building.prediction.resilient_layers

Prediction of resilient-layer performance: tapping force, floor coverings, floating floors.

The measurement modules of this domain report what a resilient layer achieved (phonometry.building.measurement.floor_covering_improvement for the ISO 16251-1 mock-up, phonometry.materials.resilient.dynamic_stiffness for the EN 29052-1 dynamic stiffness). This module is their prediction counterpart: it walks the physical chain from the material data to the improvement spectrum, so a covering or a floating floor can be sized before anything is built.

The chain is one story told in three steps.

1. The excitation (Hopkins 3.6.3). The ISO tapping machine drops a 0.5 kg hammer from 40 mm, ten impacts per second, so the impact velocity is $v_{o} = 2 g h = 0.886$ m/s (Eq. 3.85) and, for a short impact, the peak force per Fourier line is $∣ F_{n} ∣ = 2 m v_{o} / T_{i}$ (Eq. 3.90), giving the band mean-square force $F_{r m s}^{2} = 3.9 B$ (Eq. 3.92). Real floors are not that simple: the hammer, the contact stiffness K it deforms and the floor’s driving-point impedance Zdp form a mass-spring-dashpot (Fig. 3.28) whose force pulse (force_pulse, Eqs. 3.95/3.96) is over-critical when $K m \geq 4 Z_{d p}^{2}$ (a single positive pulse, no rebound) and under-critical otherwise (a rebound; only the first positive lobe is transformed). Its spectrum (tapping_force_spectrum) is flat up to the cut-off fco (Eqs. 3.101/3.102) and falls above it, and it asymptotes at low frequency between $∣ F_{n} ∣_{lower} = m v_{o} / T_{i}$ and $∣ F_{n} ∣_{upper} = 2 m v_{o} / T_{i}$ , 6 dB apart in mean square (Eqs. 3.99/3.100).

2. Soft floor coverings (Hopkins 4.4.3.1). A soft covering on a heavyweight floor changes nothing but the force input, so its improvement is the force ratio $Δ L = 20 lo g_{10} (∣ F_{n} ∣_{without} /∣ F_{n} ∣_{with})$ (Eq. 4.114). The covering’s contact stiffness $K = E π r^{2} / d$ (Eq. 3.98) sets its cut-off, against the bare plate’s $K = 2 r E / (1 - ν^{2})$ (Eq. 3.97), which is why a two-line estimate, $Δ L \approx 0$ below fco and 12 dB/octave above it, captures the whole design question (covering_improvement).

3. Floating floors (Hopkins 4.4.4, ISO 12354-2 Annex C, Vigran 8.4). Above the mass-spring resonance $f_{o} = 160 s^{'} / m^{'}$ (Formula C.2) the improvement follows one of three laws (floating_floor_improvement_spectrum): the infinite-plate result of Cremer, $Δ L = 40 lo g_{10} (f / f_{o})$ (Eq. 4.119, Vigran Eq. 8.40), the empirical $Δ L = 30 lo g_{10} (f / f_{o})$ that EN 12354-2 adopted for sand-cement screeds (Formula C.1, Eq. 4.124), and the same 40 lg law with the hammer-impedance term $10 lo g_{10} [1 + (f / f_{l imi t})^{2}]$ that a lightweight walking surface needs (Eq. 4.123, Vigran Eq. 8.48). A floating floor on discrete mounts instead of a continuous layer is a two-subsystem SEA problem (resilient_mount_improvement, Vér’s model as Hopkins Eq. 4.118 and Vigran Eq. 8.45) and rises at 30 dB/decade, not 40. Two floating floors stacked give two resonances (double_floating_floor_resonances, Eq. 4.125), and the weighted single number follows from m' and s' directly (weighted_floating_floor_improvement, Formulae C.4/C.5).

The wall and ceiling linings of ISO 12354-1 Annex D, which used to close this file, are in phonometry.building.prediction.linings: they are added to a wall rather than laid on a floor, and the annex rates them from a resonance frequency instead of from the tapping machine’s force.

Citations are to ISO 12354-2:2017, to Hopkins, Sound Insulation (2007) and to Vigran, Building Acoustics (2008). Where the two books state the same model in different algebra the test suite pins the identity rather than either transcription. One printed defect is relevant here and is recorded in docs/ERRATA.md: the carpet stiffness in the caption of Vigran’s Fig. 8.37.

