Abstract
For footsteps from upstairs, the advice is “lay a soft floor covering”. This paper counts which sounds that helps and which it does not. No new theorem or law is claimed.
First, a soft covering lengthens the contact: for a light impact $0.000222$ s becomes $0.002233$ s, a factor of $\mathbf{10.049876}$ (Section 2).
Second, longer contact trims the high bands: the light impact’s $1000$ Hz band drops $\mathbf{-27.0558}$ dB and its $2000$ Hz band $-38.8660$ dB (Section 3).
Third, the low bands are hardly trimmed: even for the light impact the $63$ Hz band drops only $-0.1856$ dB (Section 3).
Fourth, and this is the core. It barely helps against heavy impacts: contact lengthens only by $1.095445$, and no band drops by $2.3$ dB (the $63$ Hz band by $\mathbf{-2.2767}$ dB) (Section 3).
Fifth, heavy impacts already sit in the low bands: even on a hard floor the $1000$ Hz band’s average is $\mathbf{52.6212}$ dB below the $63$ Hz band’s (Section 4).
Sixth, the separator is whether one asks about a light impact or a heavy one: added softness works only when the contact was hard to begin with (Section 5).
Keywords: floor impact sound, contact time, impact force spectrum, soft floor covering, octave band, model.
Scope (what is not claimed)
No new theorem or law is claimed: that floor impact sound is treated separately for light and heavy impacts, that soft floor finishes reduce light impacts well and heavy impacts little, and that a longer contact shifts the force spectrum to lower frequencies, are all known. No design advice is given, which covering to use and how thick to make the floor are not stated. No real floor is claimed: masses of $0.5$ kg and $7$ kg and stiffnesses of $10^{8}$, $2\times10^{5}$ and $10^{6}$ N/m are model values matching no rating standard. The path from floor to sound is not included: slab vibration, radiation into the room below and room acoustics are not treated; only the force spectrum is examined. Hearing is not included: no frequency weighting for the ear is applied. Relation to earlier papers: Paper 456 showed that doubling the distance does not halve the noise, leaving sound insulation open in its scope note. This paper stands beside it: there the side where sound spreads, here the side where it is born. Paper 584 showed that more damping in a base isolator transmits more shaking on one side: a neighbour in that a soft layer’s effect splits by frequency. The contact-time ratio $10.049876$ ($\sqrt{101}$) also appears in 584, but there as resonant amplification at damping $0.05$, with a different formula and quantity: a distinct root. What is added is laying out band-by-band reductions for two impacts, measuring that no band of the heavy impact drops by $2.3$ dB, measuring with band-averaged levels that heavy impacts already sit low, and placing the separator on light versus heavy impact.
1. Notation and the Question
Setting: a mass $m$ strikes the floor with contact stiffness $k$. The blow is a half-sine of duration $\tau$, compared at fixed momentum.
$$\tau=\pi\sqrt{\frac{m}{k}},\qquad \bigl|H(f)\bigr|=\left|\frac{\cos(\pi f\tau)}{1-(2f\tau)^{2}}\right|,\qquad \frac{1}{k_{\mathrm{eff}}}=\frac{1}{k}+\frac{1}{k_{\mathrm{soft}}}$$$|H|$ equals $1$ at zero frequency. A soft covering ($k_{\mathrm{soft}}=10^{6}$ N/m) enters in series with the contact stiffness. $|H|^{2}$ is summed per octave band and compared with and without the covering.
Question: which bands of which impact does a soft covering reduce?
2. Contact Time
Computation 2.1: the contact times of the two impacts were computed.
| impact | hard floor | soft covering | ratio |
|---|---|---|---|
| light ($0.5$ kg, $k=10^{8}$) | $0.000222$ s | $0.002233$ s | $10.049876$ |
| heavy ($7$ kg, $k=2\times10^{5}$) | $0.018586$ s | $0.020360$ s | $1.095445$ |
Observation 2.2 the same covering lengthens them differently. By $10.049876$ for the light impact and $1.095445$ for the heavy: stiffnesses combine in series, so the already-soft side dominates and added softness does little.
3. Band-by-Band Reductions
Computation 3.1: the reduction in each band from laying the soft covering was computed.
| band | light impact | heavy impact |
|---|---|---|
| $63$ Hz | $-0.1856$ dB | $-2.2767$ dB |
| $125$ Hz | $-0.7312$ dB | $-1.3004$ dB |
| $250$ Hz | $-2.9300$ dB | $-0.5020$ dB |
| $500$ Hz | $-11.4530$ dB | $-2.0248$ dB |
| $1000$ Hz | $-27.0558$ dB | $-1.5762$ dB |
| $2000$ Hz | $-38.8660$ dB | $-1.4157$ dB |
Observation 3.2 for the light impact, higher bands drop more. $-27.0558$ dB at $1000$ Hz and $-0.1856$ dB at $63$ Hz: a longer contact stops producing force above roughly its inverse.
