---
title: "Structure-borne sound power of equipment (EN 15657)"
description: "The EN 15657:2018 reception-plate method for building service equipment: the plate-injected power L_Ws = 10 lg(2πfηmS) + L_v − 60 dB (Formula 14), and the plate-independent source quantities: equivalent blocked force (Formula 15), characteristic reception-plate power level (Formula 17), free velocity and source mobility (Formulae 18/19)."
references:
  - type: book
    authors: ["Cremer, L.", "Heckl, M.", "Petersson, B. A. T."]
    year: 2005
    title: "Structure-borne sound: Structural vibrations and sound radiation at audio frequencies"
    edition: "3rd ed."
    publisher: "Springer"
    doi: "10.1007/b137728"
    note: "The plate power balance and the source-receiver mobility framework behind the reception-plate method and its Formula 15-19 source quantities. ISBN 978-3-540-22696-3."
  - type: standard
    organization: "European Committee for Standardization"
    year: 2018
    title: "Acoustic properties of building elements and of buildings — Laboratory measurement of structure-borne sound from building service equipment for all installation conditions"
    designation: "EN 15657:2018"
    url: "https://www.une.org/encuentra-tu-norma/busca-tu-norma/norma?c=N0059781"
    note: "The reception-plate method (clause 7): the spatial mean velocity level (Formula 12), the plate loss factor η = 2.2/(f·Ts) (Formula 13), the plate-injected power level L_Ws (Formula 14, plate velocity levels referred to v₀ = 10⁻⁹ m/s) and the source-quantity chain: equivalent blocked force (Formula 15), characteristic reception-plate power level (Formula 17, Y_R,∞,low = 5·10⁻⁶ m/(N·s)), equivalent free velocity (Formula 18) and source mobility (Formula 19). Conformance is anchored on the resonant-plate power balance P = ω·η·(m·S)·⟨v²⟩ (of which Formula 14 is the level), the loss-factor identity, and the EN 12354-5 Annex I.3 Table I.8 conversion of the flushing-cistern source. The linked catalogue record is the UNE page for UNE-EN 15657:2018."
  - type: standard
    organization: "International Organization for Standardization"
    year: 1996
    title: "Acoustics — Characterization of sources of structure-borne sound with respect to sound radiation from connected structures — Measurement of velocity at the contact points of machinery when resiliently mounted"
    designation: "ISO 9611:1996"
    url: "https://www.iso.org/standard/17424.html"
    note: "The free-velocity source characterization that complements the reception-plate quantities: equation (9), reference velocity v₀ = 5·10⁻⁸ m/s."
---

import ThemeImage from '../../../components/ThemeImage.astro';
import ReportPreview from '../../../components/ReportPreview.astro';

Building service equipment (pumps, fans, boilers, sanitary appliances)
injects **structure-borne sound power** into the building structure it is fixed
to, which then re-radiates as airborne noise in adjoining rooms. **EN 15657:2018**
measures it with the **reception-plate method**: the source is mounted on a
plate of known mass per unit area `m` and area `S` whose structural loss factor
`η` is known, and the plate's spatial-average vibratory velocity is measured.
Formula (14) gives the power *injected into that particular plate*; the
plate-independent source quantities (the equivalent blocked force,
Formula 15; the characteristic reception-plate power level `L_Wsn`,
Formula 17; and the equivalent free velocity and source mobility,
Formulae 18/19) are derived from it and are what the EN 12354-5
installed-equipment prediction consumes.

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/structure_borne_power.svg" alt="Reception-plate structure-borne sound power level per one-third-octave band determined on a low-mobility and a high-mobility reception plate" width="82%" loading="eager" />

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import building

# The same pump-like source measured on a heavy and on a light reception plate.
bands = np.array([50.0, 100.0, 200.0, 400.0, 800.0, 1600.0, 3150.0])
lv_low = np.array([88.0, 90.0, 87.0, 84.0, 80.0, 76.0, 71.0])

low = building.reception_plate_power(lv_low, bands, mass_per_area=600.0,
                                     area=2.0, reverberation_time=0.8)
high = building.reception_plate_power(lv_low + 6.0, bands, mass_per_area=150.0,
                                      area=2.0, reverberation_time=0.5)