One relation used here carries no published worked example, so it is implemented as printed and checked only for self-consistency: the asphalt fit of Formula (C.5). The guide “Predicting Resilient-Layer Performance” says which pieces have an oracle and which do not.

Auto-generated from the source docstrings by scripts/generate_api_docs.py (make api-docs). Do not edit by hand.

combined_dynamic_stiffness

combined_dynamic_stiffness(layers: ArrayLike) -> float

Total dynamic stiffness of stacked resilient layers (Formula C.6).

$s_{t o t}^{'} = (\sum 1/ s_{i}^{'})^{- 1}$ , springs in series (Hopkins Eq. 4.121 states the same rule). ISO 12354-2:2017 warns that it holds only if every layer covers the whole floor without cuts for pipes or electrical devices.

Parameters

Name	Description
`layers`	Dynamic stiffnesses per unit area `s'i`, in N/m³ (any 1-D array-like).

Returns: The total dynamic stiffness s'tot, in N/m³.

Raises

Exception	When
ValueError	If `layers` is empty or holds a non-positive value.

covering_contact_stiffness

covering_contact_stiffness(
    youngs_modulus: float,
    thickness: float,
    *,
    radius: float = 0.015,
) -> float

Contact stiffness of a soft floor covering $K = E π r^{2} / d$ (Eq. 3.98).

The covering is treated as a linear spring of area $π r^{2}$ under the hammer, so only the ratio $E / d$ matters. Vigran’s Eq. (8.51) is the same expression written with the hammer area Sh, quoted there as 7 cm² against the 7.07 cm² of a 15 mm radius.

Parameters

Name	Description
`youngs_modulus`	Young’s modulus `E` of the covering, in Pa.
`thickness`	Covering thickness `d`, in m.
`radius`	Contact radius `r`, in m (Default: 0,015).

Returns: The contact stiffness K, in N/m.

Raises

Exception	When
ValueError	If an input is not positive and finite.

covering_improvement

covering_improvement(
    frequencies: ArrayLike,
    covering_stiffness: float,
    plate_stiffness: float,
    impedance: float,
    *,
    mass: float = 0.5,
    impact_rate: float = 10.0,
    band: BandWidth = 'third',
) -> CoveringImprovementResult

Improvement of impact sound insulation by a soft covering (Eq. 4.114).

On a heavyweight base floor a soft covering has a negligible effect on the mass, bending stiffness and total loss factor of the slab, so it alters only the force the hammer injects. The improvement is then the ratio of the two force spectra, $Δ L = 20 lo g_{10} (∣ F_{n} ∣_{without} /∣ F_{n} ∣_{with})$ , computed here from tapping_force_spectrum with the covering’s contact stiffness (Eq. 3.98) and with the plate’s (Eq. 3.97).

The tapping machine excites a line spectrum, at multiples of the 10 Hz impact rate, so Eq. (4.114) is a statement about one Fourier component and the band value is the ratio of the band mean-square forces (Eq. 3.91), that is the sum over the lines that fall in the band. improvement is that band value and line_improvement is the per-line ratio. The distinction matters: the undamped model’s transform has exact nulls at odd multiples of fco, so a band centre that happens to land on one reads tens of dB high. With the 100 Hz cut-off of Hopkins’s covering No. 2, the line ratio at 500 Hz is 66.8 dB against a two-line estimate of 27.9 dB, while the band value is 33.3 dB. Hopkins notes below Fig. 4.64 that the troughs vanish once the covering’s internal damping is included and the spectrum is averaged into bands.

two_line is Hopkins’s design estimate: $Δ L \approx 0$ below the covering’s cut-off and a straight 12 dB/octave above it, that is $40 lo g_{10} (f / f_{co})$ . Real coverings behave as non-linear springs under the tapping machine’s high force and show two or three slopes between 5 and 22 dB/octave, so the model identifies the general features rather than replacing a measurement.