Observation 3.3 this is the core. For the heavy impact it barely helps in any band. The largest reduction, at $63$ Hz, is $-2.2767$ dB, short of $2.3$ dB: contact hardly lengthens, so the spectrum hardly changes shape.
4. Heavy Impacts Are Low to Begin With
Computation 4.1: on a hard floor, the band-averaged level (divided by bandwidth, $63$ Hz band set to $0$ dB) was computed.
| band | light impact | heavy impact |
|---|---|---|
| $63$ Hz | $0.0000$ dB | $0.0000$ dB |
| $250$ Hz | $-0.0274$ dB | $-29.0829$ dB |
| $1000$ Hz | $-0.4659$ dB | $-52.6212$ dB |
| $2000$ Hz | $-1.8723$ dB | $-64.6349$ dB |
Observation 4.2 a heavy impact already sits in the low bands on a hard floor. Its $1000$ Hz average is $52.6212$ dB below its $63$ Hz average: the high bands a covering would trim hold almost nothing to trim.
Observation 4.3 a light impact is flat into the high bands. Even at $1000$ Hz it is down only $-0.4659$ dB, so there is much for a covering to remove.
5. The Separator
Observation 5.1 the separator is whether one asks about a light impact or a heavy one. What divides the effect is less the weight itself than whether the contact was already hard or soft: added softness works only when the other member of the series is hard.
Observation 5.2 placed beside Papers 456 and 584. 456 showed that doubling the distance does not halve the noise. There the side where sound spreads, here the side where it is born. 584 showed that more damping in a base isolator transmits more shaking on one side: a neighbour in that a soft layer’s effect splits by frequency. The contact-time ratio $10.049876$ appears in 584 as $\sqrt{101}$, but there it is resonant amplification at damping $0.05$, with a different formula and quantity; by the discipline of Paper 300 it is a distinct root.
Observation 5.3 what is not said. Which covering to use, slab vibration, sound in the room below and the ear’s weighting are not treated. What is shown is that the same softness helps or does not according to the kind of impact.
6. Census and Closing
- light contact lengthens $\mathbf{10.049876}$ (Computation 2.1): a lot.
- heavy lengthens $1.095445$ (Computation 2.1): hardly.
- light, $1000$ Hz: $\mathbf{-27.0558}$ dB (Computation 3.1): high bands trimmed.
- $2000$ Hz: $-38.8660$ dB (Computation 3.1): more so.
- $63$ Hz: $-0.1856$ dB (Computation 3.1): low bands stay.
- heavy, at most $\mathbf{-2.2767}$ dB (Computation 3.1): the core.
- heavy on a hard floor, $\mathbf{-52.6212}$ dB at $1000$ Hz (Computation 4.1): already low.
- light, $-0.4659$ dB (Computation 4.1): flat upward.
- $\sqrt{101}$ distinct from Paper 584 (Observation 5.2): different formula.
- light impact, or heavy (Observation 5.1): the separator; Papers 456, 584.
Closing: A soft floor covering quietens light impacts but barely touches heavy ones. For a light impact contact lengthens by $10.049876$, dropping the $1000$ Hz band by $-27.0558$ dB and the $2000$ Hz band by $-38.8660$ dB, while the $63$ Hz band falls only $-0.1856$ dB: a longer contact stops producing force above roughly its inverse. For a heavy impact contact lengthens only by $1.095445$, and no band drops by as much as $2.3$ dB: stiffnesses combine in series and the already-soft contact dominates, so added softness does little. And even on a hard floor the heavy impact’s $1000$ Hz average lies $52.6212$ dB below its $63$ Hz average, leaving little to trim. Placed among the earlier papers: beside Paper 456’s side where sound spreads, as the side where it is born, and treating the $\sqrt{101}$ shared with Paper 584 as a distinct root by the discipline of Paper 300. To be honest: only the force spectrum is examined; slab vibration and radiation into the room below are not included.
References
- Cremer, L., Heckl, M., & Petersson, B. A. T. (2005). Structure-Borne Sound. Springer.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. YOI: https://www.yamagishiyuuki.com/yoi/300; DOI: 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). Doubling the Distance Does Not Halve the Noise [Paper 456]. YOI: https://www.yamagishiyuuki.com/yoi/456; DOI: 10.5281/zenodo.22733038
- Yamagishi, Y. (2026). More Damping in a Base Isolator Transmits More Shaking on One Side [Paper 584]. YOI: https://www.yamagishiyuuki.com/yoi/584; DOI: 10.5281/zenodo.23014359
Reference
Yamagishi, Y. (2026). A Soft Floor Covering Quietens Light Impacts but Barely Touches Heavy Ones [Paper 854]. YOI: https://www.yamagishiyuuki.com/yoi/854
ORCID: 0009-0006-8587-3450
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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 854. This paper is a preprint and has not been peer reviewed.