Source: https://jmrplens.github.io/phonometry/buildings/rooms/reverberation-prediction/ # Reverberation-time prediction (Sabine · Eyring · Fitzroy · Arau-Puchades) The **reverberation time** $T$, the time for the sound-energy level to fall by 60 dB after the source stops, is predicted here from a room's **volume**, **boundary areas** and the **sound-absorption coefficients** of its surfaces, through the classical statistical-acoustics formulae. This is the design-stage counterpart of the *measured* reverberation time of [Room Acoustics](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/) (ISO 3382) and complements the EN 12354-6 model of [Sound absorption in enclosed spaces](https://jmrplens.github.io/phonometry/buildings/rooms/enclosed-space-absorption/), which specialises the same physics to that standard's Clause 4. phonometry offers five models, ordered by how much they account for a **non-uniform** absorption distribution: | Model | Absorption term in $T = k\,V / (\text{term} + 4mV)$ | Best for | |:---|:---|:---| | **Sabine** | $A = \sum_i S_i\alpha_i$ | low, uniform absorption | | **Eyring** (Norris-Eyring) | $-S\ln(1-\bar\alpha)$ | strong, uniform absorption | | **Millington-Sette** | $-\sum_i S_i\ln(1-\alpha_i)$ | a few very absorptive surfaces | | **Fitzroy** | area-weighted **arithmetic** mean of three axial Eyring times | anisotropic rooms | | **Arau-Puchades** | area-weighted **geometric** mean of the same three | anisotropic rooms (author-preferred) | with the Sabine constant $k = 24\ln 10 / c_0$ (so $k = 0.161$ for $c_0 = 343\ \mathrm{m/s}$) and the air-absorption term $4mV$.

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```python import matplotlib.pyplot as plt from phonometry import environment, room # A 10 x 7 x 3.5 m room: hard end walls, lightly treated side walls and a # very absorptive floor/ceiling pair (carpet plus an acoustic ceiling). bands = [125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0] alpha_x = [0.06, 0.07, 0.08, 0.09, 0.10, 0.10] alpha_y = [0.12, 0.14, 0.16, 0.18, 0.20, 0.20] alpha_z = [0.30, 0.50, 0.65, 0.78, 0.82, 0.80] m = environment.air_attenuation_m(bands, 20.0, 50.0) # air at 20 C / 50 % RH res = room.reverberation_time_models((10.0, 7.0, 3.5), (alpha_x, alpha_y, alpha_z), air_attenuation=m, frequencies=bands) res.plot() # the five model curves per band plt.show() ```

## 1. Sabine, Eyring and Millington-Sette The three statistical models take the room volume and a list of `(area, absorption_coefficient)` surfaces. **Sabine** is exact only for low, uniform absorption; **Eyring** replaces the absorption area by $-S\ln(1-\bar\alpha)$ and is correct where Sabine overestimates $T$ (a live room with strong absorption); **Millington-Sette** sums the Eyring term surface by surface, so a single perfectly absorbing surface drives $T$ to zero. $$ T_{\text{Sab}} = \frac{k V}{\sum_i S_i\alpha_i}, \qquad T_{\text{Eyr}} = \frac{k V}{-S\ln(1-\bar\alpha)}, \qquad T_{\text{Mil}} = \frac{k V}{-\sum_i S_i\ln(1-\alpha_i)}. $$ ```python from phonometry import room # A shoebox 8 x 5 x 3 m (V = 120 m3, S = 158 m2), uniform alpha = 0.2. surfaces = [(40.0, 0.2), (40.0, 0.2), (24.0, 0.2), (24.0, 0.2), (15.0, 0.2), (15.0, 0.2)] print(round(room.sabine_reverberation_time(120.0, surfaces), 3)) # 0.612 s print(round(room.eyring_reverberation_time(120.0, surfaces), 3)) # 0.548 s print(round(room.millington_sette_reverberation_time(120.0, surfaces), 3)) # 0.548 s ``` For a **uniform** distribution Eyring and Millington-Sette coincide, and both fall below Sabine; Sabine's over-estimate at high absorption is the reason Eyring exists. As $\alpha \to 0$, $-S\ln(1-\bar\alpha) \to \sum_i S_i\alpha_i$ and Eyring reduces to Sabine. Air absorption enters every model through the power-attenuation coefficient $m$ (in neper per metre, from the ISO 9613-1 [atmospheric absorption](https://jmrplens.github.io/phonometry/environment/propagation/outdoor-propagation/)): ```python from phonometry import environment, room m = environment.air_attenuation_m(2000.0, temperature=20.0, relative_humidity=50.0) surfaces = [(40.0, 0.3), (40.0, 0.3), (24.0, 0.3), (24.0, 0.3), (15.0, 0.3), (15.0, 0.3)] print(round(room.eyring_reverberation_time(120.0, surfaces, air_attenuation=m), 3)) ``` Every statistical model also assumes a **diffuse field**, and low frequencies break that assumption first: below the Schroeder frequency the room responds as a set of discrete modes, not as a reverberant mixture. The 2D FDTD simulation below drives a rigid 5 m by 3.5 m room exactly on its (2,1) mode and then between two modes; the standing-wave pattern that builds up on resonance is what Sabine and Eyring cannot see. