Source: https://jmrplens.github.io/phonometry/guides/panel-sound-insulation/ # Predicting Panel Sound Insulation (mass law, coincidence, double walls) 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)](https://jmrplens.github.io/phonometry/guides/insulation-prediction/) and [Field Insulation Measurement and Ratings](https://jmrplens.github.io/phonometry/guides/insulation-field/). ## 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 **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). The mass law rises **6 dB per octave and 6 dB per doubling of mass**. 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 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 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) res.plot() # predicted R(f) with the critical frequency marked (needs matplotlib) ``` The predicted spectrum plugs straight into the ISO 717-1 rating through `res.rating()`, and into EN 12354 as the "predicted" element $R$. *The predicted Sharp spectrum rated exactly like a measurement: the coincidence dip at `fc ~ 2.1 kHz` collects most of the unfavourable deviations, and the shifted reference read at 500 Hz gives the catalogue `Rw = 32 dB` of 6 mm glass.*

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```python import matplotlib.pyplot as plt 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 bp = plate_bending_stiffness(6.2e10, 0.006, 0.24) fc = coincidence_frequency(mass, bp) res = single_panel_transmission_loss(bands, mass, critical_frequency=fc, loss_factor=0.024) w = res.rating() # One line each — the predicted R(f), or the rated curve vs the reference: res.plot() w.plot() plt.show() # By hand, combining both on one axes: fig, ax = plt.subplots() ax.semilogx(bands, res.transmission_loss, "o-", label="predicted R (Sharp)") ax.semilogx(w.band_centers, w.shifted_reference, "s--", label="shifted reference") ax.fill_between(w.band_centers, w.measured, w.shifted_reference, where=w.measured < w.shifted_reference, interpolate=True, alpha=0.3, label="unfavourable deviations") ax.axvline(fc, ls=":", color="tab:green", label=f"fc = {fc:.0f} Hz") ax.set_xlabel("Frequency [Hz]") ax.set_ylabel("Sound reduction index R [dB]") ax.set_title(f"Rw = {w.rating} dB (C={w.c:+d}; Ctr={w.ctr:+d})") ax.legend() plt.show() ```

## Double wall: the mass-spring-mass resonance (Bies 7.2.6) Two leaves separated by a cavity behave as a mass-spring-mass system. 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 follows the mass law of the *combined* mass; above it the two mass laws add, boosted by the cavity, until $f_l = c_0/(2\pi d)$ where the boost saturates at 6 dB (Eq. 7.64). A porous fill 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 dw.plot() # double-wall R(f) with the mass-spring-mass resonance marked (needs matplotlib) ``` ## 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: transmission peaks at the depth's half-wavelength # resonances. 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 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 radiates per unit mean-square velocity is its **radiation efficiency** $\sigma$, the radiation factor $\varepsilon$ that [Sound Power from Surface Vibration](https://jmrplens.github.io/phonometry/reference/api/power/vibration-sound-power/) (ISO 7849) otherwise takes as a measured input. Below the critical frequency the plate radiates weakly; above it $\sigma \to 1$ (Leppington/Maidanik, Eqs 2.227-2.230): $\sigma = (1 - f_c/f)^{-1/2}$ for $f > f_c$. ```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) sig.plot() # sigma(f) with the coincidence peak (needs matplotlib) ``` ## 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. An infinite thin plate is a pure resistance $Z = 8\sqrt{B'\,m''}$ (real, frequency independent); 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. ```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'')) ```

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```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() ```

## References - Bies, D. A., Hansen, C. H., & Howard, C. Q. (2017). *Engineering Noise Control* (5th ed.). CRC Press. ISBN 978-1-4987-2405-0. [doi:10.1201/9781351228152](https://doi.org/10.1201/9781351228152). Section 7.2: the mass law, coincidence frequency, Sharp's single-panel method and the double-wall model. - Hopkins, C. (2007). *Sound insulation*. Butterworth-Heinemann. ISBN 978-0-7506-6526-1. [doi:10.4324/9780080550473](https://doi.org/10.4324/9780080550473). Section 2.9 (plate radiation efficiency) and Section 4.3.10 (slits, holes and apertures, composite transmission). - Cremer, L., Heckl, M., & Petersson, B. A. T. (2005). *Structure-Borne Sound* (3rd ed.). Springer. ISBN 978-3-540-22696-3. [doi:10.1007/b137728](https://doi.org/10.1007/b137728). Chapter 5, Table 5.1: the point impedances and mobilities of infinite structures and the injected-power relation. ## See also - [Predicting Sound Insulation (EN 12354)](https://jmrplens.github.io/phonometry/guides/insulation-prediction/): assembles predicted or measured element $R$ into the in-situ $R'_w$. - [Sound Power from Surface Vibration (ISO 7849)](https://jmrplens.github.io/phonometry/reference/api/power/vibration-sound-power/): consumes the predicted radiation efficiency as its radiation factor. - [Mechanical mobility and the FRF family (ISO 7626-1)](https://jmrplens.github.io/phonometry/reference/api/vibration/mechanical-mobility/): the measured counterpart of the theoretical point mobilities. - [Porous and multilayer absorbers](https://jmrplens.github.io/phonometry/guides/porous-absorbers/): the cavity-fill models a double wall consumes. - API reference: [`building.panel_transmission`](https://jmrplens.github.io/phonometry/reference/api/building/panel-transmission/), [`building.aperture_transmission`](https://jmrplens.github.io/phonometry/reference/api/building/aperture-transmission/), [`vibration.radiation_efficiency`](https://jmrplens.github.io/phonometry/reference/api/vibration/radiation-efficiency/) and [`vibration.point_mobility`](https://jmrplens.github.io/phonometry/reference/api/vibration/point-mobility/).