Rooms and Buildings

This page collects the theory behind rooms and buildings: impulse-response measurement and the room-acoustic parameters, background-noise criteria, airborne and impact insulation with their single-number ratings and uncertainty, and flanking and absorption prediction. It is part of the theory reference; surface scattering and acoustic material characterisation live on Materials and Surfaces.

Each section keeps its own standard’s notation, and four letters carry more than one meaning across them:

Symbol	Meaning	Where
$A$	Equivalent absorption area, m² ($A_0=10$ m² reference, $A_T$ specimen, $A_{obj}$, $A_{air}$)	throughout
$m$	Power attenuation coefficient of air, 1/m	ISO 354 and EN 12354-6
$m$	Number of repeated measurements	ISO 12999-1 uncertainty only
$m^{\prime }$ / $m^{\prime \prime }$	Mass per unit area, kg/m² — EN 12354 writes $m^{\prime }$, the panel-prediction literature $m^{\prime \prime }$	flanking and panel prediction
$C$	Clarity, dB (subscripted 50 or 80)	ISO 3382-1 parameters
$C$	Decay curvature, %	ISO 3382-2 validity
$C$, $C_{tr}$	Spectrum adaptation terms, dB	ISO 717 ratings
$T$	Reverberation time, s ($T_s$ is the centre time)	throughout

Room noise criteria (ANSI S12.2)

ANSI/ASA S12.2-2019 rates steady background noise in rooms against families of octave-band curves (16 Hz – 8 kHz). The NC rating starts from the speech interference level $\mathrm{SIL}=\frac{1}{4}(L_{500}+L_{1000}+L_{2000}+L_{4000})$ (clause 3.2), which preselects the NC-(SIL) curve: if no band exceeds it, the spectrum is designated NC-(SIL) and the procedure ends there (clause 5.2.2). The two families otherwise differ in almost every respect:

	NC	RC Mark II
Curve family	Table 1 curves NC-15 to NC-70, 16 Hz – 8 kHz	A pure −5 dB/octave line keyed to its 1 kHz value, floored at $\max (\mathrm{RC}+25,55)$ dB at 16 and 31.5 Hz (Annex D)
How the rating is picked	Tangency (clause 5.2.3): each band interpolated against the tabulated curves, the highest per-band index wins	Arithmetic mean of the 500/1000/2000 Hz levels, rounded to an integer (D.4)
What is reported	A continuous value plus the band that set it — NC-42.5 at 250 Hz, not snapped to a curve	An integer with a spectral-quality tag, e.g. RC-35(N)
Quality tag	none	Rumble “R” when any band ≤ 500 Hz exceeds the contour by more than 5 dB, hiss “H” when any band ≥ 1 kHz exceeds it by more than 3 dB, “RH” for both, else “N” (D.3)
Outside the family	Flagged: >NC-70 with the band of maximum exceedance, or <NC-15, rather than a fabricated number	—

The generated RC contours reproduce Table D.1 digit for digit, and feeding any Table 1 NC curve back returns its own tangency rating. NCB, RNC (Annex A) and the QAI (clause D.5) are deliberately out of scope.

Reading a rating. Neither number is a verdict on its own; the criterion comes from the use of the room, and Annex C (informative) is where the standard puts it. Its Table C.2 recommends, for example, NC 25-30 for a classroom or a small auditorium, 30-35 for a small private office, 40-45 for an open-plan area and 40-50 for public circulation — so a rating in the low forties is unremarkable in an open office and a clear failure in a lecture room. The RC tag is a diagnosis rather than a grade: the reference contour is a straight −5 dB/octave line because that is the slope occupants describe as bland, so R means the spectrum carries more low-frequency energy than that slope allows and points at fan and duct-borne noise or a structure-borne path, while H points at the terminal device — diffuser velocity, turbulence at the outlet. Different tags, different remedies. And a rating computed from a spectrum whose lowest bands were never actually measured is not comparable with one where they were, because those bands set both the tag and, often, the tangency point.

See the Room Noise guide for usage.

One ventilation-dominated spectrum, read by both procedures. The NC rating is set at 250 Hz — not at any of the four SIL bands — so the governing band is already telling the diagnostic story, and NC-42.5 would pass in an open-plan area and fail in a lecture room. The RC panel says the same thing in one letter: the bands below 500 Hz push through the +5 dB rumble tolerance, so the room is RC-35(R) and the remedy is on the low-frequency side, in the fan and the duct run, not at the diffuser.

Impulse response and room-acoustic parameters (ISO 18233, ISO 3382-1/-2/-3)

Deterministic-excitation impulse response (ISO 18233)

A room/transmission path is modelled as linear time-invariant, so its impulse response $h(t)$ carries everything. ISO 18233 replaces the classical noise-burst decay with a deterministic excitation that is deconvolved into $h(t)$, gaining 20–30 dB of effective signal-to-noise ratio. The exponential sine sweep (ESS, Annex B) has instantaneous frequency $f(t)=f_1(f_2/f_1{)}^{t/T}$, so its phase is the closed-form integral of $2\pi f(t)$:

$$\phi (t)=\frac{2\pi f_{1}T}{\ln (f_2/f_1)}\left[{\left(\frac{f_2}{f_1}\right)}^{t/T}-1\right].$$

A constant time-per-octave makes the ESS spectrum pink (−3 dB/octave). Deconvolution is done by linear (non-circular, zero-padded) spectral division $H=Y\overline{X}/(|X{|}^2+\epsilon )$, the Tikhonov term $\epsilon$ (a fraction of $\max |X{|}^2$) preventing noise blow-up at the band edges. Since a low-to-high sweep places harmonic-distortion products at negative arrival times, they fall in the wrapped tail and are removed by keeping the causal part (Farina). The MLS method (Annex A) instead exploits that the circular autocorrelation of a maximum-length sequence of length $2^N-1$ is a periodic delta, so $h={\mathrm{xcorr}}_{\text{circ}}(\text{recorded},\text{mls})/2^N$; synchronous averaging of $n$ periods adds $10{\log }_{10}n$ dB.