Parameters

Name	Description
`frequencies`	Band centre frequencies `f`, in Hz.
`covering_stiffness`	Contact stiffness `K` of the covering, in N/m (`covering_contact_stiffness`).
`plate_stiffness`	Contact stiffness `K` of the bare plate, in N/m (`plate_contact_stiffness`).
`impedance`	Driving-point impedance `Zdp` of the base floor, in N.s/m; unchanged by the covering.
`mass`	Hammer mass `m`, in kg (Default: 0,5).
`impact_rate`	Impact repetition rate `fi`, in Hz (Default: 10); it sets the spacing of the Fourier lines the bands average over.
`band`	`"third"` or `"octave"`.

Returns: A CoveringImprovementResult.

Raises

Exception	When
ValueError	If an input is not positive and finite, or `band` is unknown.

CoveringImprovementResult

CoveringImprovementResult(
    frequencies: np.ndarray,
    improvement: np.ndarray,
    two_line: np.ndarray,
    cut_off_frequency: float,
    bare_cut_off_frequency: float,
    lines: np.ndarray,
    line_improvement: np.ndarray,
    bare: TappingForceResult,
    covered: TappingForceResult,
)

Predicted improvement ΔL of a soft floor covering (Hopkins 4.4.3.1).

Attributes

Name	Description
`frequencies`	Band centre frequencies `f`, in Hz.
`improvement`	Band improvement `ΔL`, in dB: Eq. (4.114) evaluated over the tapping machine’s Fourier lines and summed in mean square across each band, $10 lo g_{10} (\sum ∣ F_{n} ∣_{without}^{2} / \sum ∣ F_{n} ∣_{with}^{2})$ .
`two_line`	The two-line estimate, in dB: 0 below `fco` and 12 dB/octave (40 dB/decade) above it.
`cut_off_frequency`	Cut-off frequency `fco` of the covered floor, in Hz.
`bare_cut_off_frequency`	Cut-off frequency of the bare plate, in Hz.
`lines`	Fourier line frequencies $n f_{i}$ of the tapping machine, in Hz, covering every band in `frequencies`.
`line_improvement`	The per-line ratio $Δ L = 20 lo g_{10} (∣ F_{n} ∣_{without} / ∣ F_{n} ∣_{with})$ of Eq. (4.114) at `lines`, in dB. It carries the deep troughs at odd multiples of `fco` that Hopkins notes below Fig. 4.64, which are an artefact of the undamped model and disappear from `improvement`.
`bare`	The bare-plate `TappingForceResult`, at `lines`.
`covered`	The `TappingForceResult` with the covering, at `lines`.

CoveringImprovementResult.plot()

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

Plot ΔL(f) from the force ratio beside the two-line estimate.

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

double_floating_floor_resonances

double_floating_floor_resonances(
    lower_stiffness: float,
    lower_mass_per_area: float,
    upper_stiffness: float,
    upper_mass_per_area: float,
) -> tuple[float, float]

The two resonances of a double floating floor (Hopkins Eq. 4.125).

One floating floor on top of another over a heavyweight base is a mass-spring-mass-spring system with

f_{m s m s} = \frac{1}{2 ^{3/2} π} X \pm X^{2} - \frac{4 s _{1} s _{2}}{ρ _{s 1} ρ _{s 2}}, X = \frac{s _{1}}{ρ _{s 1}} + \frac{s _{2}}{ρ _{s 1}} + \frac{s _{2}}{ρ _{s 2}}

subscript 1 being the lower floating floor (on the resilient layer that rests on the base) and 2 the upper one. The double floor avoids the single floor’s dip at fms, but the steep rise in ΔL only starts above the higher of the two resonances. For two identical floors the roots are $f_{m s} (3 \pm 5) /2$ , that is $0.618 f_{m s}$ and $1.618 f_{m s}$ .

Parameters

Name	Description
`lower_stiffness`	Dynamic stiffness per unit area `s1` of the lower resilient layer, in N/m³.
`lower_mass_per_area`	Mass per unit area `ρs1` of the lower floating floor, in kg/m².
`upper_stiffness`	Dynamic stiffness per unit area `s2` of the upper resilient layer, in N/m³.
`upper_mass_per_area`	Mass per unit area `ρs2` of the upper floating floor, in kg/m².

Returns: (lower, upper) resonance frequencies, in Hz.

Raises

Exception	When
ValueError	If an input is not positive and finite.

floating_floor_improvement_spectrum

floating_floor_improvement_spectrum(
    frequencies: ArrayLike,
    *,
    resonance_frequency: float,
    model: FloatingFloorModel = 'en12354',
    limiting_frequency: float | None = None,
    mass_per_area: float | None = None,
    dynamic_stiffness: float | None = None,
) -> FloatingFloorImprovementResult

Improvement ΔL(f) of a floating floor on a heavyweight base floor.

