
#   synthesized here (p2 = p1 delayed one sample) so the guide runs.
p1 = 0.02 * rng.standard_normal(fs)
p2 = np.concatenate(([0.0], p1[:-1]))   # p2 = p1 delayed one sample

res = emission.sound_intensity(p1, p2, fs, spacing=0.012, fraction=3,
                      limits=[100, 2500])
print(res.total_intensity_level, res.total_direction)      # LI [dB], ±1
print(res.frequency, res.intensity_level)                  # per band
res.plot()   # Lp vs LI per band + the pressure-intensity index (needs matplotlib)

Third-octave pressure and intensity levels for a plane progressive wave versus a standing wave

Left: in a plane progressive wave all pressure is transported, so $L_{I} \approx L_{p}$ . Right: a standing wave carries (almost) no net energy, so the pressure is high but the intensity collapses. The gap $L_{p} - L_{I}$ is the pressure-intensity index, the fundamental quality indicator of every intensity measurement.

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np
from scipy import signal
from phonometry import emission

fs, dr, c = 48000, 0.012, 343.0
n = int(fs * 4.0)

# Broadband noise, band-limited to 80 Hz - 6 kHz and scaled to about 70 dB SPL.
rng = np.random.default_rng(2026)
noise = signal.sosfilt(
    signal.butter(4, [80.0, 6000.0], btype="bandpass", fs=fs, output="sos"),
    rng.standard_normal(n),
)
noise *= 0.063 / np.std(noise)
spectrum = np.fft.rfft(noise)
freqs = np.fft.rfftfreq(n, 1 / fs)

# Left: a plane progressive wave. Microphone 2 sees the wave dr/c = 35 us later,
# applied as a phase shift rather than as a whole number of samples.
plane = emission.sound_intensity(
    noise, np.fft.irfft(spectrum * np.exp(-2j * np.pi * freqs * dr / c), n),
    fs, dr, fraction=3, limits=[100.0, 5000.0],
)

# Right: a standing wave, two equal counter-propagating waves, with the probe
# centred at x0 = 0.30 m from the rigid termination.
k, x0 = 2 * np.pi * freqs / c, 0.30
def at(pos):
    return np.fft.irfft(spectrum * 2.0 * np.cos(k * pos), n)
standing = emission.sound_intensity(at(x0 - dr / 2), at(x0 + dr / 2),
                                    fs, dr, fraction=3, limits=[100.0, 5000.0])

fig, (axl, axr) = plt.subplots(1, 2, figsize=(12, 5), sharey=True)
plane.plot(ax=axl)
standing.plot(ax=axr)
plt.show()

The same contrast plays out dynamically below: the pressure and velocity phasors of a progressive and a standing wave, with the instantaneous intensity averaging to a net flow in one case and to zero in the other.

Download the animation (WebM)

Knowing when to trust the number

Two physical limits bound every p-p measurement, and the result object carries both:

High frequency: the finite-difference gradient underestimates $I$ by $sin (k Δ r) / (k Δ r)$ , verified in CI against IEC 61043 Table 3. IntensityResult.bias_correction provides the factor and max_valid_frequency ( $\approx 0.1 c /Δ r$ ; 2.9 kHz for a 12 mm spacer) the practical ceiling. Larger spacers reach lower frequencies, smaller ones higher.
Reactive fields: when pressure_intensity_index ( $F_{2}$ in ISO 9614-1) approaches the probe’s residual index $δ_{p I 0}$ , phase errors dominate.

Three symbols recur from here on, and they are derived in The margin over the residual index below: $δ_{p I 0}$ is the residual pressure-intensity index, the phase-error floor of the probe-spacer-analyser chain; $K$ is the bias factor the grade allows (10 dB engineering, 7 dB survey); and $L_{d} = δ_{p I 0} - K$ is the dynamic capability, the largest field index the instrument can afford at that grade.