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_room_modes.webm) ## 2. Fitzroy and Arau-Puchades (anisotropic rooms) When the absorption is concentrated on one axis (a carpeted floor and an acoustic ceiling against otherwise hard walls), a single mean $\bar\alpha$ misrepresents the field. **Fitzroy** and **Arau-Puchades** split a rectangular (shoebox) room into the three pairs of opposing walls and combine the *axial* Eyring reverberation times $T_i$ (each using the whole surface $S$ and the mean absorption $\bar\alpha_i$ of the wall pair perpendicular to axis $i$): $$ T_{\text{Fitz}} = \sum_i \frac{S_i}{S}\,T_i \quad(\text{arithmetic}), \qquad T_{\text{Arau}} = \prod_i T_i^{\,S_i/S} \quad(\text{geometric}). $$ ```python from phonometry import room # 8 x 5 x 3 m room, absorptive x-wall pair (alpha 0.5), hard elsewhere (0.1). dims = (8.0, 5.0, 3.0) absorption = (0.5, 0.1, 0.1) # mean alpha of the (x, y, z) wall pairs print(round(room.arau_puchades_reverberation_time(dims, absorption), 3)) # 0.812 s print(round(room.fitzroy_reverberation_time(dims, absorption), 3)) # 0.974 s ``` By the arithmetic-geometric-mean inequality the Arau-Puchades time never exceeds the Fitzroy time; Fitzroy is known to over-predict when one wall pair is very reflective, which is why Arau-Puchades recommends the geometric mean. Both reduce exactly to Eyring for a uniform absorption distribution. The misrepresentation these two models exist for can be watched. The clip below simulates one flat 8 × 2.5 m section twice with the same **total** statistical absorption — spread over all four edges, or concentrated on the floor-and-ceiling pair (α ≈ 0.79) between hard ends — and races both measured energy decays through the one Sabine-Eyring band the two rooms share. The spread room decays inside the band, a straight line at T = 160 ms against Sabine's 201; the concentrated room's decay has no single slope, 183 ms early and 236 ms late, and finishes 10 dB above its twin, its surviving field a grazing pattern running parallel to the absorber. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_absorption_placement.webm) ## 3. Comparing the five models per band `reverberation_time_models` builds the six boundary surfaces of a rectangular room from its dimensions and the three wall-pair mean absorptions, then evaluates all five models on a common footing and returns a `ReverberationModelResult` whose `.plot()` draws the figure above. ```python from phonometry import room # 10 x 7 x 3.5 m room, absorptive floor/ceiling against harder walls. res = room.reverberation_time_models( (10.0, 7.0, 3.5), ( [0.06, 0.07, 0.08, 0.09, 0.10, 0.10], # x-pair: hard end walls [0.12, 0.14, 0.16, 0.18, 0.20, 0.20], # y-pair: lightly treated walls [0.30, 0.50, 0.65, 0.78, 0.82, 0.80], # z-pair: carpet + acoustic ceiling ), frequencies=[125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0], ) print(res.sabine.round(2)) # [0.74 0.47 0.37 0.31 0.3 0.3 ] print(res.arau_puchades.round(2)) # [0.79 0.51 0.38 0.29 0.26 0.27] print(res.fitzroy.round(2)) # [1.02 0.79 0.66 0.57 0.51 0.51] res.plot() # the five model curves per band (the figure above) ```

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```python import matplotlib.pyplot as plt from phonometry import environment, room m = environment.air_attenuation_m([125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0], 20.0, 50.0) room.reverberation_time_models( (10.0, 7.0, 3.5), ( [0.06, 0.07, 0.08, 0.09, 0.10, 0.10], [0.12, 0.14, 0.16, 0.18, 0.20, 0.20], [0.30, 0.50, 0.65, 0.78, 0.82, 0.80], ), air_attenuation=m, frequencies=[125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0], ).plot() plt.show() ```

