Source: https://docs.jmrp.io/phonometry/reference/theory/rooms-buildings/ # Theory: 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](https://docs.jmrp.io/phonometry/reference/theory/); surface scattering and acoustic material characterisation live on [Materials and Surfaces](https://docs.jmrp.io/phonometry/reference/theory/materials-surfaces/). ## 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** follows the two-step procedure of clause 5.2.2 on the Table 1 curves (NC-15 to NC-70): the speech interference level $\mathrm{SIL} = \tfrac14(L_{500}+L_{1000}+L_{2000}+L_{4000})$ (clause 3.2) selects the NC-(SIL) curve, and if no band exceeds it the spectrum is designated NC-(SIL); otherwise the tangency method (clause 5.2.3) applies: each measured band is interpolated against the tabulated curve values, the rating is the highest per-band index and the band that sets it is the governing band; the interpolation makes the rating continuous (an NC-42.5 is reported as such, not snapped to a curve). Spectra above NC-70 or below NC-15 fall outside the family and are flagged (>NC-70 with the band of maximum exceedance, or *The same spectrum rated both ways: NC tangency at the governing band (left) and the RC Mark II reference with the rumble excess (right).* See the [Room Noise guide](https://docs.jmrp.io/phonometry/buildings/rooms/room-noise/) for usage. ## Room and building acoustics (ISO 18233, ISO 3382, ISO 16283, ISO 10140, EN 12354, ISO 12999, ISO 717, ISO 354) ### 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 $2 \pi f(t)$: $$ \varphi(t) = \frac{2 \pi f_1 T}{\ln(f_2/f_1)} \left[ \left( \frac{f_2}{f_1} \right)^{t/T} - 1 \right] . $$ A constant time-per-octave makes the ESS spectrum pink (−3 dB/octave). Deconvolution is done by **linear** (non-circular, zero-padded) spectral division $H = Y\ \overline{X} / (|X|^2 + \varepsilon)$, the Tikhonov term $\varepsilon$ (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 $h = \operatorname{xcorr}_{\text{circ}}(\text{recorded}, \text{mls}) / 2^N$; synchronous averaging of $n$ periods adds $10 \log_{10} n$ dB. ### Schroeder backward integration (ISO 3382-1, 5.3.3) The band decay curve is the **backward-integrated** squared IR (Schroeder): $$ E(t) = \int_t^{\infty} p^2(\tau)\ d\tau = \int_0^{\infty} p^2\ d\tau - \int_0^t p^2\ d\tau , \qquad L(t) = 10 \log_{10} \frac{E(t)}{E(0)}\ \text{dB}, $$ 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 $p^2(t) = \mathrm{e}^{-a t}$ it gives $E(t) = \mathrm{e}^{-a t}/a$, an exactly straight line $L(t) = -(10 a / \ln 10)\ t$. 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 $L = a + b t$ 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 %. ### Clarity, definition and centre time (ISO 3382-1, Annex A) Splitting the energy at an early/late boundary $t_\mathrm{e}$ gives the early-to-late index and the definition ratio: $$ C_{t_\mathrm{e}} = 10 \log_{10} \frac{\int_0^{t_\mathrm{e}} p^2\ dt}{\int_{t_\mathrm{e}}^{\infty} p^2\ dt}\ \text{dB}, \qquad D_{50} = \frac{\int_0^{0.05} p^2\ dt}{\int_0^{\infty} p^2\ dt}, \qquad C_{50} = 10 \log_{10} \frac{D_{50}}{1 - D_{50}}, $$ with $t_\mathrm{e} = 50$ ms (C50, speech) or 80 ms (C80, music), and the **centre time** $T_\mathrm{S} = \int_0^{\infty} t\ p^2\ dt / \int_0^{\infty} p^2\ dt$. For a pure exponential decay these have closed forms $C_{t_\mathrm{e}} = 10 \log_{10}(\mathrm{e}^{a t_\mathrm{e}} - 1)$ and $T_\mathrm{S} = 1/a$; 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 $L_{p,\mathrm{A,S}}$ 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: $$ L = a + b\ \log_{10}(r/r_0), \qquad D_{2,\mathrm{S}} = -\log_{10}(2)\ b, \qquad L_{p,\mathrm{A,S},4\,\mathrm{m}} = a + b\ \log_{10}(4/r_0). $$ 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 fitted slope that does not fall (STI not decreasing with distance; a constant STI fits to a slope a few units in the last place either side of zero, read as no decrease) 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 $D_{2,\mathrm{S}}$ per doubling, and at $r = 4$ m for $L_{p,\mathrm{A,S},4\,\mathrm{m}}$. 