emission.intensity

Two-microphone (p-p) sound intensity per IEC 61043:1993 and the ISO 9614-1:1993 field indicators.

A p-p probe holds two pressure microphones a fixed distance spacing ( $Δ r$ ) apart. The mean of the two pressures is taken as the sound pressure at the probe reference point, while the pressure differential is used to derive the particle velocity component along the probe axis (IEC 61043:1993, definition 3.2):

p (t) = \frac{p _{1} ( t ) + p _{2} ( t )}{2}

u (t) = - \frac{1}{ρ Δ r} \int (p_{2} (t) - p_{1} (t)) d t

I = ⟨ p (t) u (t)⟩ (time average)

For stationary signals the time-averaged intensity reduces to the imaginary part of the one-sided cross spectrum $G_{12}$ of the two microphone pressures (the frequency-domain form of the same finite-difference estimator):

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

The finite-difference gradient underestimates the true plane-wave intensity by the factor $sin (k Δ r) / (k Δ r)$ with $k = 2 π f / c$ ; IEC 61043:1993 clause 7.3 specifies the probe intensity response with exactly this argument ( $F_{f} = Δ r f 2 π / c$ ) and Table 3 lists the resulting nominal response (e.g. -10.5 dB at 6.3 kHz for a 25 mm separation). Below $f = 0.1 c /Δ r$ ( $k Δ r < 0.63$ ) the bias stays under about 0.3 dB (factor $\geq 0.935$ ).

Field indicators F1 (temporal variability), F2 (surface pressure-intensity), F3 (negative partial power) and F4 (field non-uniformity) follow ISO 9614-1:1993 Annex A (normative), equations (A.1)-(A.9). F1 is measured in the initial test (clause 8.2) at one typical position on the initial measurement surface and qualifies the field, not the surface: it is the coefficient of variation of M short-time-averaged samples of the normal intensity at that fixed point, and Table B.3 calls for action code (e) when it exceeds 0.6. The dynamic capability index is $L_{d} = δ_{p I 0} - K$ (ISO 9614-1 clause 3.12, equation (10)); the instrument is adequate for a measurement when $L_{d} > F_{2}$ (criterion 1, Annex B equation (B.1)). The residual index $δ_{p I 0}$ that feeds it is classified against IEC 61043:1993 Table 2 by phonometry.emission.intensity_compliance.intensity_class_compliance.

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dynamic_capability_index

dynamic_capability_index(
    pressure_residual_intensity_index: float,
    bias_error_factor: float = 10.0,
) -> float

Dynamic capability index Ld (ISO 9614-1:1993, clause 3.12).

$L_{d} = δ_{p I 0} - K$ (equation (10)), where $δ_{p I 0}$ is the instrument pressure-residual intensity index (clause 3.11, equation (9); determined per IEC 61043:1993, which requires the Table 2 minima per class) and K the bias error factor of Table 1: 10 dB for precision (grade 1) and engineering (grade 2) measurements, 7 dB for survey (grade 3). The measurement arrangement is adequate when $L_{d} > F_{2}$ (criterion 1, Annex B equation (B.1)).

Parameters

Name	Description
`pressure_residual_intensity_index`	$δ_{p I 0}$ in decibels.
`bias_error_factor`	K in decibels (default 10.0).

Returns: Ld in decibels.

field_indicators

field_indicators(
    pressure_levels: list[float] | np.ndarray,
    normal_intensity: list[float] | np.ndarray,
    frequencies: list[float] | np.ndarray | None = None,
    *,
    temporal_intensity: list[float] | np.ndarray | None = None,
) -> FieldIndicators

ISO 9614-1:1993 Annex A field indicators F1 to F4.

Given the sound pressure level Lpi (dB) and the signed normal sound intensity Ini (W/m^2) measured at each of the N discrete positions on the measurement surface:

$F_{2} = L_{p} - L_{∣ I_{n} ∣}$ (equation (A.3)), with the surface pressure level from equation (A.4) and the level of the mean magnitude of the normal intensity from equation (A.5);
$F_{3} = L_{p} - L_{I_{n}}$ (equation (A.6)), with the algebraic surface intensity level from equation (A.7);
$F_{4} = \frac{1}{∣ I _{n} ∣} \sum_{i} (I_{ni} - \overline{I_{n}})^{2} / (N - 1)$ (equations (A.8)-(A.9)).