Sabine and Eyring are the two workhorses, and the per-band spread between them is itself a diagnostic. The diagram runs the room of this section through both, with the validity boundary every statistical formula shares. ## 4. Choosing a model, and when every model fails The five formulae are not rivals on a single axis of accuracy; each has a domain of validity: - **Sabine** is the tool for live rooms with low, reasonably even absorption (mean $\bar\alpha$ up to roughly 0.2): classrooms, halls, reverberation chambers. It is also the convention wired into measurement practice, because the ISO 354 absorption coefficient is *defined* through Sabine's formula, so feeding reverberation-room data back into Sabine is self-consistent even where the formula is strained. Its structural defect shows at high absorption: with $\alpha = 1$ on every surface (an opening in every direction) it still predicts a finite reverberation time. - **Eyring** is the choice for evenly treated rooms with substantial absorption: studios, treated offices, listening rooms. It reaches $T = 0$ for total absorption, and its correction over Sabine grows with $\bar\alpha$ (about 10 % shorter at $\bar\alpha = 0.2$, 30 % at 0.5). - **Millington-Sette** handles a mix of very absorptive and hard surfaces better than a single mean, but it is meant for measured, sub-unity coefficients: a single surface with $\alpha_i = 1$ drives the whole prediction to zero. Reverberation-room coefficients at or above 1.0 (a documented ISO 354 outcome, see the absorption section of [Room Acoustics](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/)) lie outside the domain of the logarithmic term, so phonometry enforces each formula's own domain: Sabine accepts such coefficients as supplied (its linear $A = \sum_i S_i\alpha_i$ stays finite); Eyring accepts them as long as the mean entering $\ln(1-\bar\alpha)$ stays below 1 (Fitzroy and Arau-Puchades take the wall-pair means themselves as inputs, so each must already be below 1); Millington-Sette rejects any coefficient at or above 1. To use Millington anyway, bringing such a coefficient into $[0, 1)$ is a modelling decision the formula does not prescribe: whatever adjustment you choose (limiting just below 1 is common), record it alongside the prediction. - **Fitzroy** and **Arau-Puchades** target shoebox rooms whose absorption is concentrated on one axis, the typical office or dwelling with a soft floor and ceiling between hard walls. Arau's geometric mean tempers Fitzroy's known over-prediction when one wall pair is very reflective. **When every formula fails.** All five inherit the same assumption: a diffuse field, with sound arriving equally from all directions at every point, that stays diffuse while it decays. The common breakages: - **Below the Schroeder frequency** the band holds a handful of discrete modes (the animation in §1) and a statistical reverberation time is not defined at all; each mode decays at its own rate set by the wall impedances it actually touches. - **Coupled volumes** (a hall with an open stage house, two rooms through a doorway) produce double-slope decays; no single $T$ exists, and the measured T20 and T30 disagree (the curvature diagnostic of [Room Acoustics](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/)). - **Disproportionate rooms** (corridors, low flat halls) with the absorption on one surface pair keep a grazing sound field parallel to the hard surfaces that the absorber barely touches; the measured time can be up to twice any statistical prediction, the practical experience recorded in EN 12354-6 (see [Sound absorption in enclosed spaces](https://jmrplens.github.io/phonometry/buildings/rooms/enclosed-space-absorption/)). - **Focusing geometries** (domes, curved rear walls) concentrate late energy instead of mixing it, producing position-dependent decays no single-number formula can represent. Scattering objects restore the mixing the models assume: a furnished room follows the statistical