How long the record has to be, and what breaks first. The two rules that decide whether a deconvolved impulse response is usable are in clause 6.2, and they differ by excitation type. A repetitive signal such as an MLS must have a repetition period no shorter than the reverberation time of the room, $T_{REP}\ge T$ (Formula 10), or the modes are not excited to steady state and the response wraps onto itself. A non-repetitive sweep may be of any suitable length, but it must be followed by silence, and the decay must be recorded over at least half the reverberation time (6.2.2.3); for a low-to-high sweep the silence needed is set by the reverberation time at the high frequencies, which is where padding has to be kept clear of the wrapped harmonic arrivals. A record that stops with the sweep folds the tail back onto the head and biases every $T$ and every clarity value derived from it, silently and always in the same direction.

The assumption that breaks first is time invariance, and clause 6.1 is explicit about it: moving the source or a microphone during the measurement is not acceptable, and neither air movement nor a change in the speed of sound through temperature is — the path length changes by a fraction of a wavelength between the start and the end of the excitation, and the deconvolution charges the mismatch to the noise floor. This is also what separates the two methods in practice: MLS rests on the exactness of the circular autocorrelation and degrades under both time variance and loudspeaker distortion, which it smears over the whole response, whereas the sweep pushes distortion products to negative times where they can be cut away. The sweep is therefore the default, and MLS is kept for cases where the excitation must be noise-like.

Schroeder backward integration (ISO 3382-1, 5.3.3)

The band decay curve is the backward-integrated squared IR (Schroeder):

$$E(t)={\int }_t^{\infty }{p}^2(\tau )d\tau ={\int }_0^{\infty }{p}^{2}d\tau -{\int }_0^t{p}^{2}d\tau ,L(t)=10{\log }_{10}\frac{E(t)}{E(0)}\text{dB},$$

i.e. a reversed cumulative sum in discrete time. Backward integration cancels the random fluctuation of a single squared IR: for a purely exponential energy decay $p^2(t)=e^{-at}$ it gives $E(t)=e^{-at}/a$, an exactly straight line $L(t)=-(10a/\ln 10)t$. Background noise flattens $E(t)$, so integration is truncated at the crossing $t_1$ of the fitted decay line with the noise level and the missing tail is compensated by an exponential with the fitted rate; without that term the finite integral systematically underestimates $T$.

A squared impulse response, its Schroeder backward integral and the EDT/T20/T30 regression windows of the next subsection.

Regression windows and validity (ISO 3382-2, Clause 6, Annex B/C)

Reverberation time is a least-squares fit $L=a+bt$ over a window, extrapolated to 60 dB via $T=-60/b$ (Annex C): EDT on 0 to −10 dB, T20 on −5 to −25 dB, T30 on −5 to −35 dB. A single-slope decay gives EDT = T20 = T30; a fast early / slow late double slope gives EDT < T30. Validity uses the dynamic-range rule of 5.3.3: the noise must sit at least 25 dB below the IR peak for EDT (evaluation span + 15 dB), tightened to 46 dB for T20 and 54 dB for T30 so the tail-compensation bias of a flagged-valid value stays within the 5 % JND. The curvature $C=100(T_{30}/T_{20}-1)$ % (Annex B) flags a non-straight decay above 10 %.

The regression is only half of what makes $T$ a property of the room rather than of one point in it; the other half is spatial sampling, and ISO 3382-2 grades the result by it (clause 4.3, Table 1). The survey method needs at least one source position and two source-microphone combinations, one decay each, in octave bands only, for a nominal accuracy better than 10 %. The engineering method needs at least two source positions and six independent combinations with two decays each, and is the grade the sound-insulation standards call for (better than 5 % in octaves, 10 % in thirds). The precision method needs two source positions and twelve combinations with three decays each (2.5 % and 5 %). Independence is the operative word: microphone positions should be about half a wavelength apart, in practice a minimum of some 2 m, at least about a quarter wavelength — roughly 1 m — from any reflecting surface including the floor, never symmetric, and no closer to a source than $d_{\min }=2\sqrt{V/(cT)}$, so that direct sound does not dominate the early decay. Positions packed closer than that are not the independent positions Table 1 counts, and a $T$ averaged over them carries no grade at all.

