Source: https://jmrplens.github.io/phonometry/buildings/design/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/buildings/design/insulation-prediction/) and [Field Insulation Measurement (ISO 16283)](https://jmrplens.github.io/phonometry/buildings/insulation/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) $$ f_\mathrm{c} = \frac{c_0^2}{2\pi}\sqrt{\frac{m''}{B'}} = \frac{0.55\, c_0^2}{c_\mathrm{L}\, h}, $$ 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 $f_\mathrm{c}/2$, drops through a straight line in $\log f$, and from $f_\mathrm{c}$ upward follows Eq. 7.44 with the loss factor $\eta$; the dip sits at Bies design-chart point B, $TL = 20\log_{10}(f_\mathrm{c} m'') + 10\log_{10}\eta - 44$. The whole section in one sketch: a diffuse field drives a single leaf, and the predicted $R(f)$ climbs with the mass law until the coincidence dip. A 12.5 mm plasterboard leaf puts $f_\mathrm{c}$ near 2.6 kHz and rates $R_\mathrm{w} = 27$ dB. ```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 $f_\mathrm{c} \approx 2.1$ kHz collects most of the unfavourable deviations, and the shifted reference read at 500 Hz gives the catalogue $R_\mathrm{w} = 32$ 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() ```

The clip below watches coincidence happen in a 2D [elastic FDTD field](https://jmrplens.github.io/phonometry/simulation/elastic-waves/): a 10 mm steel plate immersed in air, driven by a sustained plane wave at 45 degrees, with both frequencies picked from the library's `coincidence_frequency`. At $f_\mathrm{c}/2 = 603$ Hz the measured transmission lands on the oblique mass law; at $2f_\mathrm{c} = 2413$ Hz the 45-degree trace equals $\lambda_\mathrm{B}$ exactly and the trace-matched plate re-radiates a 45-degree beam that grows along the plate, pinning the transmitted level at the $f_\mathrm{c}/2$ figure even though the mass law demands 12 dB more blocking at four times the frequency. The air drives the heavy steel so weakly that the resonant bending wave needs tens of metres of illuminated plate to build up fully, which is why a real coincidence dip, not total transparency, is what measurements show. What the plate lets through is about 47 dB under the incident wave, so on the colour scale the standing wave above the plate sets, the transmitted field would be a black band. Both panels therefore draw the air below the plate with the display gain printed on them (x150, that is +44 dB); the gain is the same on the two so the panels stay comparable, and the level annotations are the measured, physical ones. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_elastic_coincidence.webm) ### Quick estimates: the plateau method (Norton 3.9.1) Before the physical model there is a shortcut that practitioners have drawn by hand for decades. The **plateau method** (Norton & Karczub 2003, Section 3.9.1, after Watters) approximates the whole field-incidence curve of a single panel from three numbers per material, tabulated in `PLATEAU_MATERIALS` from Norton's Table 3.1: the surface density per millimetre of thickness, the height of the coincidence plateau in decibels, and the frequency ratio $B/A$ that sets its width. The construction has three parts: 1. the **field-incidence mass law** $TL = 10\log_{10}(1 + (\pi f m''/\rho_0 c_0)^2) - 5\ \text{dB}$ (Norton Eqs. 3.104 and 3.106), rising 6 dB per octave; 2. a horizontal **coincidence plateau** at the tabulated height, with point **A** where the mass-law line reaches it; 3. point **B** at $B/A \times f_A$, above which the estimate recovers at **10 dB per octave**. It needs neither the bending stiffness nor the loss factor - the tabulated plateau absorbs both - and it assumes a diffuse field on both sides of a panel whose length and width are at least twenty times its thickness. *The two curves are the same line below point A. Above it the plateau estimate replaces the entire coincidence region with one flat value, which is exactly what it claims to be: an estimate, not the dip.