prediction distinctly better than the same room bare, beyond what the furniture's own absorption area accounts for. The clip below is that mechanism with the absorption taken out of it, so only the mixing is left: an 800 Hz wavefront enters a 4 m rigid-walled hall filled with rigid columns 10 to 17 cm across, a quarter to two fifths of the 42.9 cm wavelength. Every column diffracts the front and sheds a scattered wavelet, the wavelets interfere, and within a few passes the specular front has become energy spread over the whole hall with no preferred direction — which is the assumption Sabine and Eyring both start from, arriving here as a result rather than as a hypothesis. Nothing in the hall absorbs, so what you are watching is *only* the redistribution; the decay at the end is the energy draining out through the two open ends. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_pillar_hall.webm) In practice, quote a *band* of predictions (Sabine and Eyring, or Fitzroy and Arau-Puchades for axial cases) rather than a single value; where the models spread, the room is telling you its field is not diffuse. ## 5. Prediction report (`.report()`) `ReverberationModelResult.report(path)` renders a one-page PDF fiche of the prediction: a basis line marking it a **design-stage prediction** by the five statistical-acoustics models, an optional metadata header block (client, room, description, room volume, total surface area, climate), a per-band table with one reverberation-time column per model beside the model comparison plot (`.plot()`), and the boxed mid-frequency reverberation time from Arau-Puchades (the recommended model for a non-uniform absorption distribution) with the per-model spread alongside. It is a prediction, not a measurement: the five models bracket the reverberation time likely to occur, so no PASS/FAIL verdict is emitted. A target reverberation time supplied through the metadata's `requirement` field is printed as a reference line only, since a room reverberation time is a target range rather than a strictly higher/lower-is-better quantity. It uses the same `ReportMetadata` container (documented under [Insulation ratings](https://jmrplens.github.io/phonometry/buildings/insulation/insulation-ratings/#report-metadata-reportmetadata)) and rendering engine as the other fiches; passing `metadata=None` produces a bare prediction fiche. Rendering needs reportlab and, for the figure the fiche embeds, matplotlib (`pip install "phonometry[report,plot]"`); only `engine="reportlab"` is supported. The fiche renders in English by default; pass `language="es"` for a Spanish fiche (translated fixed strings and a comma decimal separator). ```python from phonometry import reverberation_time_models, ReportMetadata result = reverberation_time_models( (8.0, 5.0, 3.0), # a shoebox room, one treated wall pair ([0.10, 0.15, 0.30, 0.45, 0.55, 0.60], # treated wall pair, per octave band [0.08, 0.10, 0.12, 0.15, 0.18, 0.20], # side walls [0.05, 0.08, 0.10, 0.12, 0.15, 0.18]),# floor/ceiling frequencies=[125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0], ) result.report( "reverberation_fiche.pdf", metadata=ReportMetadata( specimen="Classroom, one wall lined with a broadband absorber", test_room="Classroom C1", temperature=20.0, relative_humidity=50.0, laboratory="Phonometry Reference Laboratory", requirement=0.8, # printed as a target reference line, no verdict ), ) # the five-model table + the boxed T_mid ``` The example fiche, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/reverberation_prediction_example.pdf) *Reverberation-time prediction fiche (`ReverberationModelResult.report`), the five-model table and the boxed $T_\text{mid}$.