Before any of that, ISO 9614-1 asks a question about the field rather than the surface: is it steady enough to be scanned at all? In the initial test (clause 8.2) one typical position is picked on an initial measurement surface and the normal intensity is sampled there $M$ times with a short averaging time (Note 9 suggests $M = 10$ , and, for periodic signals, 8 s to 12 s per sample or any whole number of cycles). $F_{1}$ , the temporal variability indicator, is the coefficient of variation of those samples (equations (A.1)–(A.2)), so it is zero for a perfectly steady field and grows as the extraneous intensity wanders. Table B.3 asks for action code (e) when $F_{1}$ exceeds 0.6: reduce the variability, measure during quieter periods, or lengthen the averaging time at each position. Annex B also has it evaluated immediately before and after the measurement on any one surface (B.1.4).

from phonometry import emission

# The M short-time samples of the normal intensity at one fixed position (W/m²).
samples = [1.20e-5, 0.94e-5, 1.51e-5, 1.08e-5, 1.33e-5,
           1.02e-5, 1.44e-5, 1.17e-5, 0.88e-5, 1.29e-5]

f1 = emission.temporal_variability_indicator(samples)
print(round(f1, 3))                          # 0.177, a steady field

# Or carried on the surface result alongside F2/F3/F4, by handing the same
# samples to field_indicators together with the per-position scan below:
fi = emission.field_indicators([74.1, 73.8, 74.5, 73.2],
                               [1.2e-5, 1.0e-5, 1.4e-5, 0.9e-5],
                               temporal_intensity=samples)
print(fi.field_is_stationary())              # True (Table B.3 limit 0.6)

Over a measurement surface, the remaining ISO 9614-1 Annex A field indicators grade the scan itself. $F_{2}$ , the surface pressure-intensity indicator, is the surface pressure level minus the level of the mean magnitude of the normal intensity: the larger it is, the closer the measurement sits to the probe’s phase-error floor. $F_{3}$ , the negative partial power indicator, is the same difference taken with the signed mean intensity: $F_{3} - F_{2} > 0$ reveals power flowing inward through parts of the surface. $F_{4}$ , the field non-uniformity indicator, is the normalised spread of the per-position intensities: the larger it is, the more measurement positions the surface needs.

$F_{4}$ is the one that turns into a number of positions, through criterion 2 of Annex B, $N > C F_{4}^{2}$ (Eq. B.2). The factor $C$ is not a constant: Table B.2 makes it depend on the band and on the grade being claimed — about 11 at low frequency rising to 29 in the mid band for engineering grade, and 19 rising to 57 for precision, with 8 for the A-weighted survey case. So the same field non-uniformity asks for roughly twice as many positions at grade 1 as at grade 2. Worked: $F_{4} = 0.5$ in the 1 kHz band needs $N > 29 \times 0.25$ , so at least 8 positions at engineering grade and at least 15 at precision; and when one position count is used for every band, the standard applies the criterion at the band with the largest $C F_{4}^{2}$ .

$F_{4}$ is also directly reportable, which is what makes it worth computing even when the count is already fixed: the 95 % confidence interval on a band level is $10 lo g_{10} (1 \pm 2 F_{4} / N)$ (Eq. B.3), so $F_{4} = 0.5$ over 10 positions carries +1.2 dB / −1.7 dB. A band that fails criterion 2 may still be recorded, but only accompanied by that interval.

Together with the dynamic-capability criterion the indicators are available directly:

import numpy as np
from phonometry import emission

# Per-position measurements over the ISO 9614-1 measurement surface
pressure_levels = np.array([74.1, 73.8, 74.5, 73.2])       # Lp per position (dB)
normal_intensity = np.array([1.2e-5, 1.0e-5, 1.4e-5, 0.9e-5])  # signed In per position (W/m²)

fi = emission.field_indicators(pressure_levels, normal_intensity)
print(round(fi.f2, 2), round(fi.f3, 2), round(fi.f4, 3))   # 3.41 3.41 0.197
ld = emission.dynamic_capability_index(18.0)   # δpI0 = 18 dB → Ld = δpI0 − K
print(ld, ld > fi.f2)                                      # 8.0 True (criterion 1)

With 2D (positions, bands) arrays and the band centres the indicators come back per band, and the result is plottable in one line, the form in which the criteria are actually checked (each band passes or fails on its own):

fi = emission.field_indicators(lp_bands, in_bands, freqs)   # (positions, bands)
fi.plot(dynamic_capability=ld)   # F2/F3 per band vs Ld, F4 on a twin axis (needs matplotlib)

ISO 9614-1 field indicators of a ten-position scan over six octave bands: F2 and F3 climb towards low frequency as the field turns reactive, crossing the dashed dynamic-capability line Ld at 125 Hz where F3 also rises above F2, with the field non-uniformity F4 drawn as bars on a twin axis

$F_{2}$ climbs towards low frequency as the field turns reactive, and at 125 Hz it crosses the instrument’s dynamic capability $L_{d} = δ_{p I 0} - K$ : that band fails criterion 1, and no averaging will fix it — it calls for a larger spacer, a different surface or a quieter room. $F_{3}$ rising above $F_{2}$ in the same band reveals inward-flowing (negative) partial intensity, and the $F_{4}$ bars set the number of positions the surface needs (criterion 2, $N > C \cdot F_{4}^{2}$ ).