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import plateau_transmission_loss, single_panel_transmission_loss # 6 mm float glass: Norton Table 3.1 gives 2.47 kg/m2 per mm, a 27 dB # coincidence plateau and B/A = 10; its critical frequency is 2033 Hz. bands = np.array([100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000, 6300, 8000, 10000], dtype=float) quick = plateau_transmission_loss(bands, material="glass", thickness_mm=6.0, field_correction=5.5) physical = single_panel_transmission_loss(bands, 2.47 * 6.0, critical_frequency=2033.0, loss_factor=0.02) print(round(quick.plateau_start), round(quick.plateau_end)) # 374 3742 print(quick.plateau_height) # 27.0 quick.plot() # shades the plateau between points A and B plt.show() ```

A worked check against the book: an 8 m x 3 m solid brick wall 110 mm thick at 2.1 kg/m² per mm ($m'' = 231\ \text{kg/m}^2$) gives 35.8 dB in the 63 Hz octave, on the mass-law line, then the brick plateau of 37 dB across the 125 Hz and 250 Hz octaves, then 10 dB per octave above point B - Norton's printed answer to his problem 3.11 exactly, band for band, in the two regions the construction fixes analytically. ```python from phonometry import plateau_transmission_loss octaves = [63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0] res = plateau_transmission_loss(octaves, material="brick", thickness_mm=110.0, air_density=1.21) print(res.transmission_loss.round(1)) # [35.8 37. 37. 43.3 53.3 63.3 73.3] ``` The physical model has its own Norton-flavoured variant for the region above coincidence: `coincidence_model="cremer"` replaces Sharp's Eq. 7.44 with Cremer's empirical $TL = TL_0 + 10\log_{10}(f/f_\mathrm{c} - 1) + 10\log_{10}\eta - 2\ \text{dB}$ (Norton Eq. 3.110), and runs the mass law right up to $f_\mathrm{c}$ with no interpolated bridge. Both rise at about 10 dB per octave far above coincidence; they differ in how they leave the dip. ## Orthotropic panels: a coincidence range, not a dip (Vigran 6.5.3) Ribbed and corrugated cladding is not isotropic. It is very stiff *along* the corrugations and almost as limp as a flat sheet *across* them, so instead of one coincidence frequency it has a whole **coincidence range** bounded by the two principal bending stiffnesses (Vigran Eq. 6.107, the isotropic $f_\mathrm{c} = (c_0^2/2\pi)\sqrt{m''/B}$ evaluated twice): $$ f_{\mathrm{c}1} = \frac{c_0^2}{2\pi}\sqrt{\frac{m''}{B_1}}, \qquad f_{\mathrm{c}2} = \frac{c_0^2}{2\pi}\sqrt{\frac{m''}{B_2}}, \qquad B_1 > B_2 . $$ The stiffest direction sets the *lowest* coincidence frequency, which is the sting in the tail: corrugating a sheet to gain strength drags $f_{\mathrm{c}1}$ down, often to a few hundred hertz, while $f_{\mathrm{c}2}$ runs up to 15 kHz or 30 kHz. Over that whole span the resonant transmission dominates and $R$ flattens far below the mass law of a flat plate of the same mass. The bending-wave impedance now depends on the azimuth $\vartheta$ as well as the incidence angle $\varphi$ (Heckl 1960; Hansen 1993; Vigran Eq. 6.108, which is Bies Eq. 7.30): $$ Z_\mathrm{w} = \mathrm{j}\omega m''\left[1 - \left(\frac{f}{f_{\mathrm{c}1}}\cos^2\vartheta + \frac{f}{f_{\mathrm{c}2}}\sin^2\vartheta\right)^2 (1 + \mathrm{j}\eta)\sin^4\varphi\right], $$ and the diffuse-field average of $\tau = |1 + Z_\mathrm{w}\cos\varphi/2\rho_0c_0|^{-2}$ becomes a double integral (Vigran Eq. 6.111 = Bies Eq. 7.38). Setting $f_{\mathrm{c}1} = f_{\mathrm{c}2}$ recovers Cremer's isotropic impedance exactly. `orthotropic_transmission_loss` offers both published routes: - `method="integral"` evaluates that double integral numerically. It is the only route that responds to the loss factor, and it is the one that shows how deep the coincidence region really goes. The near-grazing angles are excluded by a limiting angle: pass `area` for the size-dependent limit of