Rooms are where acoustics is most often computed, and a room is three-dimensional, many wavelengths across at most audible frequencies, and heard for a second or more. A grid for that is expensive, as Chapter 9 will measure. This chapter uses a method with no grid at all, which is exact for one idealised room and a useful model for many real ones, and uses it to test the two formulas for reverberation time that every acoustician learns first.

### Images

A wall that reflects sound specularly behaves, for a listener, like a window onto a mirror-image room containing a mirror-image source. The sound reflected once from the wall is exactly the sound the image source would send through the window. The images reflect in the other walls in their turn, and for a rectangular room the process fills space with a regular lattice of image sources, eight in each cell of size $2L_x \times 2L_y \times 2L_z$:

$$x = (1 - 2p)\,x_s + 2\ell L_x, \qquad p \in \{0, 1\},\ \ell \in \mathbb{Z},$$

and similarly in $y$ and $z$. The path from an image to the listener crosses the walls $x = 0$ and $x = L_x$ a total of $|\ell - p| + |\ell|$ times, so if every wall reflects a fraction $\beta$ of the pressure, an impulse from the source reaches the listener as a sum of delayed impulses,

$$h(t) = \sum_{\text{images}} \frac{\beta^{\,n}}{4\pi d}\,\delta\!\left(t - \frac{d}{c}\right),$$

with $d$ the image's distance and $n$ its number of reflections (Allen and Berkley 1979).

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For rigid walls, $\beta = 1$, this is not a model. It is the exact solution of the wave equation in the room with Neumann conditions on its walls, by the method of images, and Chapter 9 uses it as such. For $\beta < 1$ it is a model, one in which every wall reflects the same fraction of the pressure at every angle and every frequency. No real wall does that: a wall's reflection depends on both, and a real room also scatters sound diffusely from furniture and people. What the model keeps is the geometry of the specular reflections, exactly.

Each arrival falls between two samples. The script can put it on the nearer one, as Allen and Berkley did, or spread it over several samples as a windowed sinc, which is the impulse band-limited to the Nyquist frequency and cut off by the window. We use 8 taps by default. The choice changes the response at high frequencies, where rounding to the nearest sample misplaces each arrival by up to half a sample, but it changes the decay rate by about one per cent.

### Sabine, Eyring, and the lattice

The reverberation time $T_{60}$ is the time the sound in a room takes to decay by 60 dB after the source stops. Sabine (1900) derived the first formula for it by assuming a **diffuse** field, in which sound energy is spread uniformly through the room and travels equally in all directions. Energy then reaches the walls at a rate proportional to their area $S$, a fraction $\alpha$ of it is absorbed, and the energy decays exponentially:

$$T_{60}^{\text{Sabine}} = \frac{24\ln 10}{c}\,\frac{V}{S\alpha} \approx 0.161\,\frac{V}{S\alpha},$$

with $V$ the volume and $\alpha = 1 - \beta^2$ the absorption coefficient. Eyring (1930) observed that the loss happens in steps, one per reflection, rather than continuously. A ray in a diffuse field travels on average $4V/S$ between reflections and loses a factor $1 - \alpha$ at each, which gives

$$T_{60}^{\text{Eyring}} = \frac{24\ln 10}{c}\,\frac{V}{-S\ln(1-\alpha)}.$$

The two agree for small $\alpha$, and Eyring's is always the shorter.

The image model allows a third prediction, which we call the **lattice** prediction and which assumes nothing about diffusion. An image at distance $ct$ in direction $\mathbf{u}$ has been reflected about $ct\,g(\mathbf{u})$ times, with $g(\mathbf{u}) = |u_x|/L_x + |u_y|/L_y + |u_z|/L_z$, and the images at distance between $r$ and $r + dr$ number about $4\pi r^2 dr/V$, each contributing energy proportional to $\beta^{2n}/r^2$. So the energy arriving at time $t$ is proportional to

$$E(t) \propto \Big\langle (1 - \alpha)^{\,c t\, g(\mathbf{u})} \Big\rangle_{\mathbf{u}},$$

the average over directions of a decaying exponential whose rate depends on the direction. The average of $g$ over directions is $S/4V$, so at early times $E(t)$ decays at exactly Eyring's rate. Later it decays more slowly, because it becomes dominated by the directions with the fewest reflections per metre, along the room's longest dimension. The driver evaluates this average by quadrature on the sphere and reads a $T_{60}$ off it in the same way as off the measured response.