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np
from phonometry import emission

# A 10-position discrete-point scan over six octave bands: the surface
# pressure is nearly uniform, and the normal intensity per band is set so the
# field turns reactive towards low frequency, with two inward-flowing
# positions in the 125 Hz band (rescaled so the band mean keeps its target).
freqs = np.array([125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0])
delta_pi = np.array([10.5, 8.5, 6.0, 4.5, 3.5, 3.0])   # target Lp − L|In|
rng = np.random.default_rng(9614)
lp_bands = 78.0 + rng.normal(0.0, 0.4, (10, freqs.size))
i_mean = 10.0 ** ((78.0 - delta_pi) / 10.0) * 1.0e-12
in_bands = i_mean[None, :] * (1.0 + rng.normal(0.0, 0.18, (10, freqs.size)))
in_bands[:2, 0] = -0.35 * i_mean[0]
in_bands[2:, 0] *= (10.0 * i_mean[0] - in_bands[:2, 0].sum()) / in_bands[2:, 0].sum()

fi = emission.field_indicators(lp_bands, in_bands, freqs)
ld = emission.dynamic_capability_index(18.0)   # δpI0 = 18 dB, K = 10 dB

# One line — F2/F3 per band against Ld, with F4 on a twin axis:
fi.plot(dynamic_capability=ld)
plt.show()

# By hand, from the per-band fields the result carries — mirroring what
# FieldIndicators.plot() draws (Ld step line, merged twin-axis legend):
fig, ax = plt.subplots()
ax.plot(fi.frequency, fi.f2, "o-", label="F2 (surface pressure-intensity)")
ax.plot(fi.frequency, fi.f3, "s--", label="F3 (negative partial power)")
ax.plot(fi.frequency, np.full(fi.frequency.size, ld), ":",
        drawstyle="steps-mid", label="Dynamic capability Ld")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Indicator [dB]")
twin = ax.twinx()
twin.bar(fi.frequency, fi.f4, width=fi.frequency * 0.2, alpha=0.25,
         color="#2ca02c", label="F4 (non-uniformity)")
twin.set_ylabel("Field non-uniformity F4")
lines, labels = ax.get_legend_handles_labels()
tlines, tlabels = twin.get_legend_handles_labels()
ax.legend(lines + tlines, labels + tlabels)
plt.show()

Download the animation (WebM)

The margin over the residual index

The two channels of any real probe and analyzer are never perfectly phase matched. Feed both channels the same signal (the residual-intensity test of IEC 61043): the true intensity is exactly zero, yet the mismatch reports a small false intensity. The gap between the pressure level and that false intensity level is the residual pressure-intensity index $δ_{p I 0}$ , the instrument’s phase-error floor expressed as an index; IEC 61043 grades probes and processors (class 1 / class 2) chiefly by it.

In the field, the measured index $δ_{p I} = L_{p} - L_{I}$ says how far the pressure level stands above the level of the net flow, and the systematic error of the intensity estimate is bounded by the margin between the two indices:

ε = 10 lo g_{10} (1 \pm 1 0^{(δ_{p I} - δ_{p I 0}) /10})

A 10 dB margin keeps the bias within about 0.5 dB and a 7 dB margin within about 1 dB; these are precisely the bias factors $K$ of ISO 9614, and the dynamic capability $L_{d} = δ_{p I 0} - K$ is the largest field index the instrument can afford at a given grade. Read it as a budget: every decibel the field’s $δ_{p I}$ rises spends a decibel of margin, and when $δ_{p I}$ reaches $δ_{p I 0}$ the reading is pure phase error, of either sign. This is why the pressure-intensity index, not the microphone quality, gates the achievable accuracy of every intensity measurement.

Measuring the residual index, and checking the probe

$δ_{p I 0}$ is the number every criterion on this page and the whole scanning page rests on, and the library does not compute it: it is measured, once per probe-spacer-analyser chain, on a bench.

