---
title: "Predicting Panel Sound Insulation"
description: "Theoretical airborne sound insulation of panels from physical properties: the mass law and coincidence dip (Sharp), double-wall mass-spring-mass behaviour, transmission through slits and apertures (Gomperts), plate radiation efficiency (Leppington/Maidanik) and the point mobilities of infinite plates and beams (Cremer)."
references:
  - type: book
    authors: ["Bies, D. A.", "Hansen, C. H.", "Howard, C. Q."]
    year: 2017
    title: "Engineering Noise Control"
    edition: "5th ed."
    publisher: "CRC Press"
    doi: "10.1201/9781351228152"
    note: "Section 7.2: the mass law (Eq. 7.40/7.42), coincidence frequency (Eq. 7.3), Sharp's single-panel method (7.2.4.1) and the double-wall model (7.2.6, Eq. 7.62-7.64). ISBN 978-1-4987-2405-0."
  - type: book
    authors: ["Hopkins, C."]
    year: 2007
    title: "Sound insulation"
    publisher: "Butterworth-Heinemann"
    doi: "10.4324/9780080550473"
    note: "Section 2.9 (plate radiation efficiency, Leppington/Maidanik Eqs 2.227-2.230) and Section 4.3.10 (slits, holes and apertures, Gomperts Eq. 4.99, Wilson & Soroka Eq. 4.102, composite Eq. 4.92). ISBN 978-0-7506-6526-1."
  - type: book
    authors: ["Cremer, L.", "Heckl, M.", "Petersson, B. A. T."]
    year: 2005
    title: "Structure-Borne Sound"
    edition: "3rd ed."
    publisher: "Springer"
    doi: "10.1007/b137728"
    note: "Chapter 5, Table 5.1: the compilation of point impedances and mobilities of infinite structures (plate Z = 8 sqrt(B' m''), beam, rod) and the injected-power relation W = 0.5 |F|^2 Re{Y} (Eq. 5.23). ISBN 978-3-540-22696-3."
---

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

A laboratory test measures the sound reduction index $R$ of a finished element;
this page instead **predicts** $R(f)$ from the physical properties of the
construction, so a partition can be designed before it is built. It covers the
airborne insulation of a single panel (the mass law and the coincidence dip),
the double wall (its mass-spring-mass resonance), the transmission through slits
and apertures that caps any real construction, the radiation efficiency of a
bending plate, and the point mobilities that set the vibrational power a
structure absorbs. The measured counterparts these predictions feed live in
[Predicting Sound Insulation (EN 12354)](/phonometry/guides/insulation-prediction/)
and [Field Insulation Measurement and Ratings](/phonometry/guides/insulation-field/).

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/panel_insulation_concept.svg" alt="Four panels: the single-panel mass law with its coincidence dip, the double wall with the mass-spring-mass resonance and cavity gain, the plate radiation efficiency rising to unity above the critical frequency, and a composite wall whose 1 % open slit caps R at the open-area limit" width="92%" loading="eager" />

## Single panel: the mass law and coincidence (Bies 7.2)

A limp, non-stiff panel transmits sound by being driven bodily by the incident
pressure; the heavier it is and the higher the frequency, the more it resists.
The **normal-incidence mass law** (Bies Eq. 7.40) and its diffuse-field form
(Eq. 7.42) are

$$
TL_0 = 10 \log_{10}\!\left[ 1 + \left(\frac{\pi f\, m''}{\rho_0 c_0}\right)^2 \right],
\qquad TL = TL_0 - \Delta_{\text{band}},
$$

with $m''$ the mass per unit area and $\Delta_{\text{band}} = 5.5$ dB (one-third
octave) or $4.0$ dB (octave). Above the argument's knee the mass law rises
**6 dB per octave and 6 dB per doubling of mass**, the two facts every
insulation calculation rests on.