Three laws share the same anchor, the mass-spring resonance fo of the walking surface on the resilient layer (floating_floor_resonance_frequency), and all give $Δ L = 0$ at and below it (in the band containing fo, ΔL is in practice between −5 dB and 0 dB):

"cremer": $Δ L = 40 lo g_{10} (f / f_{o})$ , Cremer’s 1952 result for two infinite, locally reacting plates coupled by a spring layer (Hopkins Eq. 4.119, Vigran Eq. 8.40), i.e. 12 dB per octave. It holds for constructions with high internal damping, such as asphalt screeds, and is the branch ISO 12354-2 Formula (C.3) prescribes for asphalt and dry floating floors.
"en12354" (default): $Δ L = 30 lo g_{10} (f / f_{o})$ , the empirical law of ISO 12354-2 Formula (C.1) for sand-cement and calcium-sulfate screeds (Hopkins Eq. 4.124). Sand-cement screeds have a low internal loss factor and act as finite plates with a reverberant bending field, for which the 40 lg law overestimates ΔL.
"cremer_hammer": $Δ L = 40 lo g_{10} (f / f_{o}) + 10 lo g_{10} [1 + (f / f_{l imi t})^{2}]$ , the 40 lg law with the reduction in power input above the limiting frequency of the hammer’s own impedance (Hopkins Eq. 4.123, Vigran Eq. 8.48). A lightweight walking surface such as chipboard needs it, and tends to 18 dB per octave well above flimit.

The laws are stated as valid above fo, and Cremer’s derivation is reported to hold in $f_{o} < f < 4 f_{o}$ .

Parameters

Name	Description
`frequencies`	Band centre frequencies `f`, in Hz.
`resonance_frequency`	Mass-spring resonance `fo`, in Hz.
`model`	`"en12354"`, `"cremer"` or `"cremer_hammer"`.
`limiting_frequency`	Limiting frequency `flimit`, in Hz; required by `"cremer_hammer"` and ignored otherwise (`hammer_limiting_frequency`).
`mass_per_area`	Optional `m'` of the floating floor, in kg/m².
`dynamic_stiffness`	Optional `s'` of the resilient layer, in N/m³; supplied together with `mass_per_area` it adds `ΔLw` to the result, from Formula (C.4) for `"en12354"` (screeds) and Formula (C.5) for the other two models (asphalt and dry floating floors).

Returns: A FloatingFloorImprovementResult.

Raises

Exception	When
ValueError	If an input is not positive and finite, `model` is unknown, or `"cremer_hammer"` is used without a limiting frequency.

floating_floor_resonance_frequency

floating_floor_resonance_frequency(
    dynamic_stiffness: float,
    mass_per_area: float,
) -> float

Resonance $f_{o} = 160 s^{'} / m^{'}$ of a floating floor (Formula C.2).

ISO 12354-2:2017 Formula (C.2), with s' the EN 29052-1 dynamic stiffness per unit area of the resilient layer measured without pre-load and m' the mass per unit area of the floating floor. The printed constant 160 rounds the exact mass-spring value $1000/ (2 π) = 159.15$ that phonometry.materials.natural_frequency applies, so the two differ by 0.5 %; this function reproduces the standard, whose own Annex G example prints $f_{o} = 52.8$ Hz for $s^{'} = 8$ MN/m³, $m^{'} = 73.5$ kg/m².

Parameters

Name	Description
`dynamic_stiffness`	Dynamic stiffness per unit area `s'`, in N/m³ (i.e. 8e6 for the 8 MN/m³ of the standard’s example).
`mass_per_area`	Mass per unit area `m'` of the floating floor, in kg/m².

Returns: The resonance frequency fo, in Hz.

Raises

Exception	When
ValueError	If an input is not positive and finite.

FloatingFloorImprovementResult

FloatingFloorImprovementResult(
    frequencies: np.ndarray,
    improvement: np.ndarray,
    resonance_frequency: float,
    model: str,
    slope: float,
    limiting_frequency: float | None = None,
    delta_lw: float | None = None,
)

Predicted improvement ΔL(f) of a floating floor.

