Source: https://jmrplens.github.io/phonometry/buildings/design/resilient-layers/ # Predicting Resilient-Layer Performance A resilient layer is the cheapest way to buy impact sound insulation and the easiest to get wrong. Whether it is a carpet on a slab, a screed floating on mineral wool or a plasterboard lining on a masonry wall, the layer is a spring, and the only question that matters is where its resonance with the mass it carries falls. Below that frequency it does nothing, at it the construction is *worse* than bare, and above it the improvement climbs at a rate the construction type decides. This guide is the prediction side of that story, from the material data to the improvement spectrum; the measurement side lives in [Floor-Covering Impact Improvement (ISO 16251-1)](https://jmrplens.github.io/phonometry/buildings/design/impact-improvement/) and [Dynamic stiffness of resilient materials](https://jmrplens.github.io/phonometry/materials/resilient/dynamic-stiffness/). ## The excitation: the tapping machine as a mass-spring-dashpot Everything downstream depends on the force the standard tapping machine actually injects, and that force is not a property of the machine alone. Each hammer is a 0.5 kg mass dropped 40 mm, so it lands at $v_0 = \sqrt{2gh} = 0.886\ \text{m/s}$, ten times a second. If the impact were short compared with the 0.1 s between impacts, the momentum alone would fix the force, giving the line spectrum $|F_n| = 2 m v_0 / T_i$ and the band mean-square force $F_\mathrm{rms}^2 = 3.9 B$ that ISO 10140 measurements on concrete are usually reduced with. Real floors deform. The hammer, the contact stiffness $K$ it presses into and the floor's driving-point impedance $Z_\mathrm{dp}$ form a mass-spring-dashpot whose pulse splits into two cases: - **over-critical**, $K m \ge 4 Z_\mathrm{dp}^2$: a single positive pulse, no rebound. This is a lightweight walking surface (chipboard, OSB), where the spectrum tends to $|F_n|_\text{lower} = m v_0/T_i$; - **under-critical**, $K m < 4 Z_\mathrm{dp}^2$: the hammer rebounds and only the first positive lobe counts. This is a concrete slab or a screed, where the spectrum sits within 1 dB of $|F_n|_\text{upper} = 2 m v_0/T_i$ below 4 kHz and the short-pulse estimate is adequate. The two limits are 6 dB apart in mean square, which is the whole reason a tapping-machine level on a timber floor cannot be compared with one on concrete without thinking. Above the cut-off frequency $$ f_\mathrm{co} = \frac{1}{2\pi}\sqrt{\frac{K}{m}}\quad\text{(under-critical)}, \qquad f_\mathrm{co} = \frac{1}{2\pi}\left[\frac{K}{2Z_\mathrm{dp}} - \sqrt{\left(\frac{K}{2Z_\mathrm{dp}}\right)^2 - \frac{K}{m}}\right]\quad\text{(over-critical)}, $$ the force falls away, and above the limiting frequency $f_\mathrm{limit} = Z_\mathrm{dp}/(2\pi m)$ the hammer's own mass impedance, not the floor, caps the injected power. ```python import numpy as np from phonometry import ( hammer_impact_velocity, infinite_plate_impedance, plate_bending_stiffness, plate_contact_stiffness, tapping_force_spectrum, ) print(round(hammer_impact_velocity(), 3)) # 0.886 m/s (0.5 kg dropped 40 mm) # A 140 mm cast in-situ concrete slab: 2200 kg/m3, cL = 3800 m/s, nu = 0.2. rho, c_l, nu, h = 2200.0, 3800.0, 0.2, 0.14 E = rho * c_l**2 * (1 - nu**2) z = infinite_plate_impedance(plate_bending_stiffness(E, h, nu), rho * h) k = plate_contact_stiffness(E, poisson_ratio=nu) freqs = np.array([100, 200, 400, 800, 1600, 3150], dtype=float) res = tapping_force_spectrum(freqs, k, z) res.over_critical # False: the hammer rebounds off concrete round(res.cut_off_frequency) # 6948 Hz, above the building acoustics range res.peak_force # |Fn| per band, within 1 dB of res.upper_limit res.power_input_level # 10 lg(Win/1 pW), rising 3 dB per doubling res.plot() # the force spectrum with both asymptotes ``` ## Soft floor coverings: the cut-off frequency is the design A soft covering on a heavyweight floor changes nothing but the force input: its mass is negligible and it barely touches the slab's loss factor or bending stiffness. So its improvement is simply the ratio of the two force spectra, $$ \Delta L = 20 \log_{10} \frac{|F_n|_\text{without}}{|F_n|_\text{with}} \ \text{dB}, $$ and the whole design collapses to one number, the covering's own cut-off frequency, set by its contact stiffness $K = E \pi r^2 / d$ with the hammer radius $r = 15$ mm. Below $f_\mathrm{co}$ the covering does nothing; above it the improvement rises at 12 dB per octave. Two straight lines are therefore a usable estimate, and thickening a homogeneous covering lowers $f_\mathrm{co}$ and buys improvement everywhere above it. One detail decides whether the numbers are usable. The tapping machine excites a **line** spectrum, at multiples of the 10 Hz impact rate, so the formula above is a statement about one Fourier component; a band value is the ratio of the band mean-square forces over the lines the band contains. The difference is not cosmetic: the model's transform has exact nulls at odd multiples of $f_\mathrm{co}$, so a band centre landing on one reads tens of dB high. With the 100 Hz cut-off below, the single line at 500 Hz gives 66.8 dB against a design estimate of 27.9 dB, while the band value is 33.3 dB. `improvement` is the band value; `line_improvement` is the per-line ratio, and carries the nulls where they belong.