A real panel is also stiff, so at the **coincidence (critical) frequency**
(Bies Eq. 7.3)

$$
f_c = \frac{c_0^2}{2\pi}\sqrt{\frac{m''}{B'}}
    = \frac{0.55\, c_0^2}{c_L\, h},
$$

the free bending wavelength matches the trace wavelength of the incident sound
and the panel goes transparent: the **coincidence dip**. Sharp's method holds
the field-incidence mass law up to $f_c/2$, drops through a straight line in
$\log f$, and from $f_c$ upward follows (Eq. 7.44)
$TL = 10\lg[1 + (\pi f m''/\rho_0 c_0)^2] + 10\lg(2\eta f/\pi f_c)$ with the loss
factor $\eta$. The dip sits at Bies design-chart point B,
$TL = 20\lg(f_c m'') + 10\lg\eta - 44$.

```python
import numpy as np
from phonometry import (
    coincidence_frequency, plate_bending_stiffness,
    single_panel_transmission_loss,
)

# 6 mm float glass: E = 62 GPa, rho = 2500 kg/m3, nu = 0.24, eta = 0.024.
bands = np.array([100, 125, 160, 200, 250, 315, 400, 500, 630, 800,
                  1000, 1250, 1600, 2000, 2500, 3150], dtype=float)
mass = 2500.0 * 0.006                                 # 15 kg/m2
bp = plate_bending_stiffness(6.2e10, 0.006, 0.24)     # B' [N.m]
fc = coincidence_frequency(mass, bp)
print(round(fc))                                      # 2107 Hz (Hopkins declares ~2079)

res = single_panel_transmission_loss(bands, mass, critical_frequency=fc,
                                     loss_factor=0.024)
print(round(res.rating().rating))                     # 32  ->  Rw = 32 dB (catalogue 6 mm glass)
```

The predicted spectrum plugs straight into the ISO 717-1 rating through
`res.rating()`, and into EN 12354 as the "predicted" element $R$ where the
standard would otherwise demand a laboratory measurement.

## Double wall: the mass-spring-mass resonance (Bies 7.2.6)

Two leaves separated by a cavity behave as a mass-spring-mass system, with the
cavity air (or fill) the spring. Below its **resonance** (Bies Eq. 7.62,
Hopkins Eq. 4.73)

$$
f_0 = \frac{1}{2\pi}\sqrt{\frac{s''\,(m_1 + m_2)}{m_1\, m_2}}
    = 60\sqrt{\frac{m_1 + m_2}{m_1\, m_2\, d}}\quad\text{(empty air gap)},
$$

the pair moves as one and follows the mass law of the *combined* mass; above it
the two mass laws add, boosted by the cavity, until the limiting frequency
$f_l = c_0/(2\pi d)$ where the boost saturates at 6 dB (Eq. 7.64). A porous
fill (any [`PorousMediumResult`](/phonometry/guides/porous-absorbers/)) softens
the cavity and lowers $f_0$.

```python
from phonometry import double_wall_transmission_loss, mass_spring_mass_resonance
from phonometry import mass_law_transmission_loss, miki

# Two 12 kg/m2 leaves, 75 mm air gap.
f0 = mass_spring_mass_resonance(12.0, 12.0, 0.075)
print(round(f0))                                          # 89 Hz

# Below f0 the double wall equals the mass law of the total mass 24 kg/m2:
dw = double_wall_transmission_loss(bands, 12.0, 12.0, 0.075)
print(round(float(dw.transmission_loss[0]), 1),
      round(float(mass_law_transmission_loss(bands[0], 24.0)), 1))   # equal

# A mineral-wool fill (a materials porous model) lowers the resonance:
fill = miki([f0], 7000.0)
print(round(mass_spring_mass_resonance(12.0, 12.0, 0.075, cavity_medium=fill)))  # < 89 Hz
```

## Slits, holes and apertures (Hopkins 4.3.10)

A small air path is the real limit on any heavy construction. The transmission
coefficient of a straight slit (Gomperts, Hopkins Eq. 4.99) and of a circular
hole (Wilson & Soroka, Eq. 4.102) are predicted directly, with the slit's
resonances at $d + 2e = z\lambda/2$ (Eq. 4.101). They combine with the wall in
the area-weighted energy sum (Eq. 4.92)

$$
R = -10\log_{10}\!\left( \frac{1}{\sum_n S_n} \sum_n S_n\, 10^{-R_n/10} \right),
$$

so a bare opening of relative area $S_a/S$ caps the composite at
$10\lg(S/S_a)$; a 1 % opening can never do better than 20 dB, whatever the
wall.

```python
from phonometry import (
    composite_transmission_loss, slit_transmission_coefficient,
    slit_resonance_frequencies,
)

# A 2 mm x 100 mm-deep slit: its transmission peaks at the half-wavelength
# resonances of the slit depth.
print(slit_resonance_frequencies(0.1, 0.002, orders=2).round().tolist())   # [~1500, ~3100]

# A wall of Rw = 50 dB with 1 % of its area left open as a slit is capped:
print(round(float(composite_transmission_loss([0.99, 0.01], [50.0, 0.0])), 1))   # 20.0
```

