Source: https://jmrplens.github.io/phonometry/reference/api/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: ```text P = rho0 c0 S sigma [W] ``` so `sigma` is exactly the radiation factor `epsilon` that [`phonometry.sound_power_from_vibration`](/phonometry/reference/api/power/vibration-sound-power/#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`](/phonometry/reference/api/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: ```text 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): ```text 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. > Auto-generated from the source docstrings by `scripts/generate_api_docs.py` (`make api-docs`). Do not edit by hand. ## coincidence_frequency ```python 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`](/phonometry/reference/api/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 ```python 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`](/phonometry/reference/api/vibration/radiation-efficiency/#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`](/phonometry/reference/api/vibration/radiation-efficiency/#radiationefficiencyresult). **Raises** | Exception | When | | :--- | :--- | | ValueError | for a non-positive input or unknown boundary/baffle. | ## RadiationEfficiencyResult ```python 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() ```python 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.