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.* ### Image-source room impulse response (Kuttruff 4.1, Vorländer 11) A rectangular room reflects a point source in its walls; each reflection equals the free-field sound of a **mirror image** of the source. Mirroring a coordinate in a wall ($S_n = S - 2 d\,\mathbf{n}$, Vorländer Eq. 11.36) turns the source into a regular lattice of images, and the room impulse response is the sum of the direct sound and one delayed, attenuated impulse per image (Kuttruff Eqs. 4.4–4.5), $$ g(t) = \sum_n A_n\ \delta(t - t_n), \qquad A_n = \frac{1}{4\pi r_n}\ \mathrm{e}^{-m r_n / 2} \prod_{\text{walls}} R_q^{\,k_{q,n}}, \qquad t_n = \frac{r_n}{c}, $$ with the $1/(4\pi r_n)$ spherical spreading, the product of the wall **pressure reflection factors** $R_q = \sqrt{1 - \alpha_q}$ (Vorländer Eq. 11.39; $|R|^2 = 1-\alpha$ in energy) each raised to the number of reflections $k_{q,n}$ that image made off wall $q$, and the air pressure attenuation $\mathrm{e}^{-m r_n/2}$ ($m$ the *intensity* attenuation constant). Along one axis the reflection counts of the image at lattice index $n$ and parity $p$ are $|n-p|$ and $|n|$ off the two walls (Allen & Berkley 1979), so the total order is $\sum_i |2 n_i - p_i|$; a shoebox has $\tfrac{2}{3}(2 i_0^3 + 3 i_0^2 + 4 i_0)$ audible images up to order $i_0$ (Kuttruff Eq. 9.23), and the reflection density grows as $\mathrm{d}N/\mathrm{d}t = 4\pi c^3 t^2 / V$ (Kuttruff Eq. 4.6). The initial decay rate of the specular reverberant energy recovers the **Eyring** reverberation time $T = -24 V \ln 10 / (c S \ln(1 - \bar\alpha))$ (Kuttruff Eq. 5.23), because the mean reflection rate $cS/4V$ equals $\tfrac{c}{2}(1/L_x + 1/L_y + 1/L_z)$. The match is exact only near cubic geometry; an elongated room sustains energy along its long axis, so the pure specular decay runs slower than Eyring's diffuse-field estimate (the anisotropy the Fitzroy/Arau-Puchades models correct). The model is specular only (no diffraction or diffuse scattering) and exact only for real, angle-independent reflection factors. ### Steady-state room field (Bies 6.4, Kuttruff 5.6) A source of constant power sets up a steady level made of a direct field falling with distance and a diffuse **reverberant** field that is (approximately) uniform. With the **room constant** $R = S\bar\alpha/(1-\bar\alpha)$ (Bies Eq. 6.44) and the directivity factor $Q$, $$ L_p = L_W + 10 \log_{10}\!\left( \frac{Q}{4\pi r^2} + \frac{4}{R} \right) \left[ + 10 \log_{10}\frac{\rho c}{400} \right] \quad \text{(Bies Eq. 6.43)}, $$ the optional last term (about $+0.14$ dB at 20 °C) correcting a characteristic impedance away from 400 Pa·s/m. The **critical distance** $r_\mathrm{c} = \sqrt{Q R / 16\pi}$ is where the two fields cross; Kuttruff's reverberation distance (Eq. 5.44, $r_\mathrm{c} = \sqrt{A/16\pi}$ for $Q=1$) uses the Sabine area $A = S\bar\alpha$ instead of $R = A/(1-\bar\alpha)$, the two coinciding for small $\bar\alpha$. The **Schroeder frequency** $f_\mathrm{s} = 2000\sqrt{T/V}$ (Kuttruff Eq. 3.44) roughly marks the modal-to-diffuse transition, a heuristic crossover rather than a