Clarity, definition and centre time (ISO 3382-1, Annex A)

Splitting the energy at an early/late boundary $t_e$ gives the early-to-late index and the definition ratio:

$$C_{te}=10{\log }_{10}\frac{{\int }_0^{t_e}{p}^{2}dt}{{\int }_{t_e}^{\infty }{p}^{2}dt}\text{dB},D_{50}=\frac{{\int }_0^{0.05}{p}^{2}dt}{{\int }_0^{\infty }{p}^{2}dt},C_{50}=10{\log }_{10}\frac{D_{50}}{1-D_{50}},$$

with $t_e=50$ ms (C50, speech) or 80 ms (C80, music), and the centre time $T_s={\int }_0^{\infty }t{p}^{2}dt/{\int }_0^{\infty }{p}^{2}dt$. For a pure exponential decay these have closed forms $C_{te}=10{\log }_{10}(e^{a{t}_e}-1)$ and $T_s=1/a$; at $T=1$ s ($a=13.8155$) they evaluate to C80 = 3.05 dB, C50 = −0.02 dB, D50 = 0.499 and Ts = 72.4 ms, the values the implementation reproduces. Table A.1 JNDs (EDT 5 %, C80 1 dB, D50 0.05, Ts 10 ms) bound how finely each is worth reporting.

The closed forms above hold for a single exponential decay; a real room gives one set of values per band. The upper panel is the decay itself (EDT, T20 and T30 falling with frequency as air and surfaces absorb more), the lower panel the early/late split of the same impulse response, and C50 and C80 rise with frequency for the same reason the decay time falls — the later the energy, the more of it the room has already removed.

Open-plan spatial decay (ISO 3382-3, Clause 6)

The spatial decay rate of A-weighted speech is the ordinary least-squares slope of $L_{p,A,S}$ against ${\log }_{10}(r/r_0)$ ($r_0=1$ m) over the 2–16 m positions, rescaled to a per-doubling figure, and the nominal level is read off the same line at 4 m:

$$L=a+b{\log }_{10}(r/r_0),D_{2,S}=-{\log }_{10}(2)b,L_{p,A,S,4\text{m}}=a+b{\log }_{10}(4/r_0).$$

The distraction distance rD and privacy distance rP are the distances where a linear (not logarithmic) regression of STI against distance crosses 0.50 and 0.20; a non-negative fitted slope (STI not falling with distance) makes them undefined, realising the standard’s “can prove impossible to determine” note.

Two regressions on two different axes, which is what makes this clause hard to hold in the head. The level line is fitted against ${\log }_{10}(r/r_0)$ and read twice — as the slope $D_{2,S}$ per doubling, and at $r=4$ m for $L_{p,A,S,4\text{m}}$. The STI line is fitted against $r$ itself, linearly, and read where it crosses 0.50 and 0.20 for the distraction and privacy distances. If that second fit comes out flat or rising, the two distances do not exist rather than being large.

Sound insulation and absorption, measured (ISO 16283, ISO 10140, ISO 717, ISO 354)

Field insulation and weighted rating (ISO 16283-1, ISO 717-1)

Per one-third-octave band the level difference $D=L_1-L_2$ (energy-averaged over microphone positions, $L=10{\log }_{10}[(1/n){\sum }_{i}10^{L_i/10}]$) is normalised two ways: the standardized level difference $D_{nT}=D+10{\log }_{10}(T/T_0)$ with $T_0=0.5$ s (so $D_{nT}=D$ when $T=T_0$), and the apparent sound reduction index $R^{\prime }=D+10{\log }_{10}(S/A)$ with the Sabine absorption area $A=0.16V/T$, hence $R^{\prime }=D+10{\log }_{10}[ST/(0.16V)]$.

In a room below 25 m³ that default average is not the level ISO 16283-1 asks for in the three lowest bands. Clause 8 adds a corner measurement: the levels taken in the room corners are energy-averaged (Formula 12) and combined with the default average as $L_{LF}=10{\log }_{10}\left[(10^{0.1L_{corner}}+2\cdot 10^{0.1L})/3\right]$ (Formula 13), and that $L_{LF}$ replaces the default level in the 50, 63 and 80 Hz bands only, because the default positions do not sample the modal field there. The library consumes levels and does not perform this combination, so it has to be applied before calling. It matters because the extended spectrum adaptation terms $C_{50-3150}$ and $C_{tr,50-3150}$ are dominated by exactly those three bands: a small-room rating computed without the corner procedure is not comparable with one that used it.

The single-number rating (ISO 717-1, Clause 4.4) shifts the Table 3 reference curve in 1 dB steps toward the measured curve until the sum of unfavourable deviations ${\sum }_i\max (0,{\text{ref}}_i+k-{\text{meas}}_i)$ is maximal but $\le$ 32.0 dB (16 thirds) or 10.0 dB (5 octaves); the rating $R_w$ is the shifted reference at 500 Hz. The spectrum adaptation terms are $C=X_{A1}-X_w$ and $C_{tr}=X_{A2}-X_w$ with $X_{Aj}=-10{\log }_{10}{\sum }_{i}10^{(L_{ij}-X_i)/10}$ (Table 4 spectra No. 1 pink noise, No. 2 urban traffic), each rounded to an integer. The ISO 717-1 Annex C worked example ($R_w=30$, $C=-2$, $C_{tr}=-3$, unfavourable sum 31.8 dB) is reproduced exactly.