Bies Eq. 7.36 (Vigran writes it as Eq. 6.113), or leave it out for the fixed `limiting_angle` (78 degrees by default, Sharp's value, which is Vigran's $\sin^2\varphi = 0.96$). - `method="heckl"` is Heckl's closed-form approximation for $\eta = 0$, the design chart of Bies Fig. 7.9(b): field-incidence mass law below $f_{\mathrm{c}1}/2$, then $$ \tau_\mathrm{F} \approx \frac{\rho_0c_0}{2\pi^2 m''}\,\frac{f_{\mathrm{c}1}}{f^2} \left[\ln\frac{4f}{f_{\mathrm{c}1}}\right]^2 \quad (f_{\mathrm{c}1} \le f < f_{\mathrm{c}2}/2), \qquad \tau_\mathrm{F} \approx \frac{\rho_0c_0}{2m''}\,\frac{\sqrt{f_{\mathrm{c}1}f_{\mathrm{c}2}}}{f^2} \quad (f > 2f_{\mathrm{c}2}), $$ with straight lines in $\log_{10} f$ across the two gaps. It needs no loss factor and no numerical work, but it requires $f_{\mathrm{c}2} > 4f_{\mathrm{c}1}$ for its four construction points to stay in order. For the common "wavy" corrugation the equivalent stiffnesses come from Timoshenko and Woinowsky-Krieger through `corrugated_plate_stiffness` (Vigran Eq. 3.115), and the surface density has to grow with the **developed length** of the profile, which is what `corrugated_plate_mass_factor` returns. `orthotropic_plate_resonance` is the matching eigenfrequency of a simply supported orthotropic plate (Vigran Eq. 3.113 = Bies Eq. 7.27, after Hearmon 1959); its lowest mode matters because both infinite-panel models are only valid above roughly $1.5f_{1,1}$. ```python from phonometry import ( coincidence_frequency, corrugated_plate_mass_factor, corrugated_plate_stiffness, orthotropic_critical_frequencies, orthotropic_plate_resonance, plate_bending_stiffness, ) # Vigran's worked example: a 1 m x 1 m steel sheet 1 mm thick, E = 210 GPa, # nu = 0.3, m'' = 7.8 kg/m2, corrugated into a sinusoid 20 mm deep # (amplitude H = 10 mm) at a 100 mm pitch. flat_b = plate_bending_stiffness(2.1e11, 1.0e-3, 0.3) b_x, b_z, b_xz = corrugated_plate_stiffness( 1.0e-3, 0.010, 0.100, youngs_modulus=2.1e11, poisson_ratio=0.3, ) mass = 7.8 * corrugated_plate_mass_factor(0.010, 0.100) print(round(flat_b, 1), round(b_x, 1), round(b_z)) # 19.2 17.5 2202 print(round(mass, 2)) # 8.52 kg/m2 (+9 %) flat = {"length_x": 1.0, "length_z": 1.0, "mass_per_area": 7.8, "bending_stiffness_x": flat_b, "bending_stiffness_z": flat_b, "bending_stiffness_xz": flat_b} corr = {"length_x": 1.0, "length_z": 1.0, "mass_per_area": mass, "bending_stiffness_x": b_x, "bending_stiffness_z": b_z, "bending_stiffness_xz": b_xz} print(round(orthotropic_plate_resonance(1, 1, **flat), 1), round(orthotropic_plate_resonance(2, 2, **flat), 1)) # 4.9 19.7 print(round(orthotropic_plate_resonance(1, 1, **corr), 1), round(orthotropic_plate_resonance(2, 2, **corr), 1)) # 25.5 102.1 # The same sheet, flat and corrugated, in the coincidence picture: print(round(coincidence_frequency(7.8, flat_b))) # 11925 Hz print([round(f) for f in orthotropic_critical_frequencies(mass, b_x, b_z)]) # [1165, 13064] ``` The four eigenfrequencies are Vigran's own printed answers (4.9 Hz, 19.7 Hz flat; 25.5 Hz, 102 Hz corrugated), and reproducing the corrugated pair is what proves the mass factor belongs there: with the flat 7.8 kg/m² the same formulas return 26.7 Hz and 106.7 Hz. *The trade corrugating makes, on Vigran's own geometry. Below $f_{\mathrm{c}1}$ the two panels are within about 2 dB of each other (the corrugated one slightly higher: it is 9 % heavier, and the diffuse-field integral uses a 78 degree limiting angle rather than Sharp's 5.5 dB band correction). Across the coincidence range the corrugated sheet gives away up to 13 dB, and $R_\mathrm{w}$ falls from 28 dB to 25 dB, for a panel that is stiffer and barely heavier.