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import ( covering_contact_stiffness, covering_improvement, infinite_plate_impedance, plate_bending_stiffness, plate_contact_stiffness, ) rho, c_l, nu, h = 2200.0, 3800.0, 0.2, 0.14 # 140 mm concrete slab E = rho * c_l**2 * (1 - nu**2) z = infinite_plate_impedance(plate_bending_stiffness(E, h, nu), rho * h) plate = plate_contact_stiffness(E, poisson_ratio=nu) bands = np.array([50, 63, 80, 100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000.0]) d = 0.005 for label, modulus_over_thickness in (("PVC", 1.5e11), ("backed vinyl", 2.8e8)): res = covering_improvement( bands, covering_contact_stiffness(modulus_over_thickness * d, d), plate, z) plt.semilogx(bands, res.improvement, marker="o", label=f"{label}: fco = {res.cut_off_frequency:.0f} Hz") plt.semilogx(bands, res.two_line, ":") plt.legend() plt.show() ```

```python from phonometry import covering_contact_stiffness, covering_improvement # Covering No. 2 of Hopkins Fig. 4.64: E/d = 2.8e8 N/m3, a vinyl or carpet # with a resilient backing, 5 mm thick on the same 140 mm slab. d = 0.005 covering = covering_contact_stiffness(2.8e8 * d, d) res = covering_improvement([125.0, 250.0, 500.0, 1000.0], covering, plate, z) round(res.cut_off_frequency) # 100 Hz round(res.bare_cut_off_frequency) # 6948 Hz, the bare slab's own cut-off res.improvement # 3.2, 17.0, 33.3, 43.0 dB, band values res.two_line # 3.9, 15.9, 27.9, 40.0 dB, the estimate res.line_improvement # per Fourier line, nulls at 3, 5, 7 fco res.plot() ``` The model treats the covering as a *linear* spring. Under the tapping machine's high force many real coverings harden, showing two or three slopes between 5 and 22 dB per octave rather than one, which is why a laboratory $\Delta L$ remains the reference once a specimen exists. ## Floating floors: three laws above one resonance A floating floor is a rigid walking surface on a resilient layer, and its resonance follows from the two numbers known at the drawing stage, the mass per unit area $m'$ of the slab and the EN 29052-1 dynamic stiffness per unit area $s'$ of the layer: $$ f_0 = 160 \sqrt{\frac{s'}{m'}}\ \text{Hz} \qquad (s'\ \text{in MN/m}^3,\ m'\ \text{in kg/m}^2), $$ where 160 is the standard's rounding of $1000/2\pi$. Above it, which law applies is a question about damping, not about the layer: | Model | Law | When | | --- | --- | --- | | `"cremer"` | $\Delta L = 40 \log_{10} (f/f_0)$ | Cremer's infinite-plate result; asphalt screeds and dry floating floors, whose internal losses are high enough to behave as infinite plates (ISO 12354-2 Formula C.3) | | `"en12354"` | $\Delta L = 30 \log_{10} (f/f_0)$ | sand-cement and calcium-sulfate screeds, whose low loss factor makes them finite plates with a reverberant bending field (Formula C.1) | | `"cremer_hammer"` | $40 \log_{10} (f/f_0) + 10 \log_{10}[1 + (f/f_\mathrm{limit})^2]$ | a lightweight walking surface, where the hammer impedance is no longer negligible; tends to 18 dB per octave |

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import ( floating_floor_improvement_spectrum, floating_floor_resonance_frequency, ) # The worked floating floor of ISO 12354-2:2017 Annex G. f0 = floating_floor_resonance_frequency(8.0e6, 73.5) # 52.8 Hz freqs = np.logspace(np.log10(40.0), np.log10(5000.0), 400) for model, kwargs in (("en12354", {}), ("cremer", {}), ("cremer_hammer", {"limiting_frequency": 521.0})): res = floating_floor_improvement_spectrum( freqs, resonance_frequency=f0, model=model, **kwargs) plt.semilogx(freqs, res.improvement, label=model) plt.axvline(f0, ls=":") plt.legend() plt.show() ```