sharp cutoff: well below it discrete room modes dominate and these diffuse-field relations grow unreliable, well above it the modes overlap and the relations hold. Borderline rooms still warrant a band-by-band check. See the [Image sources and steady-state field guide](https://docs.jmrp.io/phonometry/buildings/rooms/room-image-sources/). ### 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 $D_{\mathrm{n}T} = D + 10 \log_{10}(T/T_0)$ with $T_0 = 0.5$ s (so $D_{\mathrm{n}T} = D$ when $T = T_0$), and the apparent sound reduction index $R' = D + 10 \log_{10}(S/A)$ with the Sabine absorption area $A = 0.16\ V / T$, hence $R' = D + 10 \log_{10}[S T / (0.16\ V)]$. 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 $R_\mathrm{w}$ is the shifted reference at 500 Hz. The **spectrum adaptation terms** are $C = X_{\mathrm{A}1} - X_\mathrm{w}$ and $C_\mathrm{tr} = X_{\mathrm{A}2} - X_\mathrm{w}$ with $X_{\mathrm{A}j} = -10 \log_{10} \sum_i 10^{(L_{ij} - X_i)/10}$ (Table 4 spectra No. 1 pink noise, No. 2 urban traffic), each rounded to an integer. The ISO 717-1 Annex C worked example ($R_\mathrm{w} = 30$, $C = -2$, $C_\mathrm{tr} = -3$, unfavourable sum 31.8 dB) is reproduced exactly. *A measured R spectrum against the shifted ISO 717-1 reference: the rating is the shifted reference read at 500 Hz.* ### Impact insulation and absorption (ISO 16283-2, ISO 717-2, ISO 354) 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 $L'_{\mathrm{n}T} = L_i - 10 \log_{10}(T/T_0)$ (the reverberation term is *subtracted*, opposite to $D_{\mathrm{n}T}$) and $L'_\mathrm{n} = L_i + 10 \log_{10}(A/A_0)$ with $A_0 = 10$ m² and $A = 0.16\ V/T$. 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 $C_\mathrm{I} = L_\mathrm{n,sum} - 15 - L_\mathrm{n,w}$ with the energetic sum $L_\mathrm{n,sum} = 10 \log_{10} \sum_i 10^{L_i/10}$ over 100–2500 Hz (thirds) or 125–2000 Hz (octaves). The ISO 717-2 Annex C examples are reproduced exactly (thirds $L_\mathrm{n,w} = 79$, $C_\mathrm{I} = -11$; 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.* Sound absorption (ISO 354) measures the equivalent absorption area from Sabine's relation applied to a reverberation room empty and with the specimen: $A = 55.3\ V/(c\ T) - 4 V m$ (the $4 V m$ term is the air absorption, $m$ the power attenuation coefficient in 1/m), so the specimen area is $A_\mathrm{T} = A_2 - A_1$ and its coefficient $\alpha_\mathrm{s} = A_\mathrm{T}/S$. With the speed of sound from Eq. (6), $c = 331 + 0.6\ t$ (°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, $\alpha_\mathrm{s}$ is left unclamped and may exceed 1.0 (Clause 3.7 NOTE 2). ### Laboratory vs field normalization (ISO 10140, ISO 16283) The field indices carry a prime because they include flanking transmission around the partition; the laboratory indices do not, because a qualified facility suppresses it. The algebra is otherwise identical, differing only in which quantity is normalised. The airborne pair is the direct laboratory sound reduction index $R = L_1 - L_2 + 10 \log_{10}(S/A)$ (ISO 10140-2) versus the apparent field index $R' = L_1 - L_2 + 10 \log_{10}(S/A)$ (ISO 16283-1), the same closed form evaluated with the facility's known $A$ or the room's measured $A = 0.16\ V/T$. The impact pair is the normalized laboratory level $L_\mathrm{n} = L_i + 10 \log_{10}(A/A_0)$ (ISO 10140-3) versus the field $L'_\mathrm{n}$ (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 $L = 10 \log_{10}(10^{L_\mathrm{sb}/10} - 10^{L_\mathrm{b}/10})$ 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, $D_{2\mathrm{m}} = L_{1,2\mathrm{m}} - L_2$, and adds a fixed angle-of-incidence correction to the element sound reduction index, $-1.5$ dB for the 45° loudspeaker method ($R'_{45°}$) and $-3$ dB for the all-angle road-traffic method ($R'_\mathrm{tr,s}$); all three carry the ISO 717-1 airborne single number. ### 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 $\ell_\mathrm{f}$ live in the formula below, and the separating element is where $S_\mathrm{s}$ is measured. Each junction contributes three flanking terms, so a room with four flanking elements sums thirteen paths.