x = np.arange(bands.size)
fig, ax = plt.subplots()
ax.bar(x - 0.2, low.power_level, width=0.4, label="low-mobility plate")
ax.bar(x + 0.2, high.power_level, width=0.4, label="high-mobility plate")
ax.set_xticks(x, [f"{b:g}" for b in bands])
ax.set(xlabel="Frequency [Hz]",
       ylabel="Structure-borne power level $L_{Ws}$ [dB re 1 pW]")
ax.legend()
plt.show()
```

</details>

## 1. The reception-plate relations

Why a plate at all? Characterising the source by its contact forces directly
would mean instrumenting every fixing point in up to six components each
(three forces, three moments), on a machine that must keep running normally.
The reception plate sidesteps the whole contact problem: let the source run
on a resonant plate whose dissipation is known, wait for the steady state,
and then the power the plate dissipates equals the power the source injects,
over all contacts and components at once. One spatial average of the plate
velocity replaces the entire force-measurement problem.

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_reception_plate.svg" alt="EN 15657 reception plate: the source machine standing on a resiliently supported plate, accelerometers averaging the plate velocity, and the plate power balance converting the velocity level into the injected structure-borne power level" width="92%" />

The power a resonant plate dissipates is `P = ω·η·(m·S)·⟨v²⟩`, so the injected
power level in one-third-octave bands is (Formula 14)

$$
L_{Ws} = 10\lg\!\left(\frac{2\pi f\,\eta\,m\,S}{f_0\,m_0\,S_0}\right)
         + L_v - 60 \;\;[\mathrm{dB\ re\ 1\ pW}],
$$

with references `f₀ = 1 Hz`, `m₀ = 1 kg`, `S₀ = 1 m²`; the `−60 dB` term is
`10 lg(v₀²/P₀)` for the EN 15657 velocity reference `v₀ = 10⁻⁹ m/s`. The plate
velocity is the energetic spatial average over the `N` positions (Formula 12)
and the loss factor comes from the structural reverberation time `Ts` (Formula
13, identical to the ISO 10848 total loss factor):

$$
L_v = 10\lg\!\Big(\tfrac{1}{N}\textstyle\sum 10^{L_{v,i}/10}\Big), \qquad
\eta = \frac{2.2}{f\,T_s}.
$$

```python
import numpy as np
from phonometry import building

bands = np.array([100.0, 200.0, 400.0, 800.0])
lv_i = np.array([88.0, 90.0, 87.0, 89.0, 86.0, 90.0])   # six plate positions @ 200 Hz
print(round(building.spatial_mean_velocity_level(lv_i), 2))    # 88.6 dB re 1 nm/s

# Power level injected into the reception plate (loss factor from Ts):
res = building.reception_plate_power(
    velocity_level=np.array([90.0, 87.0, 82.0, 77.0]),
    frequency=bands, mass_per_area=600.0, area=2.0, reverberation_time=0.8,
)
print(np.round(res.power_level, 1))     # per-band L_Ws
print(round(res.total_level, 1))        # band-summed level [dB re 1 pW]
```

## 2. Low- and high-mobility plates

Two reception plates bracket the installation conditions. On the *low-mobility*
(heavy) plate the source's own dynamics barely change the plate's point mobility
or loss factor; the *high-mobility* (light) plate is dynamically loaded by the
source, so its reverberation time and mobility are measured with the source
attached. The plate-injected power plus the plate's point mobility (see
[mechanical mobility](/phonometry/guides/mechanical-mobility/)) yield the
source description for the EN 12354-5 model through the conversion chain below.

## 3. From plate power to source quantities (Formulae 15–19)

The plate-injected `L_Ws` is **not** a source descriptor: the same source
injects a different power into a different receiver. EN 15657 derives the
plate-independent quantities: the **equivalent blocked force level**
(Formula 15, re `F₀ = 10⁻⁶ N`) from the low-mobility plate,

$$
L_{Fb,eq} = L_{Ws,\mathrm{low}} - 10\lg\frac{\mathrm{Re}\{Y_{R,\mathrm{low,eq}}\}}{Y_0},
$$

the **characteristic reception-plate power level** that EN 12354-5 consumes
(Formula 17), referred to the standard 10 cm concrete plate of characteristic
mobility `Y_R,∞,low = 5·10⁻⁶ m/(N·s)` (clause 7.2.4),

$$
L_{Wsn} = L_{Fb,eq} + 10\lg\frac{Y_{R,\infty,\mathrm{low}}}{Y_0},
$$

and, from the high-mobility plate, the **equivalent free velocity level**
(Formula 18, re `10⁻⁹ m/s`) and the **source mobility** `|Y_S,eq|`
(Formula 19). The EN 12354-5 Annex I mobility correction
(`installed_power_from_reception_plate`, see
[installed structure-borne sound](/phonometry/guides/installed-structure-borne/))
then refers `L_Wsn` to the actual receiving element.