F1, the temporal variability indicator (equation (A.1)), does not come from the surface scan: it is evaluated in the initial test at one typical position from M short-time-averaged intensity samples (clause 8.2), and again immediately before and after the measurement on any one measurement surface (Annex B, B.1.4). Passing those samples as temporal_intensity fills f1 on the result; see temporal_variability_indicator.

The inputs are either 1D per-position arrays (one frequency band, scalar indicators) or 2D (positions, bands) arrays (the indicators are evaluated band by band and returned as per-band arrays; the plottable form). If the algebraic mean intensity of any band is not positive the test conditions do not satisfy ISO 9614-1 in that band (clause A.2.3) and a ValueError is raised.

Parameters

Name	Description
`pressure_levels`	Lpi at each position, in decibels; 1D `(positions,)` or 2D `(positions, bands)`.
`normal_intensity`	Signed normal intensity Ini at each position, in W/m^2; same shape as `pressure_levels`.
`frequencies`	Band centre frequencies in Hz, one per column of the 2D input (optional for 1D input, where it is ignored beyond a length check).
`temporal_intensity`	Optional M short-time-averaged normal intensity samples `Ink` at one fixed position, in W/m^2; 1D `(samples,)` for a single band or 2D `(samples, bands)` with one column per band of the surface input. Supplying it fills `f1` on the returned result.

Returns: FieldIndicators.

FieldIndicators

FieldIndicators(
    f2: float | np.ndarray,
    f3: float | np.ndarray,
    f4: float | np.ndarray,
    frequency: np.ndarray | None = None,
    f1: float | np.ndarray | None = None,
)

ISO 9614-1:1993 Annex A field indicators over a measurement surface.

f2 is the surface pressure-intensity indicator (equation (A.3)), f3 the negative partial power indicator (equation (A.6)) and f4 the field non-uniformity indicator (equation (A.8)). $F_{3} - F_{2} > 0$ reveals negative partial power flowing through parts of the surface. The instrument’s dynamic capability index must satisfy $L_{d} > F_{2}$ (criterion 1, equation (B.1)); the number of positions N must satisfy $N > C F_{4}^{2}$ (criterion 2, equation (B.2)).

f1 is the temporal variability indicator (equation (A.1)), present only when field_indicators was given the temporal_intensity samples it is computed from; it is None otherwise. Unlike the other three it describes the field rather than the surface: it is evaluated at one typical position from M short-time-averaged samples of the normal intensity (clause 8.2, the initial test), and Table B.3 requires action when it exceeds TEMPORAL_VARIABILITY_LIMIT.

With per-position and per-band input (2D arrays passed to field_indicators) the indicators are per-band arrays and frequency carries the band centres; with 1D per-position input they are scalars and frequency is None.

FieldIndicators.field_is_stationary()

FieldIndicators.field_is_stationary(limit: float = 0.6) -> bool | np.ndarray

Whether the temporal variability stays within the Table B.3 limit.

ISO 9614-1:1993 Table B.3 lists $F_{1} > 0.6$ as the condition that calls for action code (e), i.e. reducing the temporal variability of the extraneous intensity, measuring during quieter periods or lengthening the averaging time at each position. A field at exactly the limit is not flagged.

Parameters

Name	Description
`limit`	The Table B.3 threshold (default `TEMPORAL_VARIABILITY_LIMIT`, 0.6).

Returns: True/False for a scalar f1, or a boolean array per band for a per-band f1.

Raises

Exception	When
ValueError	If the result carries no `f1` (call `field_indicators` with `temporal_intensity`, or use `temporal_variability_indicator` directly).

FieldIndicators.plot()

FieldIndicators.plot(
    ax: Axes | None = None,
    *,
    dynamic_capability: float | np.ndarray | None = None,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot the per-band indicators F2/F3, the Ld line and F4.

F1 and its Table B.3 limit of 0.6 are drawn beside F4 when the result carries them, that is when temporal_intensity was supplied to field_indicators.

