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# Sound Intensity (p-p method)

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 Δr 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):

$$
p = \frac{p_1 + p_2}{2}, \qquad
u = -\frac{1}{\rho_0\ \Delta r}\int (p_2 - p_1)\ dt, \qquad
I = \overline{p\ u}
$$

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

$$
I(f) = -\ \frac{\mathrm{Im}\lbrace G_{12}(f)\rbrace}{2\pi f\ \rho_0\ \Delta r}
$$

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_pp_probe_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_pp_probe.svg" alt="Two-microphone p-p intensity probe with the spacer distance and the measurement axis" width="92%"></picture>

```python
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)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/intensity_demo_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/intensity_demo.svg" alt="Third-octave pressure and intensity levels for a plane progressive wave versus a standing wave" width="92%"></picture>

*Left: in a plane progressive wave all pressure is transported, so
L_I ≈ 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.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
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])

# res is the IntensityResult computed in the example above.
# One line — Lp vs LI per band with the pressure-intensity index on a twin axis:
res.plot()
plt.show()

# By hand, from the per-band fields the result carries — mirroring what
# IntensityResult.plot() draws (bar label, merged twin-axis legend, δpI title):
fig, ax = plt.subplots()
ax.semilogx(res.frequency, res.pressure_level, "o-", label="Pressure level Lp")
ax.semilogx(res.frequency, res.intensity_level, "s--", label="Intensity level LI")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Level [dB]")
twin = ax.twinx()
twin.bar(res.frequency, res.pressure_intensity_index,
         width=res.frequency * 0.2, color="#2ca02c", alpha=0.25,
         label="δpI = Lp − LI")
twin.set_ylabel("Pressure-intensity index δpI [dB]")
# Merge both axes' handles into a single legend, exactly as .plot() does:
lines, labels = ax.get_legend_handles_labels()
tlines, tlabels = twin.get_legend_handles_labels()
ax.legend(lines + tlines, labels + tlabels)
ax.set_title(f"Lp vs LI  (total δpI = {res.total_pressure_intensity_index:.1f} dB)")
plt.show()
```

</details>

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.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_instantaneous_intensity_dark.gif"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_instantaneous_intensity.gif" alt="Animation: a two-microphone p-p probe with rotating pressure and velocity phasors; the instantaneous intensity arrow flips while its running average settles to a net flow for the progressive wave and to zero for the standing wave" width="640" height="360" loading="lazy"></picture>

[Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_instantaneous_intensity.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
  $\sin(k\Delta r)/(k\Delta r)$, verified in CI against IEC 61043 Table 3.
  `IntensityResult.bias_correction` provides the factor and
  `max_valid_frequency` (≈ 0.1·c/Δr; 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` (F2 in ISO 9614-1)
  approaches the probe's residual index δ_pI0, phase errors dominate.

Over a measurement surface, the ISO 9614-1 Annex A field indicators grade the
scan itself. **F2**, 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. **F3**, the negative partial power indicator, is the same
difference taken with the *signed* mean intensity: F3 − F2 > 0 reveals power
flowing inward through parts of the surface. **F4**, 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. Together with
the dynamic-capability criterion they are available directly:

```python
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)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_intensity_scan_power_dark.gif"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_intensity_scan_power.gif" alt="Animation: a p-p probe traces the serpentine scan over the top face of the measurement box while the normal-intensity arrows appear behind it, and the partial powers of the five faces accumulate into the sound power level L_W" width="640" height="360" loading="lazy"></picture>

[Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_intensity_scan_power.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** `δpI0`, 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 `δpI = Lp − LI` 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:

$$
\varepsilon = 10 \log_{10}\!\left( 1 \pm 10^{(\delta_{pI} - \delta_{pI0})/10} \right)
$$

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** `Ld = δpI0 − K` is the largest field index the
instrument can afford at a given grade. Read it as a budget: every decibel
the field's `δpI` rises spends a decibel of margin, and when `δpI` reaches
`δpI0` 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.

### 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/(kr)`; at
100 Hz and 0.25 m from the source it is already about twice the active one,
and `δpI` climbs just as it does in the standing wave of the figure above.
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 `F2` below `Ld` 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(kΔr)/(kΔr)`, so the ceiling scales as `1/Δr`:
  `max_valid_frequency ≈ 0.1·c/Δr` 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·Δr/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 `δpI0` collapses at
  low frequency. A larger spacer buys back that margin: the IEC 61043
  residual-index requirements scale as `10·lg(Δr/25 mm)`, so doubling the
  spacer is worth 3 dB of low-frequency margin.

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 `δpI0` 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 Δr (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 $(k\Delta r)/\sin(k\Delta r)$ 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](theory-signal-analysis.md) for the derivations and [Calibration](calibration.md)
for absolute scaling of the two channels.

## References

- Fahy, F. J. (1995). *Sound intensity* (2nd ed.). E&FN Spon.
  ISBN 978-0-419-19810-9.
  [doi:10.4324/9780203475386](https://doi.org/10.4324/9780203475386).
  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).
  [IEC webstore](https://webstore.iec.ch/en/publication/4353).
  The instrument standard: the cross-spectral estimator, the
  residual-intensity test behind `δpI0` and the class 1 / class 2
  requirements.
- 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).
  [iso.org catalogue](https://www.iso.org/standard/17427.html).
  The field indicators F2 to F4, the dynamic-capability criterion and the
  0.5 m surface-distance rule.

## Standards

IEC 61043:1993 (EN 61043:1994), *Electroacoustics —
Instruments for the measurement of sound intensity — Measurements with pairs
of pressure sensing microphones*: the two-microphone cross-spectral
intensity estimator, the finite-difference bias correction and the
usable-bandwidth bound (clause 7.3,
Table 3). ISO 9614-1:1993, *Acoustics — Determination of sound power levels
of noise sources using sound intensity — Part 1: Measurement at discrete
points*: the pressure-intensity index, the Annex A field indicators F2, F3
and F4, and the dynamic-capability criterion (Annex B).
