Room simulator (shoebox)

What the tool computes

The Room simulator estimates the acoustic parameters of a listening point in a rectangular room (shoebox): speech clarity (C50) and music clarity (C80), early decay time (EDT) and reverberation time (T30), per octave band from 125 to 4000 Hz. It is a hybrid model: the early reflections are computed with an image-source lattice exact within the ideal specular shoebox model — it is not “exact room acoustics”, it is exactness inside that idealisation — and the diffuse tail with the statistical Eyring and Sabine formulas.

The goal is pre-dimensioning and teaching: understanding where a room sounds good or bad and how treating its surfaces changes the result. It is not a validated solver for design.

The Image Source Method, and why it is exact on a shoebox

Each wall reflection is equivalent to a mirrored “image” source. In a rectangular room the images form a regular, infinite lattice: no edge can hide them, every path is a straight line. The computation becomes a pure nested triple sum over the indices (i, j, k) of the three axes — no mesh, no visibility test, no occlusion.

On an axis of side L with the source at s, the image of index i is at i·L + s (i even) or (i+1)·L − s (i odd). The number of reflections on each of the two sides of the axis is counted separately, faithful to the floor/ceiling asymmetry: for i > 0 the far wall takes ⌈i/2⌉ reflections, the near one ⌊i/2⌋. Each image carries an energy Π (1−α)^(reflections) / dist², summed band by band.

On a real geometry this collapses: the lattice is no longer regular and mesh, BVH and ray casting are needed. This is why the tool is shoebox-only by construction: not a limitation to work around, but the condition that makes the method exact.

The metrics

C50 and C80 — clarity

They are the ratio in dB between energy arriving “early” and “late”:

$$\begin{array}{rlrl}{C}_{50}& =10{\log }_{10}\frac{{\int }_0^{50\text{ms}}{p}^{2}dt}{{\int }_{50\text{ms}}^{\infty }{p}^{2}dt}& & \to \text{speech}\\ C_{80}& =10{\log }_{10}\frac{{\int }_0^{80\text{ms}}{p}^{2}dt}{{\int }_{80\text{ms}}^{\infty }{p}^{2}dt}& & \to \text{music}\end{array}$$

The ear integrates reflections within ~50 ms with the direct sound: they reinforce speech. Beyond that threshold they become tail that masks syllables. For speech a C50 > +2 dB indicates good intelligibility; for music a C80 around −2…+2 dB is typical of a good concert hall.

EDT and T30 — reverberation

They are derived from the Schroeder decay curve (energy back-integration). The EDT (Early Decay Time, from the first 10 dB) weights the perceived early tail and varies with position; the T30 (from −5 to −35 dB) measures the diffuse-field reverberation and is a more global room property.

Reference values by intended use

There is no “good” reverberation in the abstract: it depends on what the room is for. A concert hall lives on its reverberation, a classroom dies from it. Indicative values, meant as averages over the mid bands (500–1000 Hz):

These are orders of magnitude to orient by: the reference for speech rooms is the Italian standard UNI 11532, which ties the optimal reverberation time to the volume and the intended use, while in-situ measurement of the parameters follows ISO 3382.

Eyring versus Sabine

The diffuse tail uses two classic T60 formulas:

$$\begin{array}{rlrl}\text{Sabine:}& T_{60}=\frac{0.161V}{A}& A& =\sum _i{S}_i{\alpha }_i\\ \text{Eyring:}& T_{60}=\frac{0.161V}{-S\ln (1-\bar{\alpha })}& \bar{\alpha }& =\frac{A}{S}\end{array}$$

Sabine overestimates T60 at high absorption: in the limit ᾱ → 1 it still gives a finite value, physically absurd. Eyring starts from the per-reflection decay and correctly tends to 0. The tool uses Eyring as the reference to calibrate the tail; the two converge when absorption is low.

Why a panel raises C50

Late reflections have long paths and many bounces: they dominate the tail. An absorptive panel on a reflective wall cuts mostly that late energy, while the few early reflections that reinforce speech remain. The E_early/E_late ratio grows and C50 rises: this is the principle behind treating speech rooms.

A worked case, with the engine’s own numbers

A 12 × 8 × 4 m classroom (V = 384 m³, S = 352 m²), source 1.5 m from the back wall and a receiver moving away along the axis. Two states: bare — plaster on masonry on every surface except a heavy carpet floor — and panelled, the same envelope with an absorbing suspended ceiling (α from 0.15 at 125 Hz to 0.90 in the mid bands). The values are 500–1000 Hz averages produced by the tool’s engine.

Three readings, all quantitative.

Position matters for clarity, not for reverberation. From 2 to 10 m, C50 changes by 4.8 dB and T30 by 0.03 s. That is not an artefact: in a diffuse field the reverberation time is a property of the room, clarity is a property of the point. If T30 looks almost uniform on one of this tool’s maps, it is working as intended — why, by construction, is the subject of the next section.