Reading $C$ and $C_{tr}$. The weighted rating condenses a whole curve against one fixed reference curve, so it answers for the reference source and no other. The adaptation terms put the source dependence back by re-rating the same measured curve against two standard spectra: No. 1 for sources whose spectrum is flat-ish and mid-and-high dominated — speech, television, everyday activity — and No. 2 for sources dominated by low frequencies, such as urban road traffic, distant aircraft and bass-heavy music. Both terms are almost always negative, and the value is the number of decibels to add to $R_w$ to get the level difference that source will actually experience, which is why regulations state requirements as $R_w+C$ or $R_w+C_{tr}$ and not as $R_w$ alone. That makes $C_{tr}$ a diagnostic: near $-2$ is ordinary for a homogeneous heavy wall, while values approaching $-8$ or worse mark a construction whose insulation collapses in the lowest bands — typically a lightweight double leaf near its mass-spring-mass resonance, or a window — so two partitions with the same $R_w$ can differ by many decibels under traffic. The terms are computed over the same band range as the rating they accompany, so an adaptation term quoted over 100–3150 Hz and one extended to 50 Hz are not comparable quantities.

A measured R spectrum against the shifted ISO 717-1 reference: the rating is the shifted reference read at 500 Hz.

Impact insulation (ISO 16283-2, ISO 717-2)

Impact insulation swaps the airborne source for a standardized tapping machine and rates the receiving-room level, so the sign conventions flip. The standardized and normalized impact levels are $L_{nT}^{\prime }=L_i-10{\log }_{10}(T/T_0)$ (the reverberation term is subtracted, opposite to $D_{nT}$) and $L_n^{\prime }=L_i+10{\log }_{10}(A/A_0)$ with $A_0=10$ m² and $A=0.16V/T$. The ISO 717-2 rating shifts the Table 3 reference curve until ${\sum }_i\max (0,{\text{meas}}_i-({\text{ref}}_i+k))$ is maximal but $\le$ 32.0 dB (16 thirds) or 10.0 dB (5 octaves); the unfavourable deviation now counts where the measurement exceeds the reference (impact noise is worse when louder), the mirror image of ISO 717-1. The rating is the shifted reference at 500 Hz, reduced by a further 5 dB for octave bands, and the adaptation term is $C_I=L_{n,\text{sum}}-15-L_{n,w}$ with the energetic sum $L_{n,\text{sum}}=10{\log }_{10}{\sum }_{i}10^{L_i/10}$ over 100–2500 Hz (thirds) or 125–2000 Hz (octaves). The ISO 717-2 Annex C examples are reproduced exactly (thirds $L_{n,w}=79$, $C_I=-11$; octaves $54$, $0$), via the same monotone shift search as ISO 717-1 run on the negated curves.

The mirror image of the airborne figure above, drawn so the flip is visible rather than asserted. There the unfavourable deviations were counted where the measurement fell below the reference; here they are counted where it rises above it, because a louder receiving room is a worse floor. Everything else is the same procedure: the reference curve shifted in 1 dB steps until the unfavourable sum is as large as it can be without passing 32.0 dB, and the rating read off the shifted reference at 500 Hz.

Absorption in a reverberation room (ISO 354)

Sound absorption (ISO 354) measures the equivalent absorption area from Sabine’s relation applied to a reverberation room empty and with the specimen: $A=55.3V/(cT)-4Vm$ (the $4Vm$ term is the air absorption, $m$ the power attenuation coefficient in 1/m), so the specimen area is $A_T=A_2-A_1$ and its coefficient ${\alpha }_s=A_T/S$. With the speed of sound from Eq. (6), $c=331+0.6t$ (°C), and $m$ converted from an ISO 9613-1 attenuation coefficient by $m=\alpha /(10{\log }_{10}e)$. Because diffraction and edge scattering intercept more than the flat sample area, ${\alpha }_s$ is left unclamped and may exceed 1.0 (Clause 3.7 NOTE 2).

The two reverberation times are not free inputs. They are the output of a tightly specified arrangement, and ${\alpha }_s$ means nothing without it. The room holds at least 150 m³ (200 m³ strongly recommended for a new one), its longest dimension stays below $1.9V^{1/3}$, and it carries enough fixed or rotating diffusion for the decays to be exponential (clause 6.1). The plane specimen covers 10 to 12 m² — the upper limit scaled by $(V/200{)}^{2/3}$ above 200 m³ — is rectangular with a width-to-length ratio between 0.7 and 1, sits at least 0.75 m from any room boundary and preferably has its edges non-parallel to the nearest room edge (6.2.1). Its mounting is part of the result (Annex B): rigid backing, air gap or framed edges change the answer, which is why a catalogue ${\alpha }_s$ without its mounting code is not usable and why the same product gives different curves in different laboratories. Each $T$ is the average of at least 12 spatially independent decays, from at least 3 microphone and 2 source positions, with microphones at least 1.5 m apart, 2 m from any source and 1 m from any room surface and from the specimen (7.1.4, 7.1.3). Both measurements are made between 30 % and 90 % relative humidity and above 15 °C, at nearly the same climate (6.3.2).

That last condition is not housekeeping. The result is the difference of two large and nearly equal absorption areas, so everything that changes between the empty and the loaded measurement lands in it: a shift in temperature or humidity moves both $c$ and the air term $4Vm$, which is why the two decays belong in the same session with the climate recorded. And an ${\alpha }_s$ above 1 is not an error but the signature of edge diffraction intercepting more energy than the flat projected area — largest for small specimens and at low frequency — so the value belongs to a specimen of that size, aspect ratio and mounting and cannot be transplanted to a surface of another shape without judgement.