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import ( coincidence_frequency, corrugated_plate_mass_factor, corrugated_plate_stiffness, orthotropic_critical_frequencies, orthotropic_transmission_loss, plate_bending_stiffness, single_panel_transmission_loss, ) bands = np.array([100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000, 6300, 8000, 10000, 12500, 16000], dtype=float) eta = 0.011 flat_b = plate_bending_stiffness(2.1e11, 1.0e-3, 0.3) b_x, b_z, _ = corrugated_plate_stiffness( 1.0e-3, 0.010, 0.100, youngs_modulus=2.1e11, poisson_ratio=0.3, ) mass = 7.8 * corrugated_plate_mass_factor(0.010, 0.100) fc1, fc2 = orthotropic_critical_frequencies(mass, b_x, b_z) flat = single_panel_transmission_loss( bands, 7.8, critical_frequency=coincidence_frequency(7.8, flat_b), loss_factor=eta, ) corrugated = orthotropic_transmission_loss( bands, mass, critical_frequency_lower=fc1, critical_frequency_upper=fc2, loss_factor=eta, ) heckl = orthotropic_transmission_loss( bands, mass, critical_frequency_lower=fc1, critical_frequency_upper=fc2, method="heckl", ) # One line: R(f) with the coincidence range shaded between fc1 and fc2. corrugated.plot() plt.show() # By hand, all three on one axes: fig, ax = plt.subplots() ax.axvspan(fc1, fc2, color="0.85", zorder=0) ax.semilogx(bands, flat.transmission_loss, "-o", ms=4, label="flat sheet") ax.semilogx(bands, corrugated.transmission_loss, "-s", ms=4, label="corrugated, integral") ax.semilogx(bands, heckl.transmission_loss, "--", label="Heckl approximation") ax.set(xlabel="Frequency [Hz]", ylabel="Transmission loss TL [dB]") ax.legend() plt.show() ```

Two caveats Bies attaches to the Heckl branch are worth repeating, because no smooth model predicts either. Below about $0.7f_{\mathrm{c}1}$ the estimate underestimates $R$ on small panels, the error growing as the panel shrinks; and real corrugated panels almost always show a dip of up to 5 dB somewhere between 2 kHz and 4 kHz, which finite-element work traced to resonances of the panel sections *between* the ribs rather than to any coincidence effect. ## 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 $f_\mathrm{l} = c_0/(2\pi d)$ 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) ``` The classic lightweight case is worth drawing to scale: two 8.8 kg/m² plasterboard leaves on a 100 mm empty cavity put $f_0$ at 90 Hz. `plot_double_wall_geometry` annotates the resonance in the cavity, and a `double_wall_transmission_loss` result that retained its geometry redraws its own cross-section with `dw.plot_geometry()`. *The whole model in one section: the two thin leaves are the masses, the 100 mm air gap is the spring, and $f_0 = 90$ Hz is where the dip of every double-wall curve sits.*

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```python import matplotlib.pyplot as plt from phonometry import mass_spring_mass_resonance, plot_double_wall_geometry # Two 8.8 kg/m2 plasterboard leaves on a 100 mm cavity. f0 = mass_spring_mass_resonance(8.8, 8.8, 0.1) plot_double_wall_geometry(8.8, 8.8, 0.1, resonance_frequency=f0) plt.show() # A double-wall prediction retains its geometry and redraws it: # dw = double_wall_transmission_loss(bands, 8.8, 8.8, 0.1) # dw.plot_geometry() ```

## Masonry cavity walls: the wall-tie bridge (Hopkins 4.3.5.4) The double-wall model above treats the cavity as pure air. A real masonry cavity wall is stitched together by **wall ties** every few courses, and those ties do two things the air-only model cannot see: they add a mechanical spring in parallel with the air spring, which pushes the mass-spring-mass resonance up, and they open a structure-borne path from one leaf to the other. *Left: a soft tie rolls the structure-borne coupling off as $1/f^3$, a stiff one stays on the rigid ceiling for another two octaves. Right: the same ties lift the resonance from 26 Hz to 50 Hz, extending the combined-mass branch.* **The tie as a spring (Hopkins Eq. 4.89).