## Radiation efficiency of a bending plate (Hopkins 2.9)

How much airborne power a vibrating plate actually radiates per unit
mean-square velocity is its **radiation efficiency** $\sigma$, the radiation
factor $\varepsilon$ that
[Sound Power from Surface Vibration](/phonometry/guides/vibration-sound-power/)
(ISO 7849) otherwise takes as a measured input. Below the critical frequency the
plate radiates weakly (edge and corner modes); above it $\sigma \to 1$
(Leppington/Maidanik, Hopkins Eqs 2.227-2.230):

$$
\sigma = \frac{1}{\sqrt{1 - f_c/f}} \quad (f > f_c), \qquad \sigma \to 1 .
$$

```python
from phonometry import radiation_efficiency, sound_power_from_vibration

# The 6 mm glass pane (1.5 x 1.25 m) of the single-panel example above.
sig = radiation_efficiency(bands, 1.5, 1.25, fc)
print(sig.radiation_efficiency[bands == 2000].round(2))    # ~2.5 (peak at coincidence)

# Feed the prediction straight into ISO 7849 as the radiation factor:
lw = sound_power_from_vibration(velocity_level=80.0, area=1.875,
                                radiation_factor=sig.radiation_efficiency,
                                frequencies=bands)
```

## Point mobilities of infinite structures (Cremer Table 5.1)

The vibrational power a point force injects is $W = \tfrac12 |F|^2\,\mathrm{Re}\{Y\}$
(Cremer Eq. 5.23), so the driving-point **mobility** $Y$ (the reciprocal of the
impedance) sets how much energy a structure absorbs. For infinite structures
these are closed forms: an infinite thin plate is a pure resistance
$Z = 8\sqrt{B'\,m''}$ (real, frequency independent), while an infinite beam has
$Y = (1-\mathrm{j})/(4 m' c_B)$ (45 degrees, falling as $\omega^{-1/2}$). They
supply the receiver mobility EN 12354-5 needs when no measurement exists, and
bundle as an ISO 7626 [`MobilityResult`](/phonometry/guides/mechanical-mobility/).

```python
from phonometry import infinite_plate_impedance, infinite_beam_mobility, injected_power

z_plate = infinite_plate_impedance(bp, mass)          # Z = 8 sqrt(B' m'') [N.s/m]
print(round(z_plate))                                 # real, frequency independent
w = injected_power(force=10.0, mobility=1.0 / z_plate)
print(round(float(w) * 1e3, 3), "mW")                 # W = |F|^2 / (16 sqrt(B' m''))
```

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

```python
import numpy as np
import matplotlib.pyplot as plt
from phonometry import (
    coincidence_frequency, composite_transmission_loss,
    double_wall_transmission_loss, mass_law_transmission_loss,
    mass_spring_mass_resonance, plate_bending_stiffness,
    radiation_efficiency, single_panel_transmission_loss,
)

bands = np.array([50, 63, 80, 100, 125, 160, 200, 250, 315, 400, 500, 630, 800,
                  1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000], dtype=float)
fig, ax = plt.subplots(2, 2, figsize=(12, 9))

bp = plate_bending_stiffness(6.2e10, 0.006, 0.24)
fc = coincidence_frequency(15.0, bp)
ml = mass_law_transmission_loss(bands, 15.0, incidence="field")
sp = single_panel_transmission_loss(bands, 15.0, critical_frequency=fc, loss_factor=0.024)
ax[0, 0].semilogx(bands, ml, "--", label="field-incidence mass law")
ax[0, 0].semilogx(bands, sp.transmission_loss, "-o", ms=3, label="single panel R (Sharp)")
ax[0, 0].axvline(fc, ls=":", color="r"); ax[0, 0].set_title("Single panel")

dw = double_wall_transmission_loss(bands, 12.0, 12.0, 0.075)
ax[0, 1].semilogx(bands, mass_law_transmission_loss(bands, 24.0), "--", label="single leaf")
ax[0, 1].semilogx(bands, dw.transmission_loss, "-o", ms=3, label="double wall")
ax[0, 1].axvline(mass_spring_mass_resonance(12.0, 12.0, 0.075), ls=":", color="r")
ax[0, 1].set_title("Double wall")

sig = radiation_efficiency(bands, 1.5, 1.25, fc)
ax[1, 0].loglog(bands, sig.radiation_efficiency, "-o", ms=3, label=r"$\sigma(f)$")
ax[1, 0].axhline(1.0, ls=":"); ax[1, 0].set_title("Radiation efficiency")

wall = sp.transmission_loss
comp = [float(composite_transmission_loss([0.99, 0.01], [w, 0.0])) for w in wall]
ax[1, 1].semilogx(bands, wall, "-o", ms=3, label="solid wall")
ax[1, 1].semilogx(bands, comp, "-s", ms=3, label="wall + 1 % slit")
ax[1, 1].axhline(20.0, ls=":"); ax[1, 1].set_title("Composite with aperture")

for a in ax.flat:
    a.set_xlabel("Frequency [Hz]"); a.legend(fontsize=8); a.grid(alpha=0.3)
fig.suptitle("Theoretical panel sound insulation")
fig.tight_layout(); plt.show()
```