Laboratory vs field normalization (ISO 10140, ISO 16283)

The field indices carry a prime because the receiving-room level they start from includes flanking transmission around the partition; the laboratory indices do not, because a qualified facility suppresses it. The formulas are therefore the same, and the prime is a statement about the specimen rather than about the arithmetic: the direct laboratory sound reduction index $R=L_1-L_2+10{\log }_{10}(S/A)$ (ISO 10140-2) and the apparent field index $R^{\prime }=L_1-L_2+10{\log }_{10}(S/A)$ (ISO 16283-1) are one closed form, evaluated with the facility’s known $A$ in the laboratory and with the receiving room’s measured $A=0.16V/T$ in the field. What differs in what is normalised is not laboratory against field but airborne against impact: the airborne indices scale by the area-to-absorption ratio $S/A$, the impact levels by $A/A_0$.

What “a qualified facility suppresses it” means physically is drawn above: an ISO 10140-5 suite is two structurally decoupled reverberation rooms of roughly 50 to 60 m³ sharing a test opening of about 10 m², built so that transmission around the specimen sits far below transmission through it. The facility measures its own flanking limit, and a specimen may be reported only up to that limit — which is why a laboratory $R$ carries no prime, and why the same construction built into a building reads several decibels lower. The practical consequence is a diagnosis: when a field $R^{\prime }$ refuses to rise after the element is improved, the measurement has hit a flanking path, not the element. The impact pair is the normalized laboratory level $L_n=L_i+10{\log }_{10}(A/A_0)$ (ISO 10140-3) versus the field $L_n^{\prime }$ (ISO 16283-2), both referenced to $A_0=10$ m². Before either is formed the receiving-room level is corrected for background noise by the energy subtraction $L=10{\log }_{10}(10^{L_{sb}/10}-10^{L_b/10})$ for a 6–15 dB signal-to-background margin, capped at a fixed $1.3$ dB (the limit of measurement) at or below 6 dB and omitted at or above 15 dB (ISO 10140-4, Clause 4.3), the laboratory analogue of the 6/10 dB rule of ISO 16283-1. The façade extension (ISO 16283-3) replaces the source-room level by the level 2 m in front of the façade, $D_{2m}=L_{1,2m}-L_2$, and adds a fixed angle-of-incidence correction to the element sound reduction index, $-1.5$ dB for the 45° loudspeaker method ($R_{45°}^{\prime }$) and $-3$ dB for the all-angle road-traffic method ($R_{tr,s}^{\prime }$); all three carry the ISO 717-1 airborne single number.

Sound insulation and absorption, predicted (EN 12354-1/-2/-6, Bies, Cremer, Hopkins)

Flanking transmission prediction (EN 12354-1/2)

The apparent field index is the energetic sum of the direct path $Dd$ and, for each flanking element $F=f$ across its junction with the separating element, the three paths $Ff$, $Df$ and $Fd$ (EN 12354-1, simplified single-number model, Formula 26):

The four labels are only meaningful on this drawing. Upper case is the source side and lower case the receiving side, so $Dd$ goes straight through the separating element, $Ff$ runs along a flanking element on both sides of the junction, and $Df$ and $Fd$ cross from one to the other. The junction they cross is where $K_{ij}$ and the coupling length ${\ell }_f$ live in the formula below, and the separating element is where $S_s$ is measured. Each junction contributes three flanking terms, so a room with four flanking elements sums thirteen paths.

$$R_w^{\prime }=-10{\log }_{10}[10^{-R_{Dd,w}/10}+\sum 10^{-R_{Ff,w}/10}+\sum 10^{-R_{Df,w}/10}+\sum 10^{-R_{Fd,w}/10}].$$

The direct path is $R_{Dd,w}=R_{s,w}+\Delta R_{Dd,w}$ (Formula 27), the separating-element laboratory index plus any lining improvement. Each flanking path (Formula 28a) is

$$R_{ij,w}=\frac{R_{i,w}+R_{j,w}}{2}+\Delta R_{ij,w}+K_{ij}+10{\log }_{10}\frac{S_s}{l_0{l}_f},$$

with $R_{i,w}$, $R_{j,w}$ the laboratory indices of the two elements meeting at the junction ($i$ source side, $j$ receiving side), $\Delta R_{ij,w}$ the combined lining improvement, $S_s$ the separating-element area, $l_f$ the junction coupling length and $l_0=1$ m the reference coupling length. $K_{ij}$ is the junction vibration reduction index (Annex E), an empirical function of the mass ratio $M={\log }_{10}(m_{\perp ,i}^{\prime }/m_i^{\prime })$: for a rigid cross-junction $K_{13}=8.7+17.1M+5.7M^2$ (through) and $K_{12}=8.7+5.7M^2$ (corner), read at 500 Hz, and floored at $K_{ij,\min }=10{\log }_{10}[l_f{l}_0(1/S_i+1/S_j)]$ (Formula 29). Two linings combine as $\max (a,b)+\min (a,b)/2$ (Formulas 30/31). The impact counterpart (EN 12354-2, Formula 21) is the direct subtraction $L_{n,w}^{\prime }=L_{n,w,eq}-\Delta L_w+K$, with the bare-floor equivalent level $L_{n,w,eq}=164-35{\log }_{10}(m^{\prime }/m_0^{\prime })$ (Annex B), the covering improvement $\Delta L_w$ (ISO 717-2) and the flanking correction $K$ from Table 1. The EN 12354-1 Annex H.3 ($R_w^{\prime }=52$ dB) and EN 12354-2 Annex E.3 ($L_{n,w}^{\prime }=45$ dB) worked examples are reproduced exactly.