```python from phonometry import ( combined_dynamic_stiffness, double_floating_floor_resonances, floating_floor_improvement_spectrum, floating_floor_resonance_frequency, weighted_floating_floor_improvement, ) # 35 mm screed, m' = 73.5 kg/m2, on a resilient layer of s' = 8 MN/m3. f0 = floating_floor_resonance_frequency(8.0e6, 73.5) round(f0, 1) # 52.8 Hz bands = [50, 63, 80, 100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000.0] res = floating_floor_improvement_spectrum( bands, resonance_frequency=f0, mass_per_area=73.5, dynamic_stiffness=8.0e6) res.improvement # 0.0, 2.3, 5.4, 8.3, 11.2 ... 59.3 dB round(res.delta_lw, 1) # 32.2 dB, the Formula (C.4) weighted improvement res.plot() # Two resilient layers act as springs in series (Formula C.6), which drops the # resonance by sqrt(2) and buys about 4.5 dB everywhere above it. round(floating_floor_resonance_frequency( combined_dynamic_stiffness([8.0e6, 8.0e6]), 73.5), 1) # 37.3 Hz # One floating floor on top of another has two resonances instead of one; the # adverse dip disappears, but the steep rise only starts above the higher. double_floating_floor_resonances(7.25e6, 12.78, 7.25e6, 12.78) # (74, 194) Hz ``` The weighted single numbers come straight from the same two inputs, so a floating floor can be sized without ever drawing a spectrum: $\Delta L_\mathrm{w} = 13\log_{10} m' - 14.2\log_{10} s' + 20.8$ dB for a sand-cement screed (Formula C.4) and the steeper Formula (C.5) fit for asphalt and dry floors. Heavier slabs and softer layers rate better, and both estimates are deliberately on the safe side. A floating floor supported on **discrete mounts** rather than a continuous layer is a different problem: the walking surface carries a reverberant bending-wave field and all the transmission goes through the mounts, which Vér's two-subsystem SEA model turns into a 30 dB per decade rise rather than 40. Fewer mounts, a thicker walking surface or more internal damping all raise it. ```python from phonometry import resilient_mount_improvement # 50 mm concrete walking surface on 4 mounts per m2 of 2 MN/m stiffness. resilient_mount_improvement( [125.0, 250.0, 500.0, 1000.0], impedance=3.8e5, mass_per_area=115.0, loss_factor=0.02, mount_stiffness=2.0e6, mount_density=4.0) ``` ## Wall linings: Annex D reads the rating off the resonance A lining on a wall is the same spring problem with the sign reversed: it can *reduce* the sound insulation, and ISO 12354-1 Annex D predicts by how much from the resonance frequency alone. Which formula gives $f_0$ depends on how the layer is fixed: $$ f_0 = \frac{1}{2\pi}\sqrt{s'\left(\frac{1}{m_1'} + \frac{1}{m_2'}\right)} \ \text{(D.1, bonded directly)}, \qquad f_0 = \frac{1}{2\pi}\sqrt{\frac{0.111}{d}\left(\frac{1}{m_1'} + \frac{1}{m_2'}\right)} \ \text{(D.2, studs over a filled cavity)}. $$ Both stiffnesses are in MN/m³ and both masses per unit area in kg/m², which is how the annex prints them: the cavity term of (D.2) is $s' = 0{,}111/d$ MN/m³ with $d$ in metres, that is $0{,}111\times10^{6}/d$ in SI units. Substituting the SI value without the conversion gives a resonance about a thousand times too low. Table D.1 then reads the weighted improvement off $f_0$, rounded to the one-third-octave band it falls in. Below 200 Hz the lining helps, by $74.4 - 20\log_{10} f_0 - R_\mathrm{w}/2$ dB, and the better the bare wall the less there is to gain. From 200 Hz upwards it costs: 1 dB at 200 Hz falling to 10 dB from 630 Hz to 1600 Hz, recovering to 5 dB above. Getting the resonance well below the range of interest is therefore the entire design brief. ```python from phonometry import ( lining_improvement, lining_improvement_in_situ, lining_resonance_frequency, weighted_lining_improvement, ) # A 9.5 mm plasterboard laminated with 32 mm EPS (s' = 65 MN/m3), bonded with # adhesive dabs to a 100 mm aircrete block wall of 51 kg/m2. f0 = lining_resonance_frequency(51.0, 6.3, dynamic_stiffness=65e6) round(f0) # 542 Hz: squarely in the way round(weighted_lining_improvement(f0, 45.0)) # -9 dB, it makes things worse # The same lining on studs over a 100 mm mineral-wool-filled cavity. f0 = lining_resonance_frequency(51.0, 6.3, cavity_depth=0.100) round(f0, 