Three panels. First, the IEC 61043 residual-intensity testing device: a small coupler driving pink or white noise onto both probe capsules at the same level within plus or minus 0.1 decibel, from which the residual index is read as the pressure level minus the indicated intensity level. Second, an IEC 60942 class 0 or class 1 sound calibrator applied to one microphone at a time, with the instrument adjusted to within plus or minus 0.1 decibel in both channels. Third, the in-situ probe-reversal test: the probe on the measurement surface reading a positive normal intensity, then rotated 180 degrees about its acoustic centre and reading a negative one, accepted when the two have opposite signs and differ by less than 1.5 decibel in the band of maximum level

The residual-intensity test (IEC 61043 clause 10.2). The testing device is a small coupler that presents the same sound field to both capsules over the whole or part of 45 Hz to 7,1 kHz, driven with pink or white noise, and applying the pressure to the two microphones at the same level within ±0,1 dB at least from 45 Hz to 1 kHz, with a bounded inter-port phase difference. Because the true intensity in that field is exactly zero, whatever the instrument indicates is its own phase error, and $δ_{p I 0}$ is read band by band as the pressure level minus that indicated intensity level. Two consequences follow from what is in the coupler: the index belongs to the probe, the fitted spacer and the analyser together, so it must be re-measured whenever the spacer changes; and it is a property of the chain, not a specification of the microphones.

Before every series (clause 14). Let the instrument warm up as the manufacturer says. In sound-pressure mode, apply a sound calibrator to the two microphones, in turn or simultaneously, and adjust to the correct indication within ±0,1 dB in both channels — ISO 9614-2 clause 6.2.1 asks for a class 0 or class 1 IEC 60942 calibrator here. Then apply the residual-intensity device again and confirm that $δ_{p I 0}$ still meets the class requirement over the range the device covers, with the manufacturer’s phase compensation applied and, preferably, at a level close to the level of use.

In situ (ISO 9614-2 clause 6.2.2): the probe-reversal test. This is the one check that needs no bench and catches the error no bench can see. Place the probe on the measurement surface, axis normal, at a position where the intensity exceeds the surface mean; record $I_{n}$ in every band. Rotate the probe 180° about an axis normal to the measurement axis, keeping the acoustic centre in the same place — which in practice means mounting it on a stand rather than holding it — and measure again. For the measurement system to be acceptable, in the band of maximum level the two values of $I_{n}$ must have opposite signs and the two intensity levels must differ by less than 1,5 dB. Read the two failure modes as diagnoses: identical signs mean the channels are swapped or one is inverted, and a difference above 1,5 dB means the probe is disturbing its own field or the two channels are not matched. Clause 10.5 c) makes the result part of the test report.

That sign is worth reading in normal use too, not only in the check. A negative net intensity over part of an enclosing surface means more power entering that patch than leaving it — entirely normal on the shadow side of a machine standing beside a louder one, and completely meaningless if the channels are swapped.

Grading the instrument: IEC 61043 Table 2

$δ_{p I 0}$ is not just a number to subtract $K$ from; it is what IEC 61043 grades the hardware by. Table 2 of the standard sets a minimum $δ_{p I 0}$ in every one-third-octave band from 50 Hz to 6.3 kHz, separately for a probe, a processor and a complete instrument, in class 1 and class 2, printed for the nominal 25 mm microphone separation. Note 1 rescales the whole table for any other spacer by $+ 10 lo g_{10} (x /25)$ with $x$ in millimetres, which is the same 3 dB per doubling the previous section arrived at from the physics.

intensity_class_compliance compares a measured $δ_{p I 0}$ spectrum against both masks band by band and returns an IntensityInstrumentComplianceResult holding the class the chain actually meets: the loosest class every band clears, or None when some band clears neither.