Where the simplified model applies. It assumes homogeneous elements whose transmission is governed by their mass per unit area, joined rigidly, so that the junction attenuation can depend on the mass ratio alone; the Annex E regressions are empirical fits to exactly that family. They return no meaningful answer for framed lightweight constructions, for elastically decoupled junctions, or for elements with a strong internal loss mechanism — those need the detailed band-by-band model or measured $K_{ij}$ values. The $K_{ij,\min }$ floor exists to stop the regressions producing implausibly high junction attenuations for small elements, so a path that lands on the floor is a signal that the mass-ratio fit is at the edge of its range rather than a result to report as-is. Because the paths sum energetically, the apparent rating is pinned by whichever path is strongest: improving the separating element alone stops paying the moment one flanking path dominates, and identifying that path is the model’s chief practical value, well before the last decibel of the total. The 2 dB standard deviation the standard quotes (Clause 5) is the spread of the simplified model against measurement inside that intended population, not a general error bar.

Absorption in enclosed spaces (EN 12354-6)

EN 12354-6:2003 predicts the equivalent absorption area of a room from its parts (the normative Clause 4 model). The total (Formula 1) sums the surfaces, the objects and the air:

$$A=\sum _i{\alpha }_{s,i}{S}_i+\sum _j{A}_{obj,j}+\sum _k{\alpha }_{s,k}{S}_k+A_{air},A_{air}=4mV(1-\psi ),$$

with $m$ the power attenuation coefficient of air (Formula 2; Table 1 tabulates it for six temperature/humidity climates over the octave bands 125 Hz – 8 kHz), $\psi =\sum V_{obj}/V$ the volume fraction occupied by objects (Formula 3), and a hard irregular object approximated by $A_{obj}=V_{obj}^{2/3}$ (Formula 4). The reverberation time follows from Sabine applied to the free volume (clause 4.4, Formula 5):

$$T=\frac{55.3}{c_0}\frac{V(1-\psi )}{A},$$

with $c_0=345.6$ m/s chosen so that $55.3/c_0$ is the familiar $0.16$ (clause 4.4 NOTE). The three Annex E worked cases are reproduced: the bare 29.75 m³ room gives $A=2.26$ m² and $T=2.1$ s at 1 kHz, and adding hard objects ($\psi \approx 0.072$) raises $A$ to 5.03 m² and drops $T$ to 0.9 s. The informative Annex D method for irregular spaces and unevenly distributed absorption is out of scope.

What Formula 1 does band by band: the equivalent absorption area on the left and the reverberation time it implies through Formula 5 on the right, for the same room bare and treated. The Annex E case quoted above is the same arithmetic on a smaller room — $A$ from 2.26 to 5.03 m² and $T$ from 2.1 to 0.9 s at 1 kHz — and the figure shows why the two move in opposite directions and not proportionally.

See the Enclosed-Space Absorption guide for usage.

Predicted panel sound insulation (Bies 7.2, Hopkins 2.9/4.3.10, Cremer 5)

Where EN 12354 takes the element $R$ as a measurement, the sound reduction index of a panel can also be predicted from its physical properties. A limp panel follows the mass law $T{L}_0=10{\log }_{10}[1+(\pi f{m}^{\prime \prime }/{\rho }_0{c}_0{)}^2]$ (Bies Eq. 7.40), which rises 6 dB per octave and 6 dB per doubling of the surface mass $m^{\prime \prime }$ (the $m^{\prime }$ of the EN 12354 subsections above); the field-incidence value subtracts 5.5 dB (one-third octave). That 5.5 dB is not derived: it is the result of averaging the normal-incidence mass law over incidence angles up to the limiting angle observed in practice rather than over the full hemisphere, so it is a fixed empirical offset and not a function of anything. Stiffness adds a coincidence dip at $f_c=(c_0^2/2\pi )\sqrt{m^{\prime \prime }/B^{\prime }}$ (Eq. 7.3), where the free bending wavelength matches the acoustic trace wavelength. Sharp’s method holds the mass law to $f_c/2$, drops linearly in $\log f$ to the dip $TL=20{\log }_{10}(f_c{m}^{\prime \prime })+10{\log }_{10}\eta -44$ and rises again above $f_c$ with the loss factor $\eta$ (Eq. 7.44); the $-44$ there is an empirical fit to measured coincidence dips, and the loss factor appears explicitly because the depth of the dip is controlled by damping rather than by mass. A double wall is a mass-spring-mass system with the cavity as the spring: below $f_0=60\sqrt{(m_1+m_2)/(m_1{m}_{2}d)}$ (Eq. 7.62) it follows the mass law of the combined mass, and above it the two leaves’ mass laws add plus the cavity term $20{\log }_{10}(2kd)$, saturating at +6 dB beyond $f_l=c_0/(2\pi d)$ (Eq. 7.64); a porous fill lowers $f_0$. The 60 in $f_0$ is the numeric constant left once the stiffness of an air layer of thickness $d$ is substituted into the mass-spring-mass resonance, so it is fixed to air at ordinary conditions and changes for a gas-filled or evacuated cavity; and $f_l$ marks where half a wavelength fits across the cavity, above which the cavity stops behaving as a pure spring and standing waves in it limit what the construction can achieve. Small air paths cap any construction: the transmission coefficient of a straight slit (Gomperts, Hopkins Eq. 4.99, with resonances at $d+2e=z\lambda /2$) or a circular hole (Wilson & Soroka, Eq. 4.102) combines with the wall in the area-weighted energy sum $R=-10{\log }_{10}[(1/\sum S_n)\sum S_{n}10^{-R_n/10}]$ (Eq. 4.92), so a bare opening of relative area $S_a/S$ limits the composite to $10{\log }_{10}(S/S_a)$. The resonant transmission path and the double-wall radiation draw on the plate radiation efficiency and point mobilities of the vibration theory.