* $$ R'_\mathrm{w} = -10 \log_{10}\Big[ 10^{-R_\mathrm{Dd,w}/10} + \sum 10^{-R_\mathrm{Ff,w}/10} + \sum 10^{-R_\mathrm{Df,w}/10} + \sum 10^{-R_\mathrm{Fd,w}/10} \Big]. $$ The direct path is $R_\mathrm{Dd,w} = R_\mathrm{s,w} + \Delta R_\mathrm{Dd,w}$ (Formula 27), the separating-element laboratory index plus any lining improvement. Each flanking path (Formula 28a) is $$ R_{ij,\mathrm{w}} = \frac{R_{i,\mathrm{w}} + R_{j,\mathrm{w}}}{2} + \Delta R_{ij,\mathrm{w}} + K_{ij} + 10 \log_{10}\frac{S_\mathrm{s}}{l_0\ l_\mathrm{f}}, $$ with $R_{i,\mathrm{w}}$, $R_{j,\mathrm{w}}$ the laboratory indices of the two elements meeting at the junction ($i$ source side, $j$ receiving side), $\Delta R_{ij,\mathrm{w}}$ the combined lining improvement, $S_\mathrm{s}$ the separating-element area, $l_\mathrm{f}$ 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 $M = \log_{10}(m'_{\perp,i}/m'_i)$: for a rigid cross-junction $K_{13} = 8.7 + 17.1 M + 5.7 M^2$ (through) and $K_{12} = 8.7 + 5.7 M^2$ (corner), read at 500 Hz, and floored at $K_{ij,\min} = 10 \log_{10}[l_\mathrm{f}\ l_0 (1/S_i + 1/S_j)]$ (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 $L'_\mathrm{n,w} = L_\mathrm{n,w,eq} - \Delta L_\mathrm{w} + K$, with the bare-floor equivalent level $L_\mathrm{n,w,eq} = 164 - 35 \log_{10}(m'/m'_0)$ (Annex B), the covering improvement $\Delta L_\mathrm{w}$ (ISO 717-2) and the flanking correction $K$ from Table 1. The EN 12354-1 Annex H.3 ($R'_\mathrm{w} = 52$ dB) and EN 12354-2 Annex E.3 ($L'_\mathrm{n,w} = 45$ dB) worked examples are reproduced exactly; the simplified model is stated to have about a 2 dB standard deviation (Clause 5). ### 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: $$ A = \sum_i \alpha_{\mathrm{s},i}\ S_i + \sum_j A_{\mathrm{obj},j} + \sum_k \alpha_{\mathrm{s},k}\ S_k + A_\mathrm{air}, \qquad A_\mathrm{air} = 4\ m\ V\ (1 - \psi), $$ 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), $\psi = \sum V_\mathrm{obj} / V$ the volume fraction occupied by objects (Formula 3), and a hard irregular object approximated by $A_\mathrm{obj} = V_\mathrm{obj}^{2/3}$ (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](https://docs.jmrp.io/phonometry/buildings/rooms/enclosed-space-absorption/) 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 $\sigma_\mathrm{R}$; situation **B** (same location, different teams) the in-situ $\sigma_\mathrm{situ}$; situation **C** (same location, operator and equipment, repeated) the repeatability $\sigma_\mathrm{r}$. The per-band and single-number values are tabulated for airborne $R$/$R'$/$D_\mathrm{n}$/$D_{\mathrm{n}T}$ (Tables 2/3), impact $L_\mathrm{n}$/$L'_\mathrm{n}$ (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 $U = k\ u$ (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 $u_\mathrm{c} = \sqrt{\sum u_i^2}$ (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](https://docs.jmrp.io/phonometry/buildings/rooms/room-acoustics/) and [Field Insulation Measurement (ISO 16283)](https://docs.jmrp.io/phonometry/buildings/insulation/insulation-field/) guides 