from phonometry import metrology

# The band centres Table 2 is defined on, and the measured residual index of
# the chain: one value per band, taken with the spacer that will be fitted in
# the field. Replace the placeholder with your own residual-intensity test.
freqs, _, _ = metrology.residual_index_limits("instrument", spacing=0.012)
measured_delta_pi0 = [11.0, 11.9, 11.2, 10.0, 13.2, 15.9, 17.0, 18.0,
                      19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 24.0, 24.0,
                      24.0, 24.0, 24.0, 24.0, 24.0, 24.0]   # dB, 22 bands

res = metrology.intensity_class_compliance(measured_delta_pi0, freqs,
                                           device="instrument", spacing=0.012)
print(res.overall_class)          # 2: one band misses the class 1 minimum
print(res.binding_margin())       # smallest per-band margin to that class [dB]
print(res.failing_bands(1))       # the band centres that cost it class 1 [Hz]
res.plot()                        # measured δpI0 over the two Table 2 masks
res.report("verification.pdf")    # one-page verification fiche (PDF)

Both Table 2 masks come down by $10 lo g_{10} (12/25) = - 3.2 dB$ for the 12 mm spacer. The measured index climbs 10 dB per decade at low frequency, parallel to the requirement, because a channel phase mismatch that is constant in degrees buys exactly that slope; it flattens above 1 kHz where the mismatch of a real chain starts growing with frequency instead. A vent resonance around 100 Hz costs 4 dB and drops that one band below the class 1 minimum, so the whole chain is graded class 2.

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np
from phonometry import metrology

# A complete instrument with the common 12 mm spacer. The measured index is
# modelled from the physics behind Table 2: a residual phase mismatch φs reads
# as δpI0 = 10 lg(kd/φs), so a mismatch that is constant in degrees already
# climbs 10 dB per decade, and above 1 kHz the mismatch of a real chain grows
# with frequency, so the index levels off.
spacing = 0.012
freqs, _, _ = metrology.residual_index_limits("instrument", spacing=spacing)
phase_mismatch = 0.05 * np.maximum(1.0, freqs / 1000.0)          # degrees
measured = metrology.residual_index_from_phase_mismatch(phase_mismatch, freqs,
                                                        spacing)
measured = measured - 4.0 * np.exp(-((np.log(freqs / 100.0) / 0.25) ** 2))

res = metrology.intensity_class_compliance(measured, freqs, spacing=spacing)
res.plot()
plt.show()

Two companion rules of the standard come with it. Clause 6.1 fixes the frequency range a class attests: 45 Hz to 7.1 kHz in one-third octaves, which class 1 requires and class 2 may also use, or 45 Hz to 5.6 kHz in octaves, which is offered to class 2 as an alternative. A verdict computed over fewer bands is flagged range_limited so it cannot be read as a full-range claim. The Spanish translation UNE-EN 61043:1999 records only the octave alternative for class 2 and drops the one-third-octave one; the library follows the EN/IEC text (see the errata registry). Clause 8 combines separately supplied components: instrument_class_from_components(probe_class, processor_class) returns 1 only when both are class 1, and 2 for every other pairing.

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

IEC 61043 sound intensity instrument class verification example report (PDF) — Instrument class verification fiche (IntensityInstrumentComplianceResult.report), the achieved class with its binding margin and the measured δpI0 over the Table 2 minima rescaled to the 12 mm spacer.

Reading `δpI0` as a phase error

The requirement is really a phase-matching requirement in disguise. In an axially propagating plane wave the true phase difference across the spacer is $k d$ , so a residual intensity produced by a channel mismatch $φ_{s}$ gives

δ_{p I 0} = 10 lo g_{10} (\frac{k d}{φ _{s}})

and the two conversions run both ways:

from phonometry import metrology

# 20 dB of residual index at 1 kHz over a 25 mm spacer:
phi = metrology.phase_mismatch_from_residual_index(20.0, 1000.0, 0.025)
print(round(float(phi), 2))     # 0.26 degrees, a hundredth of kd

# And back, for a chain whose channels are matched to 0.05°:
print(round(float(metrology.residual_index_from_phase_mismatch(
    0.05, 1000.0, 0.012)), 1))  # 24.0 dB

That is why the low-frequency end of Table 2 rises 1 dB per third-octave band and then flattens: below the knee the standard is asking for a constant phase match, and above it for a constant index. The knee sits at 250 Hz for the three class 1 columns and for the class 2 processor, and at 630 Hz for the class 2 probe and the class 2 complete instrument, which is also where the table stops stepping in whole decibels (the instrument goes 14 dB at 315 Hz, 14.5 dB at 400 Hz, 15 dB at 500 Hz). It is also why a tenth of a degree of channel mismatch is a demanding specification, and why a probe must be verified with the spacer it will actually be used with.