The four behaviours of the paragraph above, one per panel. Top left, the mass law rising 6 dB per octave with Sharp’s coincidence dip cut into it at $f_c$. Top right, the double wall: no better than the combined mass below $f_0$, then the cavity term climbing until it saturates. Bottom left, the radiation efficiency that decides how much of the plate’s vibration becomes sound. Bottom right, the ceiling a leak imposes: a 1 % open area holds the composite at $10{\log }_{10}(S/S_a)=20$ dB however good the wall is, which is the panel worth showing a client.

See the Predicting Panel Sound Insulation guide for usage.

Measurement uncertainty (ISO 12999-1)

ISO 12999-1 supplies the uncertainty of the quantities above from inter-laboratory (ISO 5725) reproducibility and repeatability rather than a GUM functional model. Three measurement situations fix the standard uncertainty $u$: situation A (laboratory characterisation) uses the reproducibility standard deviation ${\sigma }_R$; situation B (same location, different teams) the in-situ ${\sigma }_{situ}$; situation C (same location, operator and equipment, repeated) the repeatability ${\sigma }_r$. The per-band and single-number values are tabulated for airborne $R$/$R^{\prime }$/$D_n$/$D_{nT}$ (Tables 2/3), impact $L_n$/$L_n^{\prime }$ (Table 4 bands, situations B/C only; Table 5 ratings adding a situation-A estimate) and the covering reduction $\Delta L$ (Tables 6/7, situation A only). The expanded uncertainty is $U=ku$ (Formula 2) with the coverage factor $k$ of Table 8 (at 95 %, $k=1.96$ two-sided, $k=1.65$ one-sided; a minimum $k=1$ is enforced). A two-sided interval $Y=y\pm U$ reports a value (Formula 3); a one-sided factor declares conformity, $y-U>$ requirement for a lower limit (Formula 5) or $y+U<$ requirement for an upper limit (Formula 4). Uncorrelated components combine in quadrature $u_c=\sqrt{\sum u_i^2}$ (Formula C.2), $m$ independent measurements reduce $u$ to $u/\sqrt{m}$ (Formula A.7), and the uncorrelated single-number uncertainty is the energy-weighted quadrature sum of the band uncertainties (Formula B.2).

See the Room Acoustics and Field Insulation Measurement (ISO 16283) guides for usage.