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 $TL_0 = 10\log_{10}[1 + (\pi f m''/\rho_0 c_0)^2]$ (Bies Eq. 7.40), which rises 6 dB per octave and 6 dB per doubling of the surface mass $m''$; the field-incidence value subtracts 5.5 dB (one-third octave). Stiffness adds a **coincidence dip** at $f_\mathrm{c} = (c_0^2/2\pi)\sqrt{m''/B'}$ (Eq. 7.3), where the free bending wavelength matches the acoustic trace wavelength. Sharp's method holds the mass law to $f_\mathrm{c}/2$, drops linearly in $\log f$ to the dip $TL = 20\log_{10}(f_\mathrm{c} m'') + 10\log_{10}\eta - 44$ and rises again above $f_\mathrm{c}$ with the loss factor $\eta$ (Eq. 7.44). 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 $f_\mathrm{l} = c_0/(2\pi d)$ (Eq. 7.64); a porous fill lowers $f_0$. 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 $S_\mathrm{a}/S$ limits the composite to $10\log_{10}(S/S_\mathrm{a})$. The resonant transmission path and the double-wall radiation draw on the plate radiation efficiency and point mobilities of the [vibration theory](https://docs.jmrp.io/phonometry/reference/theory/vibration/). The prediction is clean-room from Bies, Hansen & Howard (2017), Hopkins (2007) and Cremer, Heckl & Petersson (2005). *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 $f_\mathrm{c}$. 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 $10\log_{10}(S/S_\mathrm{a}) = 20$ dB however good the wall is, which is the panel worth showing a client.* See the [Predicting Panel Sound Insulation](https://docs.jmrp.io/phonometry/buildings/design/panel-sound-insulation/) guide for usage. ## References - Kuttruff, H. (2016). *Room acoustics* (6th ed.). CRC Press. [doi:10.1201/9781315372150](https://doi.org/10.1201/9781315372150). The statistical decay theory behind backward integration and the Sabine relations used throughout this page, the image-source construction (Section 4.1), the Eyring reverberation and reverberation distance (Sections 5.5–5.6) and the Schroeder frequency (Section 3.6). - Vorländer, M. (2020). *Auralization: Fundamentals of acoustics, modelling, simulation, algorithms and acoustic virtual reality* (2nd ed.). Springer. [doi:10.1007/978-3-030-51202-6](https://doi.org/10.1007/978-3-030-51202-6). The image-source / mirror-source model of the impulse-response section (Chapter 11). - Allen, J. B., & Berkley, D. A. (1979). Image method for efficiently simulating small-room acoustics. *The Journal of the Acoustical Society of America*, 65(4), 943-950. [doi:10.1121/1.382599](https://doi.org/10.1121/1.382599). The reflection-count decomposition of the rectangular-room image lattice. - Bies, D. A., Hansen, C. H., & Howard, C. Q. (2017). *Engineering noise control* (5th ed.). CRC Press. [doi:10.1201/9781351228152](https://doi.org/10.1201/9781351228152). The steady-state room field and the room constant (Section 6.4). - Schroeder, M. R. (1965). New method of measuring reverberation time. *The Journal of the Acoustical Society of America*, 37(3), 409-412. [doi:10.1121/1.1909343](https://doi.org/10.1121/1.1909343). The backward-integration method of the decay-curve section. - 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. [doi:10.3813/aaa.918574](https://doi.org/10.3813/aaa.918574). The INR decay-range analysis behind the tightened T20/T30 validity thresholds. - Beranek, L. L. (1957). Revised criteria for noise in buildings. *Noise Control*, 3(1), 19-27. [doi:10.1121/1.2369239](https://doi.org/10.1121/1.2369239). The original NC curves rated by the tangency method of the room-noise section. - 