vibration.radiation_efficiency

Radiation efficiency of a plate in bending (Hopkins 2007, Sound Insulation, Section 2.9; Leppington et al. 1982; Maidanik 1962).

The radiation efficiency sigma of a vibrating plate relates the airborne sound power it radiates to its mean-square surface velocity:

P = rho0 c0 S <v**2> sigma                                  [W]

so sigma is exactly the radiation factor epsilon that phonometry.sound_power_from_vibration (ISO/TS 7849) otherwise takes as a measured input: this module predicts it from the plate geometry and its coincidence (critical) frequency, closing the ISO 7849 chain without a power measurement, and it supplies the resonant transmission path of the single- and double-leaf sound-reduction-index predictions in phonometry.building.panel_transmission.

Coincidence (critical) frequency (Hopkins Eq. 2.201). Below it the free bending wavelength is shorter than the acoustic wavelength, so the plate radiates weakly; above it the bending wave is supersonic and radiates efficiently:

fc = (c0**2 / (2 pi)) sqrt(m'' / B')

with m'' the mass per unit area (kg/m^2) and B' the bending stiffness per unit width (N.m). This is the closed form fc = 0.55 c0**2 / (cL h) in terms of the plate longitudinal wave speed cL and thickness h.

Frequency-averaged efficiency (Hopkins 2.9.4, “method no. 1”). With mu = sqrt(fc / f) (Eq. 2.228), the perimeter U, area S, the boundary constant C_BC (1 simply supported, 2 clamped) and the baffle-orientation constant C_OB (1 plate flush in an infinite baffle, 2 baffles perpendicular to the edges):

• below fc (Eq. 2.227):

    sigma = U / (2 pi mu k S sqrt(mu**2 - 1))
            * [ ln((mu + 1)/(mu - 1)) + 2 mu/(mu**2 - 1) ]
            * [ C_BC C_OB - mu**-8 (C_BC C_OB - 1) ]

with ``k = 2 pi f / c0`` the acoustic wavenumber;

• above fc (Eq. 2.229): sigma = 1 / sqrt(1 - mu**2) = (1 - fc/f)**-0.5, so sigma -> 1 well above coincidence;
• in the band that contains fc (Eq. 2.230): sigma ~= (0.5 - 0.15 L1/L2) sqrt(k fc L1) with k fc = 2 pi fc / c0, L1 the smaller and L2 the larger plate dimension.

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coincidence_frequency

coincidence_frequency(
    mass_per_area: float,
    bending_stiffness: float,
    *,
    speed_of_sound: float = 343.0,
) -> float

Coincidence (critical) frequency fc of a thin plate (Hopkins 2.201).

fc = (c0**2 / 2 pi) sqrt(m'' / B') (identical to Bies Eq. 7.3).

Parameters

Name	Description
mass_per_area	Mass per unit area m'', in kg/m^2.
bending_stiffness	Bending stiffness per unit width B', in N.m (see phonometry.vibration.point_mobility.plate_bending_stiffness).
speed_of_sound	Speed of sound in air c0 (Default: 343 m/s).

Returns: The coincidence frequency fc, in hertz.

Raises

Exception	When
ValueError	for a non-positive input.

radiation_efficiency

radiation_efficiency(
    frequency: ArrayLike,
    length_x: float,
    length_y: float,
    critical_frequency: float,
    *,
    boundary: str = 'simply_supported',
    baffle: str = 'infinite',
    speed_of_sound: float = 343.0,
) -> RadiationEfficiencyResult

Frequency-averaged radiation efficiency of a plate (Hopkins 2.9.4).

Implements Hopkins “method no. 1” (Eqs 2.227, 2.229, 2.230): the below-, above- and at-coincidence expressions of Leppington/Maidanik. The band whose centre lies closest (on a log scale) to critical_frequency uses the at-coincidence expression (Eq. 2.230); all others use the below/above expressions.

Parameters

Name	Description
frequency	Band centre frequencies f, in hertz (array, > 0).
length_x	Plate dimension Lx, in m (> 0).
length_y	Plate dimension Ly, in m (> 0).
critical_frequency	Coincidence frequency fc, in hertz (> 0); see coincidence_frequency.
boundary	"simply_supported" (C_BC = 1) or "clamped" (C_BC = 2).
baffle	"infinite" (C_OB = 1, plate flush in a rigid baffle) or "perpendicular" (C_OB = 2, baffles perpendicular to the edges).
speed_of_sound	Speed of sound in air c0 (Default: 343 m/s).

Returns: A RadiationEfficiencyResult.

Raises

Exception	When
ValueError	for a non-positive input or unknown boundary/baffle.

RadiationEfficiencyResult

RadiationEfficiencyResult(
    frequencies: np.ndarray,
    radiation_efficiency: np.ndarray,
    critical_frequency: float,
    length_x: float,
    length_y: float,
    boundary: str,
    baffle: str,
)

Frequency-averaged plate radiation efficiency (Hopkins 2.9.4).

Attributes

Name	Description
frequencies	Band centre frequencies, in hertz.
radiation_efficiency	Radiation efficiency sigma per band.
critical_frequency	Coincidence frequency fc, in hertz.
length_x	Plate dimension Lx, in m.
length_y	Plate dimension Ly, in m.
boundary	Boundary condition ("simply_supported" / "clamped").
baffle	Baffle orientation ("infinite" / "perpendicular").

RadiationEfficiencyResult.plot()

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

Plot the radiation efficiency sigma(f) on log-log axes.

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

RadiationEfficiencyResult.radiation_index

property

Radiation index 10 lg(sigma) per band, in dB.