Choosing the probe

Two decisions are made before any data is taken: where the measurement surface can be placed, and which spacer goes on the probe.

Reactive fields near sources

Close to a source the field turns reactive: pressure and particle velocity drift toward quadrature, so a large pressure carries little net flow. For a small source the quadrature component grows as $1/ (k r)$ ; at 100 Hz and 0.25 m from the source it is already about twice the active one, and $δ_{p I}$ climbs just as it does in the standing wave of the two-panel figure at the top of this page. The same happens between a machine and a hard reflecting surface, and in reverberant rooms where the diffuse field raises pressure without transporting energy outward. This is why ISO 9614-1 keeps the measurement surface on average more than 0.5 m away from the source, and why, when a scan fails the dynamic-capability criterion, moving the surface outward or adding absorption to the room usually lowers $F_{2}$ below $L_{d}$ more cheaply than better hardware.

Choosing the spacer

The spacer sets both ends of the usable band, in opposite directions:

The top end is geometry. The finite difference underestimates the gradient by $sin (k Δ r) / (k Δ r)$ , so the ceiling scales as $1/Δ r$ : max_valid_frequency $\approx 0.1 c /Δ r$ keeps the bias within about 0.3 dB, giving roughly 5.7 kHz for a 6 mm spacer, 2.9 kHz for 12 mm and 690 Hz for 50 mm (bias_correct=True undoes the known bias somewhat beyond that).
The bottom end is phase. A progressive wave puts only $360 f Δ r / c$ degrees of true phase across the spacer, 0.8° at 63 Hz over 12 mm, while the channel mismatch stays fixed. Lowering the frequency shrinks the signal, not the error, so the margin over $δ_{p I 0}$ collapses at low frequency. A larger spacer buys back that margin: the IEC 61043 residual-index requirements scale as $10 lo g_{10} (Δ r /25 mm)$ , so doubling the spacer is worth 3 dB of low-frequency margin.

The two limits close in from opposite sides. The upper panel is geometry: each spacer’s max_valid_frequency is exactly where its own bias curve crosses −0.3 dB. The lower panel is the phase budget: the requirement moves with the spacer by $10 lo g_{10} (Δ r /25 mm)$ , so a 50 mm probe starts 3 dB better off than a 25 mm one and 9 dB better off than a 6 mm one. Where the two panels leave a spacer usable is the band it may be trusted over.

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np

# The bias comes out of sound_intensity itself: bias_correction is the
# compensating factor (k dr)/sin(k dr), so the bias is its reciprocal in dB.
fs, c = 48000, 343.0
rng = np.random.default_rng(61043)
x = rng.standard_normal(fs)
spectrum, bins = np.fft.rfft(x), np.fft.rfftfreq(fs, 1 / fs)

fig, (axt, axb) = plt.subplots(2, 1, sharex=True)
for dr in (0.006, 0.012, 0.050):
    p2 = np.fft.irfft(spectrum * np.exp(-2j * np.pi * bins * dr / c), fs)
    res = emission.sound_intensity(x, p2, fs, dr, fraction=3,
                                   limits=[20.0, 12500.0])
    usable = res.frequency < c / (4.0 * dr)      # drop the clamped bands
    axt.semilogx(res.frequency[usable],
                 -10.0 * np.log10(res.bias_correction[usable]),
                 label=f"{dr * 1000:.0f} mm")
    axt.axvline(res.max_valid_frequency, linestyle=":")
    axb.semilogx(res.frequency,
                 np.full(res.frequency.shape, 10.0 * np.log10(dr / 0.025)))
axt.axhline(-0.3, linestyle="--")
axt.set(ylabel="Bias [dB]", ylim=(-2.4, 0.35))
axb.set(xlabel="Frequency [Hz]", ylabel="δpI0 margin re 25 mm [dB]")
axt.legend()
plt.show()

No single spacer covers the full audio range: 6 mm suits high-frequency work, 50 mm low-frequency work, and the common 12 mm covers the mid band. Wide-band surveys are measured twice with two spacers and the band results merged; whichever spacer is fitted, verify $δ_{p I 0}$ with that spacer in place, since the index belongs to the probe-spacer-analyzer chain, not to the microphones alone.