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. [doi:10.3397/1.2828446](https://doi.org/10.3397/1.2828446). The RC Mark II contour and spectral-quality tag codified by ANSI/ASA S12.2 Annex D. - Hopkins, C. (2007). *Sound insulation*. Butterworth-Heinemann. ISBN 978-0-7506-6526-1. [doi:10.4324/9780080550473](https://doi.org/10.4324/9780080550473). The measurement chains, flanking transmission and EN 12354 prediction framework of the insulation sections. - Vigran, T. E. (2008). *Building acoustics*. CRC Press. ISBN 978-0-415-42853-8. [doi:10.1201/9781482266016](https://doi.org/10.1201/9781482266016). Sound transmission in buildings, from single and double constructions to floating floors. - Acoustical Society of America. (2019). *Criteria for evaluating room noise* (ANSI/ASA S12.2-2019). [ANSI webstore](https://webstore.ansi.org/standards/asa/ansiasas122019). The normative NC tangency method and the Annex D RC Mark II rating with its spectral tag. - International Organization for Standardization. (2006). *Acoustics — Application of new measurement methods in building and room acoustics* (ISO 18233:2006). [iso.org catalogue](https://www.iso.org/standard/40408.html). The swept-sine and MLS deconvolution of the deterministic-excitation section. - International Organization for Standardization. (2009). *Acoustics — Measurement of room acoustic parameters — Part 1: Performance spaces* (ISO 3382-1:2009). [iso.org catalogue](https://www.iso.org/standard/40979.html). Backward integration, the parameter definitions and the Annex A clarity family. - International Organization for Standardization. (2008). *Acoustics — Measurement of room acoustic parameters — Part 2: Reverberation time in ordinary rooms* (ISO 3382-2:2008). [iso.org catalogue](https://www.iso.org/standard/36201.html). The regression windows, dynamic-range rules and curvature check of the validity section. - International Organization for Standardization. (2012). *Acoustics — Measurement of room acoustic parameters — Part 3: Open plan offices* (ISO 3382-3:2012; since revised as [ISO 3382-3:2022](https://www.iso.org/standard/77437.html)). [iso.org catalogue](https://www.iso.org/standard/46520.html). The open-plan spatial decay and the distraction and privacy distances. - 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). [iso.org catalogue](https://www.iso.org/standard/55997.html). The field level differences and normalizations of the insulation sections. - 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). [iso.org catalogue](https://www.iso.org/standard/77435.html). The reference-curve shift and the spectrum adaptation terms C and Ctr. - International Organization for Standardization. (2003). *Acoustics — Measurement of sound absorption in a reverberation room* (ISO 354:2003). [iso.org catalogue](https://www.iso.org/standard/34545.html). The reverberation-room absorption measurement and its air-absorption term. - 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). [BSI Knowledge record (BS EN 12354-6:2003)](https://knowledge.bsigroup.com/products/building-acoustics-estimation-of-acoustic-performance-of-buildings-from-the-performance-of-elements-sound-absorption-in-enclosed-spaces). The Clause 4 absorption model and Annex E worked cases of the enclosed-space section. - International Organization for Standardization. (2020). *Acoustics — Determination and application of measurement uncertainties in building acoustics — Part 1: Sound insulation* (ISO 12999-1:2020). [iso.org catalogue](https://www.iso.org/standard/73930.html). The measurement situations, tabulated uncertainties and coverage factors of the uncertainty section.