### Measuring the decay

The reverberation time is read off the **Schroeder curve**, the backward integral $\int_t^\infty h^2\,dt'$, in decibels, by a straight line through its $-5$ and $-25$ dB points, scaled to 60 dB; this is the standard $T_{20}$. We measure it in the band from 0.7 to 5.7 kHz, the three octaves around 1, 2 and 4 kHz together, and the reason for the band is itself instructive. Every image has the same sign, so arrivals that land within a sample or two of each other add coherently at low frequencies, and the broadband response carries a slowly decaying low-frequency component. For rigid walls it would never decay at all: a point source that injects air into a closed box leaves its pressure raised. The lattice prediction is the high-frequency limit, in which arrivals add incoherently, and that is also what Sabine's and Eyring's formulas describe.

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In the default room, $6 \times 4.5 \times 3$ m with $\alpha = 0.2$, the measured $T_{60}$ in the band is 0.59 s. The lattice predicts 0.59 s, Sabine 0.56 s and Eyring 0.50 s. The broadband curve, drawn faintly, decays more slowly, with a $T_{60}$ of 0.78 s, which is the low-frequency build-up and not the room's reverberation in the usual sense.

The phrase is a C major arpeggio of plucked notes, synthesised as in Chapter 1 and convolved with the impulse response by FFT. At $\alpha = 0.1$ it is heard in a hard, live room with a reverberation time of about 1.3 s; at $\alpha = 0.6$ it is nearly dry, at about 0.15 s. The impulse response itself sounds something like a hand clap in the same room, which is how reverberation times are often measured in practice.

### Reverberation time against absorption

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| $\alpha$ | 0.05 | 0.1 | 0.2 | 0.3 | 0.5 | 0.7 |
|---|---|---|---|---|---|---|
| measured | 2.47 s | 1.28 s | 0.59 s | 0.37 s | 0.21 s | 0.12 s |
| lattice | 2.55 s | 1.24 s | 0.59 s | 0.37 s | 0.19 s | 0.11 s |
| Sabine | 2.23 s | 1.12 s | 0.56 s | 0.37 s | 0.22 s | 0.16 s |
| Eyring | 2.17 s | 1.06 s | 0.50 s | 0.31 s | 0.16 s | 0.09 s |

Three things are visible in the table and in the ratio plot.

The measured values follow the lattice prediction to within about 3 per cent up to $\alpha = 0.3$. Above that they drift long, by 10 per cent at $\alpha = 0.5$ and 14 per cent at $0.7$. At high absorption the sound has died after a few dozen reflections, and a lattice average over infinitely many images is a poor description of the few that matter; the early reflections of this particular source and receiver dominate.

The lattice prediction is a fixed multiple of Eyring's, 1.17 for this room, at every absorption. That is because it depends on $\alpha$ only through $\ln(1 - \alpha)$, exactly as Eyring's formula does; the factor depends only on the room's shape, and measures how far a shoebox with specular walls is from being diffuse. A cube would presumably have a smaller factor and a long corridor a larger one.

Sabine's formula is the closest of the two to the measurements over much of the range, which is a coincidence of two errors in opposite directions. Its assumption of continuous absorption overestimates the decay time, by a factor $-\ln(1-\alpha)/\alpha$, and the non-diffuse geometry of a specular shoebox lengthens the true decay time by a factor of 1.17. The two cancel near $\alpha = 0.3$. Below that, Sabine's formula is short of the lattice prediction by up to 13 per cent; above it, long by up to 46 per cent at $\alpha = 0.7$, where it is the worst of the three.

### What this chapter establishes

The image method computes a rectangular room's response exactly for rigid walls, and as a well-defined model for uniformly absorbing ones, at a cost set by the number of images, which grows as the cube of the length of the response. For decay times, it confirms Eyring's formula as the correct early-time limit for specular reflection, and it shows that the shape of the room matters by a factor that neither formula contains. It cannot model what it leaves out: frequency-dependent absorption, diffuse scattering, and anything that is not a rectangular box. The last of these is what finite-difference methods are for, and Chapter 9 measures what they cost.