** A tie is characterised by a single number, its dynamic stiffness $s_{X\,\text{mm}}$ at a cavity width $X$, measured on two nominally identical 100 mm concrete cubes (Section 3.11.3.2, Eq. 3.202). $N$ ties over a plate of area $S$ add $Nk/S$ to the cavity air stiffness: $$ f_\mathrm{msm} = \frac{1}{2\pi} \sqrt{\frac{s_\mathrm{a} + Nk/S}{\rho_{\mathrm{s}1}\rho_{\mathrm{s}2}/(\rho_{\mathrm{s}1}+\rho_{\mathrm{s}2})}}. $$ Below $f_\mathrm{msm}$ the two leaves act as one plate of the combined mass, so stiff ties are doubly bad: they raise the resonance into the rating range and bridge the cavity. `WALL_TIE_STIFFNESS` carries Hopkins' Table A4, whose 50 mm rows come from Hopkins, Wilson & Craik (1999) and whose 100 mm row from Hall & Hopkins (2001). | Wall tie | Cavity width (mm) | $s_{X\,\text{mm}}$ (MN/m) | |---|---|---| | Butterfly tie (BS 1243:1978) | 50 | 1.7 | | Double-triangle tie (BS 1243:1978) | 50 | 16.1 | | Vertical-twist tie (BS 1243:1978) | 50 | 94.0 | | Vertical-twist tie (proprietary) | 100 | 43.4 | ```python from phonometry import ( double_wall_transmission_loss, mass_spring_mass_resonance, wall_tie_stiffness, wall_tie_stiffness_per_area, ) print(wall_tie_stiffness("butterfly")) # (0.05 m, 1.7e6 N/m) # Hopkins Fig. 4.35: two 140 kg/m2 leaves across an empty 75 mm cavity. print(round(mass_spring_mass_resonance(140.0, 140.0, 0.075))) # 26 Hz # Add 2.5 ties per m2 of s_75mm = 2 MN/m: the resonance nearly doubles. ties = wall_tie_stiffness_per_area(2.5, 2.0e6) print(round(mass_spring_mass_resonance(140.0, 140.0, 0.075, tie_stiffness_per_area=ties))) # 50 Hz dw = double_wall_transmission_loss(bands, 140.0, 140.0, 0.075, tie_stiffness_per_area=ties) ``` **The tie as a point connection (Hopkins Eqs. 4.84 to 4.88).** Each tie transmits structure-borne power between the leaves. With the driving-point mobilities $Y_i$, $Y_j$ of the two leaves (infinite thin plates, $Y = 1/(8\sqrt{B'm''})$, Eq. 2.190) and the connector mobility of a linear spring $Y_\mathrm{c} = \mathrm{i}\omega/k$ (Eq. 4.88), $N$ identical uncorrelated connections give the coupling loss factor (Eq. 4.87) $$ \eta_{ij} = \frac{N}{\omega m_i} \frac{\mathrm{Re}\{Y_j\}}{|Y_i + Y_j + Y_\mathrm{c}|^2}. $$ The plate area cancels ($N/m_i = n/\rho_{\mathrm{s}1}$ with $n$ ties per m²), so only the tie density enters. A rigid connection (a screw, a nail, a bolt, or a tie stiff enough never to yield) is the limit $Y_\mathrm{c} = 0$; a resilient tie rolls the coupling off two powers faster once $|Y_\mathrm{c}| = \omega/k$ overtakes the plate mobilities: $\eta_{ij}$ then falls as $1/f^3$ against the rigid ceiling's $1/f$, so the *ratio* between them goes as $1/f^2$. That is exactly why a butterfly tie at 1.7 MN/m and a vertical-twist tie at 94 MN/m behave so differently. ```python import numpy as np from phonometry import plate_bending_stiffness, wall_tie_coupling_loss_factor freq = np.logspace(np.log10(50.0), np.log10(5000.0), 60) b1 = plate_bending_stiffness(2.0e10, 0.1, 0.2) # 100 mm masonry leaves res = wall_tie_coupling_loss_factor( freq, 150.0, 170.0, b1, b1, ties_per_area=2.5, tie="butterfly" ) print(res.coupling_loss_factor[0], res.rigid_coupling_loss_factor[0]) res.plot() # eta_ij against the rigid-connection ceiling (needs matplotlib) ``` The **inputs** of this model are printed data: Table A4, confirmed value for value by Hopkins, Wilson & Craik (1999) Table 1, which prints the same 1.7 / 16.1 / 94.0 MN/m at a 50 mm cavity. Craik & Wilson (1995) Table 1 measures the same tie *types* at an 85 mm cavity and reports 1.1 and 4.3 MN/m for the butterfly and double-triangle ties, so it corroborates the ordering but not the values: the dynamic stiffness is defined at a given cavity width and changes with it. The **output** is not printed anywhere: every published sound reduction index of a bridged masonry cavity wall is a figure, so the per-band transmission-loss penalty of the ties has no printed numeric oracle. The resonance shift does: Hopkins Fig. 4.35 prints 26 Hz without ties and 50 Hz with them for the same wall. ## 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 $S_\mathrm{a}/S$ caps the composite at $10\log_{10}(S/S_\mathrm{a})$: 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 ``` The leak that undoes a heavy wall is almost invisible, and `.plot_geometry()` makes the point by drawing it to scale: a 2 mm gap through 100 mm of masonry. *The tiny geometry behind a large leak: the gap is 50 times deeper than it is wide, which is why it behaves as a short tube with half-wavelength depth resonances near 1.5 and 3.1 kHz rather than as a simple open area.