Source: https://jmrplens.github.io/phonometry/reference/api/power/intensity/ # 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): $$ I(f) = -\ \frac{\mathrm{Im}\lbrace G_{12}(f)\rbrace}{2\pi f\ \rho_0\ \Delta r} $$ ```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) ``` *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.*

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```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() ```

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. [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) ``` 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): ```python 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) ``` *F2 climbs towards low frequency as the field turns reactive, and at 125 Hz it crosses the instrument's dynamic capability `Ld = δpI0 − 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. F3 rising above F2 in the same band reveals inward-flowing (negative) partial intensity, and the F4 bars set the number of positions the surface needs (criterion 2, `N > C·F4²`).*

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```python 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() ```

[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](https://jmrplens.github.io/phonometry/reference/theory/signal-analysis/) for the derivations and [Calibration](https://jmrplens.github.io/phonometry/reference/api/levels/calibration/) 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).