* ## See also - [Sound Absorption Measurement and Rating](https://jmrplens.github.io/phonometry/materials/absorbers/absorption-measurement/): the ISO 354 measurement every $\alpha_\mathrm{s}$ on this page comes from, its mounting dependence, and the ISO 11654 rating that must *not* be substituted for the per-band table (§0). - [Room acoustic parameters (ISO 3382-1/2)](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/#how-much-decay-range-do-i-need-for-a-valid-t20-or-t30): the measured counterpart, and the curvature diagnostic that tells you a prediction's diffuse-field assumption has failed. - [Modes of a rectangular room](https://jmrplens.github.io/phonometry/buildings/rooms/room-image-sources/#3-modes-of-a-rectangular-room): what is below the Schroeder frequency, where none of these five formulae applies. - API reference: [`room.reverberation_prediction`](https://jmrplens.github.io/phonometry/reference/api/rooms/reverberation-prediction/) and [`environment.propagation.air_absorption`](https://jmrplens.github.io/phonometry/reference/api/environment/air-absorption/). - Theory: [Steady-state room field](https://jmrplens.github.io/phonometry/reference/theory/rooms-buildings/#steady-state-room-field-bies-64-kuttruff-56): the diffuse-field assumption every Sabine-family formula makes, and where it stops holding. ## References - Sabine, W. C. (1922). *Collected papers on acoustics*. Harvard University Press. [Free scan at the Internet Archive](https://archive.org/details/collectedpaperso00sabi). The original reverberation experiments and the $T = 0.161\,V/A$ law of §1. - Eyring, C. F. (1930). Reverberation time in "dead" rooms. *The Journal of the Acoustical Society of America*, 1(2A), 217-241. [doi:10.1121/1.1915175](https://doi.org/10.1121/1.1915175). The mean-free-path derivation behind the $-S\ln(1-\bar\alpha)$ term of §1. - Millington, G. (1932). A modified formula for reverberation. *The Journal of the Acoustical Society of America*, 4(1), 69-82. [doi:10.1121/1.1915588](https://doi.org/10.1121/1.1915588). The per-surface logarithmic absorption term of §1. - Fitzroy, D. (1959). Reverberation formula which seems to be more accurate with nonuniform distribution of absorption. *The Journal of the Acoustical Society of America*, 31(7), 893-897. [doi:10.1121/1.1907814](https://doi.org/10.1121/1.1907814). The axial split into three wall-pair decays of §2. - Arau-Puchades, H. (1988). An improved reverberation formula. *Acustica*, 65(4), 163-180. [Publisher record at Ingenta](https://www.ingentaconnect.com/content/dav/aaua/1988/00000065/00000004/art00003). The geometric-mean combination of §2 (its Formula 18). - Kuttruff, H. (2016). *Room acoustics* (6th ed.). CRC Press. [doi:10.1201/9781315372150](https://doi.org/10.1201/9781315372150). The diffuse-field theory, its limits and the modern assessment of the classical formulae behind §4. - Everest, F. A. (2001). *Master handbook of acoustics* (4th ed.). McGraw-Hill. ISBN 978-0-07-136097-5. [Open Library record](https://openlibrary.org/isbn/9780071360975). The Fig. 7-22 worked example the conformance suite reproduces. - Carrión Isbert, A. (1998). *Diseño acústico de espacios arquitectónicos*. Edicions UPC. ISBN 978-84-8301-252-9. [Open Library record](https://openlibrary.org/books/OL23159935M). A Spanish-language textbook treatment of the reverberation models and their use in room design. ## Standards The classical reverberation formulae predate the normative world; they enter it through EN 12354-6:2003, whose Clause 4 model is a Sabine calculation with object and air terms (see [Sound absorption in enclosed spaces](https://jmrplens.github.io/phonometry/buildings/rooms/enclosed-space-absorption/)), and through ISO 354:2003, which defines the measured absorption coefficient via Sabine's formula. Air absorption follows ISO 9613-1:1993, *Acoustics — Attenuation of sound during propagation outdoors — Part 1: Calculation of the absorption of sound by the atmosphere* (see [Outdoor propagation](https://jmrplens.github.io/phonometry/environment/propagation/outdoor-propagation/)). The conformance suite is anchored on a real worked example, Everest's Fig. 7-22 Example 1 (an untreated 23.3 × 16 × 10 ft room), whose six printed Sabine reverberation times the SI implementation reproduces to ≤ 0.02 s, reinforced by hand-computed closed-form values and the model identities (every model collapses to Eyring for uniform absorption; Eyring collapses to Sabine as $\alpha \to 0$), which transitively carry that real-data anchor to the whole family.