`sound_intensity()` parameters

Parameter	Type	Units	Range / default	Notes
`p1`, `p2`	1D arrays	Pa	equal length	Microphone closer to the source first; reversing them flips the sign
`fs`	int	Hz	> 0
`spacing`	float	m	> 0	Microphone separation $Δ r$ (typ. 6/12/50 mm)
`rho`	float	kg/m³	default `1.204`	Air density
`c`	float	m/s	default `343.0`	Speed of sound (bias/validity estimates)
`fraction`	int, optional	—	`1`, `3` or `None` (default)	Octave/third-octave band integration
`limits`	list, optional	Hz	default library band range	Band analysis limits
`bias_correct`	bool	—	default `False`	Apply the per-bin $(k Δ r) / sin (k Δ r)$ correction (IEC 61043 §7.3) before summing, so band/broadband totals stop under-reading as $f \to$ `max_valid_frequency`; bins past the first null are left uncorrected. The per-band `bias_correction` factor is reported either way

See Theory for the derivations and Calibration for absolute scaling of the two channels.

What this guide covers

Covered
IEC 61043:1993 (EN 61043:1994) for the two-microphone cross-spectral intensity estimator of sound_intensity, the finite-difference bias correction and its usable-bandwidth bound (clause 7.3, Table 3), the residual pressure-intensity index $δ_{p I 0}$ as the instrument’s phase-error floor, and the instrument grading built on it: the Table 2 minima per band for probes, processors and instruments with the Note 1 separation rule, the clause 6.1 frequency ranges and the clause 8 component-combination rule (intensity_class_compliance, residual_index_limits, instrument_class_from_components), plus the $δ_{p I 0}$ to phase-mismatch conversion in both directions. ISO 9614-1:1993 for the Annex A field indicators $F_{1}$ to $F_{4}$ (temporal_variability_indicator, field_indicators) with the Table B.3 limit on $F_{1}$ , and the Annex B dynamic-capability criterion (dynamic_capability_index).
Not covered
ISO 9614-1’s own sound-power determination, clauses 8 and 9, which sums the partial powers $I_{i} \cdot S_{i}$ of the discrete measurement points into $L_{W}$ , is not implemented as a function here: sound_intensity and field_indicators report intensity and field quality, not a summed power. The scanning routes that do compute $L_{W}$ from intensity (ISO 9614-2 and ISO 9614-3) live in Sound Power by Intensity Scanning. The residual-intensity test of IEC 61043 is not performed by the library either: $δ_{p I 0}$ is a value the caller measures on their own probe-analyzer chain, with the spacer that will be fitted, and supplies — the procedure is in Measuring the residual index above, and what phonometry does with the number is grade it against Table 2. The before-use check of clause 14 and the ISO 9614-2 probe-reversal test are likewise procedures, not functions.

References

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 the subject: active and reactive intensity, the p-p estimator and the phase-mismatch error budget behind this page.
International Electrotechnical Commission. (1993). Electroacoustics — Instruments for the measurement of sound intensity — Measurements with pairs of pressure sensing microphones (IEC 61043:1993). Adopted in Europe as EN 61043:1994. The instrument standard: the two-microphone cross-spectral estimator, the residual-intensity test behind δpI0, the Table 2 minimum residual index per band for probes, processors and instruments in class 1 and class 2 with its +10 lg(x/25) separation rule (Note 1), the processor frequency ranges (clause 6.1), the component-combination rule (clause 8), and the finite-difference bias correction with its usable-bandwidth bound (clause 7.3, Table 3).
International Organization for Standardization. (1993). Acoustics — Determination of sound power levels of noise sources using sound intensity — Part 1: Measurement at discrete points (ISO 9614-1:1993). The pressure-intensity index, the Annex A field indicators F1 to F4 (F1 from equations (A.1)-(A.2), evaluated in the initial test of clause 8.2 and again per B.1.4), the Table B.3 temporal-variability limit, the dynamic-capability criterion (Annex B) and the 0.5 m surface-distance rule.

Created and maintained by José Manuel Requena Plens ORCID All projects

Sound Intensity (p-p)

The two-microphone principle (IEC 61043)

Knowing when to trust the number

The margin over the residual index

Measuring the residual index, and checking the probe

Grading the instrument: IEC 61043 Table 2

Reading δpI0 as a phase error

Choosing the probe

Reactive fields near sources

Choosing the spacer

sound_intensity() parameters

What this guide covers

See also

References

Reading `δpI0` as a phase error

`sound_intensity()` parameters