Attributes

Name	Description
`frequencies`	Band centre frequencies `f`, in Hz.
`improvement`	Improvement `ΔL` per band, in dB (0 at and below `resonance_frequency`).
`resonance_frequency`	Mass-spring resonance `fo`, in Hz.
`model`	`"en12354"`, `"cremer"` or `"cremer_hammer"`.
`slope`	Slope of the law, in dB per decade (30 or 40).
`limiting_frequency`	Limiting frequency `flimit` of the hammer term, in Hz, or `None` when the term is not applied.
`delta_lw`	Weighted improvement `ΔLw`, in dB, or `None` when the floor data were not supplied: Formula (C.4) for the `"en12354"` model, Formula (C.5) for the other two.

FloatingFloorImprovementResult.plot()

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

Plot ΔL(f) with the resonance and the asymptotic slope marked.

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

force_pulse

force_pulse(
    time: ArrayLike,
    contact_stiffness: float,
    impedance: float,
    *,
    mass: float = 0.5,
    impact_velocity: float | None = None,
) -> np.ndarray

Force pulse F1(t) of a single hammer impact (Eqs. 3.95/3.96).

Lindblad’s solution of the mass-spring-dashpot of Hopkins Fig. 3.28, the hammer mass m on the contact stiffness K in series with the floor’s driving-point impedance Zdp. For an over-critical oscillation ( $K m \geq 4 Z_{d p}^{2}$ ) the pulse decays to zero without changing sign (Eq. 3.95); for an under-critical one it is a decaying sinusoid (Eq. 3.96) whose first positive lobe is the impact proper. Hopkins’s rule is stated in terms of the sign of the force rather than of any mechanism: “only the initial force pulse that has zero or positive force values is used to determine the force spectrum, with all subsequent values of F1(t) due to the oscillations set to zero before taking the Fourier transform”, the hammer having rebounded from the plate. That truncation is applied here, so the under-critical pulse is returned as zero beyond its first zero crossing at $t = π / β$ ; it is the same truncation tapping_force_spectrum transforms, so integrating this pulse over $0 \leq t \leq π / β$ reproduces that spectrum.

The over-critical pulse has no such cut and decays for all t; it is evaluated in a form that stays finite over the whole 0,1 s between impacts rather than one that overflows partway through it.

Parameters

Name	Description
`time`	Time `t` since the impact, in s (scalar or array, $\geq 0$ ).
`contact_stiffness`	Contact stiffness `K`, in N/m.
`impedance`	Driving-point impedance `Zdp`, in N.s/m.
`mass`	Hammer mass `m`, in kg (Default: 0,5).
`impact_velocity`	Impact velocity `vo`, in m/s (Default: `hammer_impact_velocity`).

Returns: The force F1(t), in N, with the same shape as time; never negative.

Raises

Exception	When
ValueError	If an input is not positive and finite, or `time` contains a negative value.

hammer_impact_velocity

hammer_impact_velocity(
    drop_height: float = 0.04,
    *,
    gravity: float = 9.81,
) -> float

Hammer velocity at impact $v_{o} = 2 g h$ (Hopkins Eq. 3.85).

The ISO tapping machine’s nominal 40 mm drop gives $v_{o} = 0.886$ m/s.

Parameters

Name	Description
`drop_height`	Drop height `h`, in m (Default: 0,04).
`gravity`	Acceleration of free fall `g`, in m/s² (Default: 9,81).

Returns: The impact velocity vo, in m/s.

Raises

Exception	When
ValueError	If an input is not positive and finite.

hammer_limiting_frequency

hammer_limiting_frequency(impedance: float, *, mass: float = 0.5) -> float

Limiting frequency $f_{l imi t} = Z_{d p} / (2 π m)$ (Hopkins Eq. 3.106).

The frequency at which the floor’s driving-point impedance equals the magnitude of the hammer’s own mass impedance $∣ Z_{h} ∣ = ω m$ ; above it the hammer mass, not the floor, limits the injected power, and the power input stops rising at 3 dB per doubling of frequency. Vigran’s Eq. (8.48) writes the same frequency as $f_{z} = 4 m_{1} B_{1} / (π m_{h})$ .