Requires per-band data (call field_indicators with 2D (positions, bands) arrays and frequencies) and matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

Name	Description
`ax`	Existing axes, or `None` to create a figure.
`dynamic_capability`	Optional instrument dynamic capability index `Ld` in dB (scalar or per band), drawn as the criterion-1 reference line ( $L_{d} > F_{2}$ , equation (B.1)); see `dynamic_capability_index`.
`language`	Label language, `"en"` (default) or `"es"`.
`kwargs`	Forwarded to the F2 curve `plot` call.

Raises

Exception	When
ValueError	If the result carries no per-band data.

IntensityResult

IntensityResult(
    frequency: np.ndarray | None,
    intensity: np.ndarray | None,
    intensity_level: np.ndarray | None,
    pressure_level: np.ndarray | None,
    pressure_intensity_index: np.ndarray | None,
    direction: np.ndarray | None,
    bias_correction: np.ndarray | None,
    total_intensity: float,
    total_intensity_level: float,
    total_pressure_level: float,
    total_pressure_intensity_index: float,
    total_direction: int,
    max_valid_frequency: float,
    spacing: float | None = None,
)

Result of a p-p sound intensity measurement.

Per-band arrays are None unless a band fraction was requested. intensity is signed (positive along the probe axis, from microphone 1 towards microphone 2); intensity_level is computed from the magnitude, $10 lo g_{10} (∣ I ∣/1 0^{- 12})$ dB, with the sign reported separately in direction (+1/-1). pressure_intensity_index is $L_{p} - L_{I}$ (the single-position form of the ISO 9614-1:1993 F2 indicator, equation (A.3)). bias_correction is the multiplicative factor $(k Δ r) / sin (k Δ r)$ compensating the finite-difference underestimation at each band centre (IEC 61043:1993, 7.3); it is NaN at and beyond the first null $k Δ r \geq π$ . max_valid_frequency is the usable-bandwidth bound $0.1 c /Δ r$ (bias < ~0.3 dB). spacing retains the microphone separation the measurement was reduced with so plot_geometry can draw the probe; it is appended after the original fields and None for hand-built results.

IntensityResult.plot()

IntensityResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot Lp vs LI per band with the pressure-intensity index.

Requires per-band data (call sound_intensity(..., fraction=...)) and matplotlib (pip install phonometry[plot]); returns the Axes.

IntensityResult.plot_geometry()

IntensityResult.plot_geometry(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the p-p probe with its spacer to scale.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Raises

Exception	When
ValueError	If the result does not retain its geometry.

plot_pp_probe_geometry

plot_pp_probe_geometry(
    spacing: float = 0.012,
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the face-to-face p-p intensity probe to scale.

Two phase-matched microphones separated by the solid spacer, with the intensity axis through both; default is the classic 12 mm spacer.

Parameters

Name	Description
`spacing`	Microphone separation `dr`, in metres.
`ax`	Existing axes, or `None` to create a figure.
`language`	Label language, `"en"` (default) or `"es"`.
`kwargs`	Forwarded to the spacer rectangle.

Returns: The axes.

sound_intensity

sound_intensity(
    p1: list[float] | np.ndarray,
    p2: list[float] | np.ndarray,
    fs: int,
    spacing: float,
    rho: float = 1.204,
    c: float = 343.0,
    fraction: int | None = None,
    limits: list[float] | None = None,
    bias_correct: bool = False,
) -> IntensityResult

Sound intensity from a two-microphone (p-p) probe (IEC 61043:1993).

The one-sided cross spectrum G12 of the two microphone pressures is estimated with Welch-averaged, Hann-windowed segments (scipy.signal.csd) and converted to the intensity spectral density along the probe axis, with $Δ r$ the microphone spacing (definition 3.2 of IEC 61043:1993 gives the underlying p-p formulation):

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

Positive intensity flows from microphone 1 towards microphone 2. The mean-square pressure is taken from the mean signal $(p_{1} + p_{2}) /2$ at the probe reference point. When fraction is given, both quantities are integrated into octave (1) or one-third octave (3) bands using the ANSI S1.11/IEC 61260-1 band edges of phonometry.nominal_frequencies; bands without any spectral bin are dropped. Broadband totals are always computed (over limits when provided, otherwise over all positive frequencies).