```python
from phonometry import building

# EN 12354-5 Annex I.3 (flushing cistern, wall contact, 63 Hz): measured on a
# plate of Y = 5.34e-6 m/(N·s); the wall's characteristic mobility is 24.1e-6.
lfb = building.equivalent_blocked_force_level(61.7, 5.34e-6)      # Formula (15)
lwsn = building.characteristic_reception_plate_power(lfb)         # Formula (17)
inst = building.installed_power_from_reception_plate(lwsn, 24.1e-6)  # Annex I
print(round(float(lwsn), 1), round(float(inst), 1))         # 61.4 68.2  (Table I.8)

# Free velocity (Formula 18) + blocked force close the source mobility (19):
lvf = building.equivalent_free_velocity_level(70.0, 1.0e-2)
print(float(building.source_mobility_from_levels(lvf, lfb)))      # |Y_S,eq| in m/(N·s)
```

The direct source-side counterpart is the ISO 9611 free velocity level (re
`v₀ = 5·10⁻⁸ m/s`) measured at the contact points of resiliently mounted
machinery; its equation (9) position average is `mean_free_velocity_level()`.

## 4. The characterization report (`.report()`)

A characterization ends as a *document*. The `StructureBornePowerResult`
exposes a `.report()` method that writes a one-page PDF fiche laid out like a
sound-power test sheet: the standard-basis line naming the EN 15657:2018
reception-plate method (Formula 14), an optional metadata header (client,
source equipment, test environment, instrumentation, climate, date), a per-band
table (nominal octave/one-third-octave frequency, the spatial mean plate
velocity level $L_v$ and the injected structure-borne sound power level
$L_{Ws}$), the $L_{Ws}(f)$ spectrum with a nominal band axis, and a boxed
band-summed total $L_{Ws}$ (dB re 1 pW) with the plate mass per area $m$ and
area $S$.

The metadata is supplied through a `ReportMetadata`, whose applicable fields
here are the **source equipment** (`specimen`), the **test environment**
(`test_room`), the **client**, the **instrumentation**, the **temperature**,
**relative humidity** and **ambient pressure**, the **date of test**
(`test_date`) and the footer identity (`laboratory`, `operator`, `report_id`,
`notes`); the plate mass and area come from the result itself. Supplying
`requirement` adds a PASS/FAIL verdict against a declared upper limit on the
total $L_{Ws}$ (a power emission is a quantity where less is better, so the
source passes at or below the limit). `verbose=True` adds the plate loss factor
$\eta$ column to the table. `language="es"` renders the Spanish fiche with comma
decimals. The basis strip states Formula 14 and reminds that the plate-injected
level must be converted to the plate-independent source quantities (Formulae
15/17) before it feeds EN 12354-5.

```python
import numpy as np
from phonometry import ReportMetadata, reception_plate_power

freqs = np.array([125, 250, 500, 1000, 2000, 4000], float)
lv = np.array([88.0, 90, 86, 82, 78, 73])   # spatial mean plate velocity level [dB]
res = reception_plate_power(
    lv, freqs, mass_per_area=25.0, area=1.2, reverberation_time=0.3,
)

res.report(
    "structure_borne_power.pdf",
    metadata=ReportMetadata(
        client="Example building services contractor",
        specimen="Circulation pump (wall-mounted)",
        test_room="Reception-plate test rig (heavy concrete plate)",
        laboratory="Phonometry reference example",
        report_id="EXAMPLE-15657",
    ),
)   # total L_Ws ~ 65 dB re 1 pW
```

The example fiche is regenerated with `make reports` and kept rendered in the
repository; click the preview to open the PDF.

<ReportPreview
	name="en15657_structure_borne_power_example"
	title="EN 15657 structure-borne sound power example report (PDF)"
	description="One-page EN 15657:2018 reception-plate structure-borne sound power fiche: a header with the client, the source equipment, the reception-plate test rig and the accelerometer and climate, the octave-band table (125 Hz to 4 kHz) of spatial mean plate velocity levels Lv and injected structure-borne sound power levels L_Ws, the L_Ws(f) spectrum with a nominal band axis, and the boxed band-summed total L_Ws (dB re 1 pW) with the plate mass per area m = 25 kg/m2 and area S = 1.20 m2, closed by a basis strip stating the Formula 14 relation and the conversion to the plate-independent source quantities required before EN 12354-5."
	caption="Structure-borne sound power fiche (StructureBornePowerResult.report), an EN 15657 reception-plate characterization with the boxed total L_Ws."
/>

## See also

- API reference: [`building.structure_borne_power`](/phonometry/reference/api/building/structure-borne-power/).