1) # 70.8 Hz round(weighted_lining_improvement(f0, 45.0), 1) # +13.8 dB (80 Hz band) # External thermal insulation systems have their own Annex D fits. res = lining_improvement(100.0, system="mineral_wool") # Formula (D.3) res.delta_rw, res.delta_ra, res.delta_ratr # (10.5, 8.0, 9.7) dB lining_improvement(100.0, system="mineral_wool", anchors=True).delta_rw # (D.5) res.plot() # the Annex D ratings against fo, with this system marked # A laboratory rating transfers to the field through Formula (D.8). round(lining_improvement_in_situ(10.0, 100.0, 60.0), 1) # 4.4 dB on a Rw 60 wall ``` ## What is anchored on a published number, and what is not Not every formula on this page has a worked example behind it, and it is worth being explicit about which. **Anchored on printed values.** The tapping-machine cut-off frequencies (about 7000 Hz bare, 2300 Hz and 100 Hz for Hopkins's two coverings), the over/under-critical classification of his four walking surfaces, the double floating floor's two resonances, the lining resonances of his Fig. 4.48, and the floating-floor spectrum of ISO 12354-2:2017 Table G.4, whose 21 printed values from 0.0 dB at 50 Hz to 59.3 dB at 5 kHz the library reproduces to their printed precision of 0.1 dB. Three things are worth knowing about that Table G.4 anchor. Its column is labelled $\Delta L_\mathrm{situ}$, the *in-situ* improvement, and it stands in for the laboratory $\Delta L$ only because Clause 4.2.2.1 Formula (4) states $\Delta L_\mathrm{situ} = \Delta L$ outright. The resonance $f_0 = 52.8$ Hz is an *input* of the example, printed in Clause G.1.1's construction data and repeated in the box under the table, not a value derived from the grid. And the leading 0.0 dB at 50 Hz needs a clamp below $f_0$ that no printed formula states: Formula (C.1) is written as $\Delta L = 30\log_{10}(f/f_0)$ with no range of validity at all, and the only condition printed anywhere is the informal "for $f > f_0$" inside that same input box. ISO 12354-1:2017 Table L.4 prints the identical 21 numbers as $\Delta R_\mathrm{d,situ}$, which confirms the transcription in a second published document, but its own box says the column comes from "ISO 12354-2:2017, Formula (C.1)" with the same $m'$, $s'$ and $f_0$, so it is the same calculation republished rather than an independent one. **Implemented as printed, with no published number to check against.** The cavity stiffness $0.111/d$ of Formula (D.2); the asphalt and dry-floor weighted fit of Formula (C.5), whose source is the Figure C.2 nomogram; and the exterior-system, anchor, glued-area, stud and laboratory-to-field fits of Formulae (D.3) to (D.8). For these the library restates the printed expression and the tests pin the transcription and the qualitative relations the standard asserts, which is all that can honestly be done without a worked example. ## What this guide covers **Covered.** Hopkins 3.6.3's tapping-machine force pulse (Eqs. 3.85, 3.90 to 3.92, 3.95 to 3.106) through `tapping_force_spectrum`, `force_pulse`, `plate_contact_stiffness`, `covering_contact_stiffness`, `tapping_cut_off_frequency`, `hammer_limiting_frequency` and `short_pulse_mean_square_force`; the soft-covering improvement of Hopkins 4.4.3.1 (Eq. 4.114 and the two-line estimate) through `covering_improvement`; ISO 12354-2:2017 Annex C (Formulae C.1 to C.6) with Hopkins 4.4.4 and Vigran 8.4 through `floating_floor_resonance_frequency`, `floating_floor_improvement_spectrum`, `weighted_floating_floor_improvement`, `combined_dynamic_stiffness`, `double_floating_floor_resonances` and `resilient_mount_improvement`; and ISO 12354-1:2017 Annex D (Formulae D.1 to D.8 and Table D.1) through `lining_resonance_frequency`, `weighted_lining_improvement`, `lining_improvement` and `lining_improvement_in_situ`. **Not covered.