vibration.structural.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, in watts, to its mean-square surface velocity:

$$P={\rho }_0{c}_{0}S⟨v^2⟩\sigma$$

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.prediction.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:

$$f_{\mathrm{c}}=\frac{c_0^2}{2\pi }\sqrt{\frac{m^{\prime \prime }}{B^{\prime }}}\text{\hspace{1em}(Eq. 2.201)}$$

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 $f_{\mathrm{c}}=0.55c_0^2/(c_{\mathrm{L}}h)$ in terms of the plate longitudinal wave speed cL and thickness h.

Frequency-averaged efficiency (Hopkins 2.9.4, “method no. 1”). With $\mu =\sqrt{f_{\mathrm{c}}/f}$ (Eq. 2.228), the perimeter U, area S, the boundary constant $C_{\mathrm{BC}}$ (1 simply supported, 2 clamped) and the baffle-orientation constant $C_{\mathrm{OB}}$ (1 plate flush in an infinite baffle, 2 baffles perpendicular to the edges), the efficiency below fc is

$$\sigma =\frac{U}{2\pi \mu kS\sqrt{{\mu }^2-1}}\left[\ln \frac{\mu +1}{\mu -1}+\frac{2\mu }{{\mu }^2-1}\right]\left[C_{\mathrm{BC}}{C}_{\mathrm{OB}}-{\mu }^{-8}\left(C_{\mathrm{BC}}{C}_{\mathrm{OB}}-1\right)\right]\text{\hspace{1em}(Eq. 2.227)}$$

with $k=2\pi f/c_0$ the acoustic wavenumber. Above fc, $\sigma =1/\sqrt{1-{\mu }^2}=(1-f_{\mathrm{c}}/f{)}^{-0.5}$ (Eq. 2.229), so $\sigma \to 1$ well above coincidence. In the band that contains fc,

$$\sigma \approx \left(0.5-0.15\frac{L_1}{L_2}\right)\sqrt{k_{fc}{L}_1},k_{fc}=\frac{2\pi f_{\mathrm{c}}}{c_0}\text{\hspace{1em}(Eq. 2.230)}$$

with 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).

$f_{\mathrm{c}}=(c_0^2/2\pi )\sqrt{m^{\prime \prime }/B^{\prime }}$ (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.structural.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.

plot_plate_geometry

plot_plate_geometry(
    length_x: float,
    length_y: float,
    ax: Axes | None = None,
    *,
    boundary: str = 'simply_supported',
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the baffled rectangular plate of the radiation model, to scale.

Plate a x b inside its baffle frame, boundary condition in the title.

Parameters

Name	Description
length_x	Plate length a, in metres.
length_y	Plate width b, in metres.
ax	Existing axes, or None to create a figure.
boundary	Boundary-condition label of the model.
language	Label language, "en" (default) or "es".
kwargs	Forwarded to the plate rectangle.

Returns: The axes.

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" ($C_{\mathrm{BC}}=1$) or "clamped" ($C_{\mathrm{BC}}=2$).
baffle	"infinite" ($C_{\mathrm{OB}}=1$, plate flush in a rigid baffle) or "perpendicular" ($C_{\mathrm{OB}}=2$, 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.plot_geometry()

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

Draw the baffled plate to scale.

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

RadiationEfficiencyResult.radiation_index

property

Radiation index $10{\log }_{10}(\sigma )$ per band, in dB.