The panel acts on the denominator. At the 6 m receiver the suspended ceiling takes C50 from −7.2 to −2.0 dB (+5.2) and C80 from −3.8 to +1.1: +4.9 dB of clarity gained without touching the early reflections, purely by removing energy from the tail. In the same move EDT falls from 2.86 to 1.42 s and T30 from 2.35 to 0.72 s.

Treatment is not enough to cover the whole room. Even panelled, the room passes the C50 > +2 dB threshold only at 2 m: halfway down it is already at −0.4 dB and at the back −2.8. It is the result the arithmetic of reverberation time hints at, seen here point by point: past a certain depth the answer is no longer absorption but useful early reflections (oriented reflective surfaces) or electroacoustic reinforcement.

Truncating the lattice, and what follows from it

The image lattice is infinite, the computation is not. The tool truncates at a time tMax, that is at a radius c·tMax around the receiver. The cost grows as the cube:

Doubling tMax multiplies the images by seven or eight — and a heat map wants one lattice per cell. Leaving tMax free up to the real T60 is not viable: in the bare room of the example the T60 at 125 Hz is 11.7 s, which at this rate means billions of images per point. The tool therefore caps tMax at 0.35 s (in practice 0.6·T60 of the mid bands, clamped between 0.15 and 0.35 s).

The part of the echogram beyond tMax is not thrown away, though: it is continued analytically by the Eyring diffuse tail, calibrated on the discrete energy density just before the cut. The model is hybrid in this precise sense: exactly geometric at the start, statistical afterwards.

Hence the most important consequence to know before reading the results. T30 is measured on the −5 to −35 dB stretch of the Schroeder curve, and with tMax ≤ 0.35 s that stretch almost always falls inside the analytic tail, whose slope is by construction −60/T60 dB/s. The computed T30 therefore tends to reproduce the global Eyring T60 and to vary little in space — which is the physically expected behaviour, but it needs saying: the T30 map is not a point-by-point measurement. The spatial information lives in C50, C80 and EDT, which come from the geometric early reflections. A genuinely cell-by-cell T30 needs a wave solver, or ray tracing carrying a discrete tail down to −35 dB, beyond the scope of a pre-sizing tool.

You can also see where the two halves of the model hand over. In the bare room, band by band:

Where the T60 is long, the −5…−35 dB stretch lies entirely in the analytic tail and the two values agree to the hundredth. Where the T60 shortens — high bands, more absorption — that stretch starts falling into the discrete part and T30 departs from it by up to a quarter of a second. It is the same direction as the classic Sabine-versus-Eyring result: the more absorption rises, the less the diffuse-field assumption holds.

Where the geometric model stops reaching

Among the limits is “no waves below ~250 Hz”. The real threshold is not a fixed number: it is the room’s Schroeder frequency, below which the room modes separate and the field is no longer statistically dense.

$$f_S\approx 2000\sqrt{\frac{T_{60}}{V}}$$

For the classroom of the example, using the 125 Hz T60:

The result is counter-intuitive and worth keeping in mind: the bare room has the higher Schroeder frequency. At 349 Hz not only the 125 Hz band is in the modal regime but the 250 Hz one as well, and partly the 500 Hz one. The very room that would most need studying is the one the geometric model says least about. Treat it, and f_S drops to 185 Hz, leaving only the 125 Hz band out of reach.

The practical reading is that the 125 and 250 Hz columns should be taken as indications of trend rather than predictions, the more so the more reverberant the room. The criterion is the same one discussed in the reverberation time wiki.

How to use it

1. Geometry — set the length, width and height of the shoebox.
2. Source and receiver — place the two points in the room; the clarity metrics (C50, C80) and the EDT depend on position, the T30 far less.
3. Absorption — assign each surface (floor, ceiling, four walls) an α coefficient per octave band, from a material library or manually.
4. Read the results — C50, C80, EDT and T30 per band from 125 to 4000 Hz; compare against the reference values for the intended use and check the effect of a treatment by moving the α of one surface.

Limitations

• Rectangular geometries only. Do not extend to “nearly” rectangular rooms: the model keeps producing plausible but wrong numbers.
• No waves below ~250 Hz. Geometric model: the axial modes of low frequencies are invisible to it.
• Random-incidence α, tabulated (ISO 354), not angle-dependent impedance; in situ they differ.
• Pre-dimensioning and teaching, not a validated solver. Design needs dedicated software, in-situ measurements and a competent acoustic technician.

References

• Allen J.B., Berkley D.A. (1979) — Image method for efficiently simulating small-room acoustics. J. Acoust. Soc. Am. 65(4).
• Eyring C.F. (1930) — Reverberation Time in “Dead” Rooms. J. Acoust. Soc. Am. 1.
• ISO 3382-1/-2 — room acoustic parameters (T30, EDT, C50, C80).
• ISO 354:2003 — measurement of sound absorption in a reverberation room.
• Acustica per chi progetta — speech/music rooms chapter.