Source: https://jmrplens.github.io/phonometry/reference/api/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/reference/api/rooms/room-acoustics/) (ISO 3382) and complements the EN 12354-6 model of [Sound absorption in enclosed spaces](https://jmrplens.github.io/phonometry/reference/api/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 environmental, 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 = environmental.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/reference/api/environment/outdoor-propagation/)): ```python from phonometry import environmental, room m = environmental.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. ## 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 environmental, room m = environmental.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() ```

## 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/reference/api/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/reference/api/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/reference/api/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. 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 [Field insulation](https://jmrplens.github.io/phonometry/guides/insulation-field/#report-metadata-reportmetadata)) and rendering engine as the other fiches; passing `metadata=None` produces a bare prediction fiche. Rendering needs reportlab (`pip install phonometry[report]`); 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 $T_\text{mid}$.* ## 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/reference/api/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/reference/api/environment/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.