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import slit_transmission_coefficient f = np.geomspace(100.0, 5000.0, 200) result = slit_transmission_coefficient(f, 0.002, 0.1) # One line: the wall section with the slit to scale. result.plot_geometry() plt.show() ```

The clip below puts the two regimes side by side in a 2D FDTD field: a plane front meets a rigid 0.10 m wall with a 25 mm slit ($\lambda/20$ at 686 Hz) and with a 0.50 m opening (one wavelength). The narrow slit re-radiates what it swallows as a cylindrical wave, the Gomperts transmission of the model annotated; the wavelength-sized opening lets the front through nearly intact and casts sharp-edged shadows. What the slit passes is about 23 dB under the standing wave that faces the wall, so in the instantaneous panels the half space behind the wall rides the display gain each panel prints (x10, that is +20 dB, for the slit and none for the wavelength-sized opening); the RMS maps below keep one shared scale, which is where the two openings are compared level for level. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_aperture_slit.webm) ## 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/devices/emission/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): $\sigma = (1 - f_\mathrm{c}/f)^{-1/2}$ for $f > f_\mathrm{c}$. ```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 $Y = (1-\mathrm{j})/(4 m' c_\mathrm{B})$ (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() ```

## Function parameters Every argument of this page in one place, with its units and its admissible range: | 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 $f_\mathrm{c}$, or $B'$ to compute it | | `loss_factor` | float | — | > 0; default `0.01` single panel, `0.1` double wall | 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` (slit) | float | m | > 0 | Slit width $w$. **Second** positional argument of both `slit_transmission_coefficient` and `slit_resonance_frequencies` | | `depth` (slit) | float | m | > 0 | Slit depth $d$ (the wall thickness). **Third** positional argument of `slit_transmission_coefficient`, **first** of `slit_resonance_frequencies` — the two functions take the pair in opposite orders | | `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()`). **Where the closed forms stop being valid.** Nothing in the code refuses an argument outside these ranges, so the three bounds below are the reader's to enforce. Sharp's coincidence method is stated for frequencies above $1.5\times$ the first panel resonance. Gomperts' slit is inviscid and holds for $w < 0.3\lambda$; a narrower slit at a higher frequency leaves the model behind. The plate radiation efficiency is Leppington's "method no. 1", one of several in that paper and not interchangeable with the others. ## What this guide covers **Covered.** The closed forms themselves, not the measurements behind them: Bies' §7.2 mass law and Sharp's coincidence method (Eqs 7.40/7.42/7.44) for a single panel; the double-wall mass-spring-mass resonance with its saturating cavity boost (Eqs 7.62-7.64); Hopkins' Gomperts slit and the Wilson & Soroka circular hole with the area-weighted composite sum (Eq. 4.92); the Leppington/Maidanik plate radiation efficiency (Eqs 2.227-2.230); and the Cremer Table 5.1 point impedances and mobilities of infinite plates and beams with the injected-power relation — through `single_panel_transmission_loss`, `double_wall_transmission_loss`, `slit_transmission_coefficient`, `composite_transmission_loss`, `radiation_efficiency`, `infinite_plate_impedance`, `infinite_beam_mobility` and `injected_power`. Vigran's orthotropic chapter joins them: the coincidence *range* of a ribbed or corrugated panel and its flattened $R$ by the diffuse-field integral or Heckl's closed form (Eqs 6.107-6.113), the Timoshenko equivalent stiffnesses of a sinusoidal corrugation (Eq. 3.115) and the orthotropic plate eigenfrequency (Eq. 3.113). Masonry cavity walls add Hopkins' wall-tie bridge (Sections 3.11.3.2 and 4.3.5.4.1): the Table A4 dynamic stiffnesses, the tie array in parallel with the cavity air (Eq. 4.89, through the `tie_stiffness_per_area` keyword) and the point-connection coupling loss factor with a linear-spring connector (Eqs 4.87, 4.88). **Not covered.** Every prediction stops where its source stops, and the three numerical bounds are stated in [Function parameters](#function-parameters) above: Sharp below $1.5\times$ the first panel resonance, Gomperts above $w = 0.3\lambda$, and Leppington's other formulations. The orthotropic models are infinite-panel models: below about $0.7 f_{\mathrm{c}1}$ Heckl's branch underestimates $R$ on small panels, and the 2 kHz to 4 kHz dip real corrugated cladding shows comes from resonances of the sections between the ribs, which neither route predicts. Only the sinusoidal corrugation has equivalent stiffnesses here; trapezoidal cladding needs its own, from the literature Vigran cites. On the wall-tie side the inputs are printed data but no published $R$ of a bridged masonry cavity wall is tabulated, so the per-band penalty the ties cause has **no numeric oracle** — only the resonance shift is anchored (Hopkins Fig. 4.35, 26 Hz to 50 Hz). The coupling loss factor is not fed into a full statistical energy analysis: combining it with the airborne path would need the whole room-plate-cavity-plate-room chain. ## 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. - Vigran, T. E. (2008). *Building Acoustics*. Taylor & Francis. ISBN 978-0-415-42853-8. [doi:10.1201/9781482266016](https://doi.org/10.1201/9781482266016). Sections 3.7.3.3 (orthotropic plate eigenfrequencies, Eqs 3.113-3.115 after Timoshenko & Woinowsky-Krieger) and 6.5.3 (orthotropic panel transmission, Eqs 6.107-6.113 after Heckl 1960 and Hansen 1993). - Hopkins, C., Wilson, R., & Craik, R. J. M. (1999). Dynamic stiffness as an acoustic specification parameter for wall ties used in masonry cavity walls. *Applied Acoustics*, 58, 51-68. [doi:10.1016/S0003-682X(98)00068-1](https://doi.org/10.1016/S0003-682X(98)00068-1). The measurement behind the 50 mm rows of Hopkins' Table A4. ## See also - [Predicting Sound Insulation (EN 12354)](https://jmrplens.github.io/phonometry/buildings/design/insulation-prediction/): assembles predicted or measured element $R$ into the in-situ $R'_\mathrm{w}$. - [Sound Power from Surface Vibration (ISO 7849)](https://jmrplens.github.io/phonometry/devices/emission/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/vibration/structural/mechanical-mobility/): the measured counterpart of the theoretical point mobilities. - [Porous and multilayer absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/porous-absorbers/): the cavity-fill models a double wall consumes. - API reference: [`building.prediction.panel_transmission`](https://jmrplens.github.io/phonometry/reference/api/building/panel-transmission/), [`building.prediction.masonry_cavity_wall`](https://jmrplens.github.io/phonometry/reference/api/building/masonry-cavity-wall/), [`building.prediction.aperture_transmission`](https://jmrplens.github.io/phonometry/reference/api/building/aperture-transmission/), [`vibration.structural.radiation_efficiency`](https://jmrplens.github.io/phonometry/reference/api/vibration/radiation-efficiency/) and [`vibration.structural.point_mobility`](https://jmrplens.github.io/phonometry/reference/api/vibration/point-mobility/).