Parameters

Name	Description
`impedance`	Driving-point impedance `Zdp`, in N.s/m.
`mass`	Hammer mass `m`, in kg (Default: 0,5).

Returns: The limiting frequency flimit, in Hz.

Raises

Exception	When
ValueError	If an input is not positive and finite.

plate_contact_stiffness

plate_contact_stiffness(
    youngs_modulus: float,
    *,
    poisson_ratio: float = 0.2,
    radius: float = 0.015,
) -> float

Contact stiffness of a plate material $K = 2 r E / (1 - ν^{2})$ (Eq. 3.97).

The stiffness the hammer deforms when it lands on the bare walking surface (Timoshenko and Goodier’s Hertzian contact for a flat circular punch), as opposed to covering_contact_stiffness for a soft covering laid on top of it.

Parameters

Name	Description
`youngs_modulus`	Young’s modulus `E` of the plate, in Pa.
`poisson_ratio`	Poisson’s ratio `ν` of the plate (Default: 0,2, the value Hopkins Table A2 estimates for concrete and masonry).
`radius`	Contact radius `r`, in m (Default: 0,015).

Returns: The contact stiffness K, in N/m.

Raises

Exception	When
ValueError	If an input is not positive and finite, or $∣ ν ∣ \geq 1$ .

resilient_mount_improvement

resilient_mount_improvement(
    frequencies: ArrayLike,
    *,
    impedance: float,
    mass_per_area: float,
    loss_factor: ArrayLike,
    mount_stiffness: float,
    mount_density: float,
) -> np.ndarray

Improvement of a floating floor on discrete resilient mounts (Vér).

Vér’s two-subsystem SEA model of a walking surface carrying a reverberant bending-wave field, connected to a heavyweight base floor by N mounts per unit area of stiffness k each, with all transmission through the mounts and none through the cavity. Hopkins Eq. (4.118) writes it as

Δ L \approx 10 lo g_{10} (\frac{2.3 ρ _{s 1}^{2} c _{L 1} h _{1} η _{1} S _{1} ω ^{3}}{N k ^{2}})

where k is the dynamic stiffness of each mount, N the number of mounts and S1 the area of the walking surface. Since $2.3 ρ_{s 1}^{2} c_{L 1} h_{1} = Z_{d p 1} ρ_{s 1}$ for $Z_{d p 1} = 2.3 ρ c_{L} h^{2}$ (Eq. 2.190), the same expression reads $10 lo g_{10} (Z_{d p 1} ρ_{s 1} η_{1} ω^{3} / (N / S_{1} \cdot k^{2}))$ , which is the form evaluated here: this function takes the mount density N/S1, not the count.

Vigran’s Eq. (8.45) is a sum of three terms, $Z_{1} / Z_{2}$ , $m_{1} η_{1} / (m_{2} η_{2})$ and $Z_{1} η_{1} N f^{3} / (2 π m_{1} f_{o}^{4})$ with $f_{o} = N k / m_{1} / (2 π)$ . Only the third of them is the model implemented here, and that term is algebraically identical to Hopkins Eq. (4.118); the first two are the low-frequency floor, negligible once the third dominates, which is the regime Vigran states the 9 dB per octave slope for. The dominant-term form used here therefore rises at 30 dB per decade (9 dB per octave), against the 40 dB per decade of a continuous resilient layer: fewer mounts, a thicker walking surface or more internal damping all raise ΔL.

Vigran’s simplified Eq. (8.46) inserts Z1 into that third term and prints the coefficient as $2/ (3 π) = 0.3676$ , which is the same number as the $2.3094/ (2 π) = 0.3676$ the substitution gives; the two forms agree.

Parameters

Name	Description
`frequencies`	Band centre frequencies `f`, in Hz.
`impedance`	Driving-point impedance `Zdp1` of the walking surface, in N.s/m.
`mass_per_area`	Mass per unit area `ρs1` of the walking surface, in kg/m².
`loss_factor`	Total loss factor `η1` of the walking surface (scalar or per band).
`mount_stiffness`	Dynamic stiffness `k` of one mount, in N/m.
`mount_density`	Number of mounts per unit area `N/S1`, in 1/m² (Vigran’s `N`, which is already a density).

Returns: The improvement ΔL per band, in dB, and 0 dB at and below fo.