** The force model assumes a frequency-independent driving-point impedance, so a joisted or battened lightweight floor, whose impedance changes with where the hammer lands, is outside it; Hopkins points to numerical methods there. Soft coverings are treated as linear springs, and their measured non-linearity under the tapping machine is not modelled. There is no per-band prediction of a lining's $\Delta R(f)$: Annex D is a single-number method, and where ISO 12354 needs a lining spectrum it either measures it (ISO 10140-1 Annex G) or, for a floating floor, takes $\Delta R(f) = \Delta L(f)$, which its own Annex L calls rough. Heavy impact sources (the rubber ball) are not covered by any of these models. ## References - Hopkins, C. (2007). *Sound insulation*. Butterworth-Heinemann. ISBN 978-0-7506-6526-1. [doi:10.4324/9780080550176](https://doi.org/10.4324/9780080550176). Section 3.6.3 for the tapping-machine contact model and the force spectrum, Section 4.3.8 for the covering cut-off frequency and the two-line estimate, and Sections 4.3.9 and 4.3.10 for the floating-floor improvement laws. - Vigran, T. E. (2008). *Building acoustics*. CRC Press. ISBN 978-0-415-42853-8. [doi:10.1201/9781482266016](https://doi.org/10.1201/9781482266016). Section 8.4 for the resilient-mount improvement and the hammer-on-covering resonances used here as spot checks. - International Organization for Standardization. (2017). *Building acoustics — Estimation of acoustic performance of buildings from the performance of elements — Part 1: Airborne sound insulation between rooms* (ISO 12354-1:2017). [iso.org catalogue](https://www.iso.org/standard/70242.html). Annex D, the additional-layer ratings this page reads off the resonance frequency. - International Organization for Standardization. (2017). *Building acoustics — Estimation of acoustic performance of buildings from the performance of elements — Part 2: Impact sound insulation between rooms* (ISO 12354-2:2017). [iso.org catalogue](https://www.iso.org/standard/70243.html). Annex C, the floating-floor improvement and its weighted rating, and the Annex G worked example whose per-band column anchors Formula (C.1). ## Standards ISO 12354-2:2017, Annex C: the floating-floor resonance (Formula C.2), the improvement spectrum of Formula (C.1) and the weighted ratings of Formulae (C.4) and (C.5), with the combination rule of Formula (C.6). The per-band column of the Annex G worked example anchors Formula (C.1) band for band. ISO 12354-1:2017, Annex D: the additional-layer resonance (Formulae D.1 and D.2), the Table D.1 ratings read at the rounded one-third-octave band of the resonance, and the corrections of Formulae (D.3) to (D.8). Two defects of the printed annex are recorded in `docs/ERRATA.md`. What each formula rests on is set out in "What is anchored on a published number, and what is not" above. ## See also - [Floor-Covering Impact Improvement (ISO 16251-1)](https://jmrplens.github.io/phonometry/buildings/design/impact-improvement/): the measurement these predictions are checked against. - [Dynamic stiffness of resilient materials (EN 29052-1)](https://jmrplens.github.io/phonometry/materials/resilient/dynamic-stiffness/): where $s'$ comes from. - [Predicting Sound Insulation (EN 12354)](https://jmrplens.github.io/phonometry/buildings/design/insulation-prediction/): the simplified model that consumes $\Delta L_\mathrm{w}$ and $\Delta R_\mathrm{w}$. - [Detailed Per-Band Prediction (ISO 12354)](https://jmrplens.github.io/phonometry/buildings/design/detailed-prediction/): the per-band model whose floating-floor term is `floating_floor_improvement`. - [Predicting Panel Sound Insulation](https://jmrplens.github.io/phonometry/buildings/design/panel-sound-insulation/): the mass-spring-mass resonance of a double leaf, the airborne counterpart of the lining resonance. - API reference: [`building.prediction.resilient_layers`](https://jmrplens.github.io/phonometry/reference/api/building/resilient-layers/).