References

• Acoustical Society of America. (2019). Criteria for evaluating room noise (ANSI/ASA S12.2-2019). The normative NC tangency method and the Annex D RC Mark II rating with its spectral tag.
• Beranek, L. L. (1957). Revised criteria for noise in buildings. Noise Control, 3(1), 19-27. https://doi.org/10.1121/1.2369239The original NC curves rated by the tangency method of the room-noise section.
• Bies, D. A., Hansen, C. H., & Howard, C. Q. (2018). Engineering noise control (5th ed.). CRC Press. https://doi.org/10.1201/9781351228152ISBN 978-1-138-30690-5. The mass law, coincidence, double-wall and composite-partition equations (7.3, 7.40, 7.44, 7.62, 7.64) of the panel-prediction subsection.
• Blazier, W. E. (1997). RC Mark II: A refined procedure for rating the noise of heating, ventilating, and air-conditioning (HVAC) systems in buildings. Noise Control Engineering Journal, 45(6), 243-250. https://doi.org/10.3397/1.2828446The RC Mark II contour and spectral-quality tag codified by ANSI/ASA S12.2 Annex D.
• Cremer, L., Heckl, M., & Petersson, B. A. T. (2005). Structure-borne sound: Structural vibrations and sound radiation at audio frequencies (3rd ed.). Springer. https://doi.org/10.1007/b137728ISBN 978-3-540-22696-3. The structure-borne transmission and radiation theory the panel-prediction subsection draws on, shared with the vibration theory page.
• European Committee for Standardization. (2000). Building acoustics — Estimation of acoustic performance of buildings from the performance of elements — Part 1: Airborne sound insulation between rooms (EN 12354-1:2000). The simplified single-number flanking model (Formulae 26-31), the Annex E junction regressions and the Annex H.3 worked example of the flanking subsection.
• European Committee for Standardization. (2000). Building acoustics — Estimation of acoustic performance of buildings from the performance of elements — Part 2: Impact sound insulation between rooms (EN 12354-2:2000). The impact counterpart of the same subsection: Formula 21, the Annex B bare-floor equivalent level and the Annex E.3 worked example.
• European Committee for Standardization. (2003). Building acoustics — Estimation of acoustic performance of buildings from the performance of elements — Part 6: Sound absorption in enclosed spaces (EN 12354-6:2003). The Clause 4 absorption model and Annex E worked cases of the enclosed-space section. The linked catalogue record is the BSI Knowledge page for BS EN 12354-6:2003.
• Hak, C. C. J. M., Wenmaekers, R. H. C., & van Luxemburg, L. C. J. (2012). Measuring room impulse responses: Impact of the decay range on derived room acoustic parameters. Acta Acustica united with Acustica, 98(6), 907-915. https://doi.org/10.3813/aaa.918574The INR decay-range analysis behind the tightened T20/T30 validity thresholds.
• Hopkins, C. (2007). Sound insulation. Butterworth-Heinemann. https://doi.org/10.4324/9780080550473ISBN 978-0-7506-6526-1. The measurement chains, flanking transmission and EN 12354 prediction framework of the insulation sections.
• International Organization for Standardization. (2003). Acoustics — Measurement of sound absorption in a reverberation room (ISO 354:2003). The reverberation-room absorption measurement and its air-absorption term.
• International Organization for Standardization. (2006). Acoustics — Application of new measurement methods in building and room acoustics (ISO 18233:2006). The swept-sine and MLS deconvolution of the deterministic-excitation section.
• International Organization for Standardization. (2008). Acoustics — Measurement of room acoustic parameters — Part 2: Reverberation time in ordinary rooms (ISO 3382-2:2008). The regression windows, dynamic-range rules and curvature check of the validity section.
• International Organization for Standardization. (2009). Acoustics — Measurement of room acoustic parameters — Part 1: Performance spaces (ISO 3382-1:2009). Backward integration, the parameter definitions and the Annex A clarity family.
• International Organization for Standardization. (2012). Acoustics — Measurement of room acoustic parameters — Part 3: Open plan offices (ISO 3382-3:2012). The open-plan spatial decay and the distraction and privacy distances. Since revised as ISO 3382-3:2022 (https://www.iso.org/standard/77437.html); the 2012 edition is the implemented one.
• International Organization for Standardization. (2014). Acoustics — Field measurement of sound insulation in buildings and of building elements — Part 1: Airborne sound insulation (ISO 16283-1:2014). The field level differences and normalizations of the insulation sections.
• International Organization for Standardization. (2016). Acoustics — Field measurement of sound insulation in buildings and of building elements — Part 3: Façade sound insulation (ISO 16283-3:2016). The façade extension of the laboratory-versus-field subsection: the 2 m reference position and the angle-of-incidence corrections.
• International Organization for Standardization. (2020). Acoustics — Determination and application of measurement uncertainties in building acoustics — Part 1: Sound insulation (ISO 12999-1:2020). The measurement situations, tabulated uncertainties and coverage factors of the uncertainty section.
• International Organization for Standardization. (2020). Acoustics — Field measurement of sound insulation in buildings and of building elements — Part 2: Impact sound insulation (ISO 16283-2:2020). The field impact-level measurement behind L'nT and L'n in the impact-insulation subsection.
• International Organization for Standardization. (2020). Acoustics — Rating of sound insulation in buildings and of building elements — Part 1: Airborne sound insulation (ISO 717-1:2020). The reference-curve shift and the spectrum adaptation terms C and Ctr.
• International Organization for Standardization. (2020). Acoustics — Rating of sound insulation in buildings and of building elements — Part 2: Impact sound insulation (ISO 717-2:2020). The impact single-number rating and the CI adaptation term, with the Annex C worked examples the impact-insulation subsection reproduces.
• International Organization for Standardization. (2021). Acoustics — Laboratory measurement of sound insulation of building elements — Part 2: Measurement of airborne sound insulation (ISO 10140-2:2021). The laboratory sound reduction index R of the laboratory-versus-field subsection.
• International Organization for Standardization. (2021). Acoustics — Laboratory measurement of sound insulation of building elements — Part 3: Measurement of impact sound insulation (ISO 10140-3:2021). The laboratory normalized impact level Ln of the same subsection.
• International Organization for Standardization. (2021). Acoustics — Laboratory measurement of sound insulation of building elements — Part 4: Measurement procedures and requirements (ISO 10140-4:2021). Clause 4.3, the background-noise correction and its 6/15 dB margins and 1.3 dB cap, quoted in the laboratory-versus-field subsection.
• International Organization for Standardization. (2021). Acoustics — Laboratory measurement of sound insulation of building elements — Part 5: Requirements for test facilities and equipment (ISO 10140-5:2021). The qualified transmission suite drawn in the laboratory-versus-field subsection, and the flanking limit that lets a laboratory index drop its prime.
• Kuttruff, H. (2016). Room acoustics (6th ed.). CRC Press. https://doi.org/10.1201/9781315372150The statistical decay theory behind backward integration and the Sabine relations used throughout this page.
• Schroeder, M. R. (1965). New method of measuring reverberation time. The Journal of the Acoustical Society of America, 37(3), 409-412. https://doi.org/10.1121/1.1909343The backward-integration method of the decay-curve section.
• Vigran, T. E. (2008). Building acoustics. CRC Press. https://doi.org/10.1201/9781482266016ISBN 978-0-415-42853-8. Sound transmission in buildings, from single and double constructions to floating floors.