Sound Intensity (p-p)

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):

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

from phonometry import sound_intensity

res = 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

Left: in a plane progressive wave all pressure is transported — L_I ≈ L_p. Right: a standing wave carries (almost) no net energy — 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.

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. The ISO 9614-1 Annex A field indicators and the dynamic-capability criterion are available directly:

from phonometry import field_indicators, dynamic_capability_index

fi = field_indicators(pressure_levels, normal_intensity)   # F2, F3, F4
ld = dynamic_capability_index(18.0)   # δpI0 = 18 dB → Ld = δpI0 − K
ok = ld > fi.f2                                            # criterion 1

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

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