The pressure-intensity index $L_{p} - L_{I}$ is reported per band and broadband; in a free plane progressive wave it equals $10 lo g_{10} (ρ c /400) = 0.14$ dB (IEC 61043:1993 clause 5 note), while large values flag reactive or noisy fields (compare with the instrument dynamic capability, ISO 9614-1:1993 criterion 1).

Usable bandwidth: the finite-difference gradient biases the result by the factor $sin (k Δ r) / (k Δ r)$ (IEC 61043:1993, 7.3); results are essentially unbiased (< ~0.3 dB) below max_valid_frequency $= 0.1 c /Δ r$ , and bias_correction provides the per-band compensation factor.

Parameters

Name	Description
`p1`	Pressure signal of microphone 1, in pascals (1D).
`p2`	Pressure signal of microphone 2, in pascals (1D).
`fs`	Sample rate in Hz.
`spacing`	Microphone separation $Δ r$ , in metres.
`rho`	Air density, in kg/m^3. Default 1.204 (20 degC).
`c`	Speed of sound, in m/s. Default 343.0.
`fraction`	`None` (broadband only), 1 (octave bands) or 3 (one-third octave bands).
`limits`	[f_min, f_max] band limits in Hz (default [12, 20000], as in `phonometry.nominal_frequencies`).
`bias_correct`	If True, apply the per-bin finite-difference correction $(k Δ r) / sin (k Δ r)$ (IEC 61043:1993, 7.3) to the intensity spectral density before summing the band and broadband totals, so the totals no longer under-read as the frequency approaches `max_valid_frequency`. The reciprocal diverges as $k Δ r \to π$ (the first spatial-aliasing null at $c / (2Δ r)$ , inside the default band range for close spacings), so it is applied only over the probe’s usable range (up to $k Δ r = π /2$ ) and held constant beyond, keeping the totals bounded instead of letting a few near-null bins dominate them. Default False keeps the exact legacy totals; the per-band `bias_correction` factor (same clamped definition) is reported either way.

Returns: IntensityResult.

temporal_variability_indicator

temporal_variability_indicator(
    short_time_intensity: list[float] | np.ndarray,
) -> float | np.ndarray

ISO 9614-1:1993 temporal variability indicator F1 (equation (A.1)).

In the initial test a “typical” measurement position is chosen on an initial measurement surface and the normal sound intensity is sampled there M times with a short averaging time (clause 8.2). The indicator is the coefficient of variation of those samples:

F_{1} = \frac{1}{I _{n}} \frac{\sum _{k} ( I _{nk} - I _{n} ) ^{2}}{M - 1}

with $\overline{I_{n}}$ the mean of the M samples (equation (A.2)). It is dimensionless, and it is zero for a perfectly steady field, so it qualifies the stationarity of the field, not the uniformity of the surface (that is F4). Note 9 of Annex A recommends M = 10 samples, with a short averaging time of 8 s to 12 s (or a whole number of cycles) for periodic signals. Table B.3 requires corrective action when $F_{1} > 0.6$ (TEMPORAL_VARIABILITY_LIMIT).

Parameters

Name	Description
`short_time_intensity`	The M short-time-averaged signed normal intensity samples `Ink` at the fixed position, in W/m^2; 1D `(samples,)` for one frequency band, or 2D `(samples, bands)` to evaluate every band at once (Annex A.1 requires F1 in each band used for the sound-power determination).

Returns: F1 as a float for 1D input, or a per-band numpy.ndarray for 2D input.

Raises

Exception	When
ValueError	If fewer than two samples are supplied, the input is not 1D or 2D, or the mean intensity of a band is not positive. A.2.1 states no positivity condition on the M samples, but F1 normalizes by that mean, so a non-positive one leaves the indicator meaningless and the Table B.3 criterion vacuous; this library rejects it rather than return a negative F1.

TEMPORAL_VARIABILITY_LIMIT

Constant (float).

TEMPORAL_VARIABILITY_LIMIT = 0.6

Created and maintained by José Manuel Requena Plens ORCID All projects