Raises

Exception	When
ValueError	If an input is not positive and finite.

short_pulse_mean_square_force

short_pulse_mean_square_force(
    frequencies: ArrayLike,
    *,
    band: BandWidth = 'third',
) -> np.ndarray

Band mean-square force of a short impact $F_{r m s}^{2} = 3.9 B$ (Eq. 3.92).

The limiting case in which the impact is short enough that the hammer’s momentum alone sets the force: combining $∣ F_{n} ∣ = 2 m v_{o} / T_{i}$ (Eq. 3.90) with $F_{r m s}^{2} = ∣ F_{n} ∣^{2} B / (2 f_{i})$ (Eq. 3.91) gives 3.925 B, printed as 3.9 B. Hopkins finds it adequate for bare concrete slabs of at least 100 mm.

Parameters

Name	Description
`frequencies`	Band centre frequencies `f`, in Hz.
`band`	`"third"` ( $B = 0.23 f$ ) or `"octave"` ( $B = 0.707 f$ ).

Returns: The band mean-square force F²rms, in N².

Raises

Exception	When
ValueError	If an input is not positive and finite, or `band` is unknown.

tapping_cut_off_frequency

tapping_cut_off_frequency(
    contact_stiffness: float,
    impedance: float,
    *,
    mass: float = 0.5,
) -> float

Cut-off frequency fco of the force spectrum (Eqs. 3.101/3.102).

Above fco the tapping machine’s force spectrum is no longer flat and the force falls away. For an under-critical oscillation ( $K m < 4 Z_{d p}^{2}$ , the case of a concrete slab with or without a soft covering) it is the undamped mass-spring value $f_{co} = K / m / (2 π)$ (Eq. 3.102); for an over-critical one (a lightweight walking surface) it is the lower root $[K / (2 Z_{d p}) - (K / (2 Z_{d p}))^{2} - K / m] / (2 π)$ (Eq. 3.101).

Parameters

Name	Description
`contact_stiffness`	Contact stiffness `K`, in N/m (see `plate_contact_stiffness` / `covering_contact_stiffness`).
`impedance`	Driving-point impedance `Zdp` of the floor, in N.s/m (for a homogeneous plate, `phonometry.vibration.infinite_plate_impedance`).
`mass`	Hammer mass `m`, in kg (Default: 0,5).

Returns: The cut-off frequency fco, in Hz.

Raises

Exception	When
ValueError	If an input is not positive and finite.

TAPPING_DROP_HEIGHT

Constant (float).

TAPPING_DROP_HEIGHT = 0.04

tapping_force_spectrum

tapping_force_spectrum(
    frequencies: ArrayLike,
    contact_stiffness: float,
    impedance: float,
    *,
    mass: float = 0.5,
    impact_rate: float = 10.0,
    impact_velocity: float | None = None,
    band: BandWidth = 'third',
) -> TappingForceResult

Force spectrum of the ISO tapping machine on a floor (Hopkins 3.6.3.1).

The Fourier transform of the single-impact force pulse (force_pulse), scaled by the impact repetition rate. Writing $a = K / (2 Z_{d p})$ and $ω_{o}^{2} = K / m$ , the transform of Eqs. (3.95)/(3.96) is the same rational function in both critical cases, $\hat{F} (ω) = v_{o} K / (ω_{o}^{2} - ω^{2} + 2 iaω)$ , multiplied for the under-critical case by $1 + e^{- aπ / β} e^{- iω π / β}$ because only the first positive lobe (of duration $π / β$ ) is transformed. That truncation is what produces the deep troughs at $n f_{co}$ , $n = 3, 5, 7$ that Hopkins notes below Fig. 4.64; they vanish once the covering’s internal damping is included and the spectrum is averaged into bands.

The transform is normalised so that the low-frequency asymptote is $m v_{o} / T_{i}$ for an over-critical impact (no rebound) and $2 m v_{o} / T_{i}$ for a lightly damped under-critical one (full rebound), the two limits of Eqs. (3.99)/(3.100).

Parameters

Name	Description
`frequencies`	Band centre frequencies `f`, in Hz.
`contact_stiffness`	Contact stiffness `K`, in N/m.
`impedance`	Driving-point impedance `Zdp` of the floor, in N.s/m.
`mass`	Hammer mass `m`, in kg (Default: 0,5).
`impact_rate`	Impact repetition rate `fi`, in Hz (Default: 10).
`impact_velocity`	Impact velocity `vo`, in m/s (Default: `hammer_impact_velocity`).
`band`	`"third"` or `"octave"`, the band width of Eq. (3.91).