</details>

### Function parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `mass_per_area` | float | kg/m² | > 0 | Surface density $m''$ |
| `critical_frequency` / `bending_stiffness` | float | Hz / N·m | give one | Coincidence $f_c$, or $B'$ to compute it |
| `loss_factor` | float | — | > 0, default `0.01` | Total loss factor $\eta$ |
| `band` | str | — | `'third'` (−5.5 dB) / `'octave'` (−4.0 dB) | Field-incidence correction width |
| `mass1` / `mass2` | float | kg/m² | > 0 | Double-wall leaf surface densities |
| `gap` | float | m | > 0 | Cavity depth $d$ |
| `cavity_medium` | `PorousMediumResult` | — | default `None` | Porous fill; lowers $f_0$ |
| `width` / `depth` (slit) | float | m | > 0 | Slit width $w$ / depth $d$ |
| `field` / `position` (slit) | str | — | `'diffuse'`/`'normal'`, `'mid'`/`'edge'` | Incident field and slit location |
| `radius` / `depth` (hole) | float | m | > 0 | Circular-aperture radius $a$ / depth $d$ |
| `areas` / `reduction_indices` | seq | m² / dB | length $N$ | Composite elements (1-D or (N, bands)) |
| `length_x` / `length_y` | float | m | > 0 | Plate dimensions (radiation efficiency) |
| `boundary` / `baffle` | str | — | `'simply_supported'`/`'clamped'`, `'infinite'`/`'perpendicular'` | Plate edge and baffle constants |

`single_panel_transmission_loss` and `double_wall_transmission_loss` return a
`SoundReductionResult` (`transmission_loss`, `transmission_coefficient`,
`critical_frequency` / `resonance_frequency`, `.rating()`, `.plot()`); the slit
and hole functions return an `ApertureTransmissionResult`
(`transmission_coefficient`, `transmission_loss`, `.plot()`);
`radiation_efficiency` a `RadiationEfficiencyResult` (`radiation_efficiency`,
`radiation_index`, `.plot()`). The predictions carry the modelling caveats of
their sources (Sharp's method above $1.5\times$ the first panel resonance;
Gomperts' inviscid slit valid for $w < 0.3\lambda$; Leppington "method no. 1").

## See also

- [Predicting Sound Insulation (EN 12354)](/phonometry/guides/insulation-prediction/):
  assembles predicted or measured element $R$ into the in-situ $R'_w$.
- [Sound Power from Surface Vibration (ISO 7849)](/phonometry/guides/vibration-sound-power/):
  consumes the predicted radiation efficiency as its radiation factor.
- [Mechanical mobility and the FRF family (ISO 7626-1)](/phonometry/guides/mechanical-mobility/):
  the measured counterpart of the theoretical point mobilities.
- [Porous and multilayer absorbers](/phonometry/guides/porous-absorbers/): the
  cavity-fill models a double wall consumes.
- API reference: [`building.panel_transmission`](/phonometry/reference/api/building/panel-transmission/), [`building.aperture_transmission`](/phonometry/reference/api/building/aperture-transmission/), [`vibration.radiation_efficiency`](/phonometry/reference/api/vibration/radiation-efficiency/) and [`vibration.point_mobility`](/phonometry/reference/api/vibration/point-mobility/).