Returns: A TappingForceResult.

Raises

Exception	When
ValueError	If an input is not positive and finite, or `band` is unknown.

TAPPING_HAMMER_MASS

Constant (float).

TAPPING_HAMMER_MASS = 0.5

TAPPING_HAMMER_RADIUS

Constant (float).

TAPPING_HAMMER_RADIUS = 0.015

TAPPING_IMPACT_RATE

Constant (float).

TAPPING_IMPACT_RATE = 10.0

TappingForceResult

TappingForceResult(
    frequencies: np.ndarray,
    peak_force: np.ndarray,
    mean_square_force: np.ndarray,
    power_input: np.ndarray,
    cut_off_frequency: float,
    limiting_frequency: float,
    over_critical: bool,
    contact_stiffness: float,
    impedance: float,
    lower_limit: float,
    upper_limit: float,
    band: str = 'third',
)

Force spectrum of the ISO tapping machine on one walking surface.

Attributes

Name	Description
`frequencies`	Band centre frequencies `f`, in Hz.
`peak_force`	Magnitude of the Fourier force component `\|Fn\|`, in N (Hopkins Fig. 3.32).
`mean_square_force`	Band mean-square force `F²rms`, in N² (Eq. 3.91).
`power_input`	Power injected into the floor $W_{in} = F_{r m s}^{2} / Z_{d p}$ , in W (Eq. 3.103).
`cut_off_frequency`	Cut-off frequency `fco`, in Hz (Eqs. 3.101/3.102).
`limiting_frequency`	Limiting frequency `flimit`, in Hz (Eq. 3.106).
`over_critical`	`True` when $K m \geq 4 Z_{d p}^{2}$ , i.e. the hammer does not rebound.
`contact_stiffness`	Contact stiffness `K` used, in N/m.
`impedance`	Driving-point impedance `Zdp` used, in N.s/m.
`lower_limit`	Low-frequency asymptote $∣ F_{n} ∣_{lower} = m v_{o} / T_{i}$ , in N (Eq. 3.99).
`upper_limit`	Low-frequency asymptote $∣ F_{n} ∣_{upper} = 2 m v_{o} / T_{i}$ , in N (Eq. 3.100); 6 dB above `lower_limit` in mean square.
`band`	Band width used for `mean_square_force`.

TappingForceResult.plot()

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

Plot the force spectrum |Fn| with its asymptotes and fco.

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

TappingForceResult.power_input_level

property

Power input level $10 lo g_{10} (W_{in} /1 pW)$ , in dB (Hopkins Fig. 3.33).

weighted_floating_floor_improvement

weighted_floating_floor_improvement(
    mass_per_area: float,
    dynamic_stiffness: float,
    *,
    floor: FloorType = 'screed',
) -> float

Weighted improvement ΔLw of a floating floor (Formulae C.4/C.5).

The single number that feeds the simplified prediction (phonometry.predicted_impact_insulation), read directly from the floating floor’s mass per unit area and the resilient layer’s dynamic stiffness. ISO 12354-2:2017 gives it as the two nomograms of Figures C.1 and C.2 and prints the fits:

floor="screed" (sand-cement or calcium-sulfate screeds, Formula C.4): $Δ L_{w} = 13 lo g_{10} (m^{'}) - 14.2 lo g_{10} (s^{'}) + 20.8$ ;
floor="asphalt" (asphalt or dry floating floors, Formula C.5): $Δ L_{w} = (- 0.21 m^{'} - 5.45) lo g_{10} (s^{'}) + 0.46 m^{'} + 23.8$ .

Parameters

Name	Description
`mass_per_area`	Mass per unit area `m'` of the floating floor, in kg/m².
`dynamic_stiffness`	Dynamic stiffness per unit area `s'`, in N/m³.
`floor`	`"screed"` (Formula C.4) or `"asphalt"` (Formula C.5).

Returns: The weighted improvement ΔLw, in dB.

Raises

Exception	When
ValueError	If an input is not positive and finite, or `floor` is unknown.

Created and maintained by José Manuel Requena Plens ORCID All projects