A room is a linear system: its impulse response, how fast it dies away, how loud it stays, and the modes it rings in.
A flat wall acts as a mirror: each reflection comes from an image of the source behind it. Shoebox room, every wall with the same absorption α; drag source (orange) and listener (blue) in the plan.
Sound: ready
The room is a linear, time-invariant system, so everything it does to a sound is in one function: its impulse response, the Green's function of the room between this source and this listener. Left: the plan with the first-order images (the lateral walls; floor and ceiling images are in the response too). Right: the first 200 ms, each image a spike at its arrival time d/c with energy ∝ (1 − α)^bounces / d². First the direct sound, then a few distinct early reflections (within 20 ms: intimacy; within about 35 ms the ear fuses them with the direct sound and still hears the source where it is: the Haas or precedence effect), then reflections so dense they become the reverberant tail. Bottom: the whole decay in dB (Schroeder's backward integral of the response), with the −60 dB line that defines T. Listen: "In this room" is the dry signal convolved with this response, y = x ∗ h. Raise α and the room turns from boomy to dry; move the listener far from the source and clarity drops. Readouts: C₈₀ = 10 log(E₀₋₈₀ ms / E₈₀ ms₋∞), the early-to-late energy ratio used for music; D₅₀ = E₀₋₅₀ ms / E_total, the useful fraction for speech (times counted from the direct sound). The first reflection is usually the floor.
Sabine: T = 0.161 V / Σαᵢ Sᵢ
Eyring: T = 0.161 V / [−S ln(1 − ᾱ)]
(0.161 s/m = 55.3/c)
The sound energy hits a wall after travelling, on average, the mean free path ⟨d⟩ = 4V/S (for a diffuse field in any room shape; for a cube 4V/S = ⅔ d, the factor on the slide). Each hit keeps a fraction (1 − α) of the energy, so after a time t, with c t/⟨d⟩ hits, the energy is E₀(1 − α)^{ct/⟨d⟩}. Setting this to 10⁻⁶ gives T = 6 ln10 · 4V / [−c S ln(1 − α)] = 0.161 V / [−S ln(1 − α)]: that is Eyring's formula. For small α, −ln(1 − α) ≈ α and it becomes Sabine's (the slide's 13.8/α is the same step). Sabine is excellent in live rooms but overestimates T in absorbing ones: with α = 1 everywhere (an open window, Sabine's own unit) Eyring gives T = 0, Sabine still 0.161 V/S.
Shoot the rays: 2000 sound particles leave the centre, bounce diffusely off the walls and lose (1 − α) of their energy at each hit. Their measured mean free path lands on 4V/S, and with the same α on every wall their energy decay (black, right) follows Eyring (try the cube); when the absorption is concentrated, as with an audience on the floor, it falls between Sabine and Eyring, a known limit of both formulas. Two caveats a room designer adds.
Air absorption. The air itself absorbs, more at high frequency: along a path x the energy falls as e^{−mx}. In a time t the sound travels ct, so the air multiplies the energy by e^{−mct}; written like the wall losses, e^{−c S_e t/4V}, this is an extra absorbing area 4mV: T = 0.161 V / (S_e + 4mV). Typical m (energy attenuation coefficient, 20 °C, 50 % humidity): ≈ 0.0001 m⁻¹ at 125 Hz, 0.0006 m⁻¹ at 500 Hz, 0.0024 m⁻¹ at 2 kHz, strongly dependent on humidity (tables in dB/km: m = dB per metre / 4.34). Tick the box: it matters only in large halls and at high frequency, where it shortens T by 10 % or more.
Schroeder frequency. Each room mode is a damped resonance of width Δf ≈ 2.2/T Hz (the decay rate 6.9/T divided by π), and the number of modes per hertz grows as 4πVf²/c³ (Weyl, tab 4). Schroeder (1954) called the field diffuse when about three modes fall within one resonance width; solving 3 = (4πVf²/c³)(2.2/T) gives f_S ≈ 2000 √(T/V), T in s, V in m³. Above f_S the modes overlap, the response is smooth and nearly the same everywhere, and everything on this tab applies; below it the room answers as a few separate modes, with level changing from seat to seat and note to note. The 3 × 3 × 3 m concrete cube has f_S ≈ 770 Hz, so at 500 Hz it is still modal; a concert hall has f_S ≈ 20–30 Hz, diffuse over the whole musical range.
SPL(r) = L_W + 10 log[ Q/(4πr²) + 4/R ]
L_W: sound power level of the source, 10 log(W / 10⁻¹² W). Q: directivity. R: room constant (explained below the plot).
r_d = √(QR/16π). Room from tab 2, at the band chosen below.
Direct sound. A source of power W spreads it over a sphere: intensity QW/(4πr²), with Q = 1 in all directions, 2 for a source on the floor (half-space), 4 in an edge, 8 in a corner or for a directional speaker.
Reverberant sound, the room constant. Direct sound becomes reverberant only after its first bounce, and that bounce already absorbs a fraction ᾱ: the reverberant field is fed with W(1 − ᾱ). It loses energy at the walls: in a diffuse field of energy density w, the energy hitting the walls per second is wcS/4 (the same ¼ as in 4V/S), and a fraction ᾱ of it is absorbed. In steady state W(1 − ᾱ) = wcSᾱ/4, so the reverberant intensity is wc = 4W/R with R = Sᾱ/(1 − ᾱ). For small ᾱ, R ≈ Sᾱ = Sₑ, Sabine's absorbing area; for ᾱ → 1, R → ∞: no reverberant field, as outdoors or in an anechoic room.
Adding them (energies add) gives the total intensity W[Q/(4πr²) + 4/R]; its level re 10⁻¹² W/m², which in air is practically the SPL, is SPL(r) = L_W + 10 log[Q/(4πr²) + 4/R], with r in metres and R in m². The direct-to-reverberant ratio is D/R = 10 log[(Q/4πr²)/(4/R)] = 10 log[QR/(16πr²)]: 0 dB at the critical distance r_d (that is its definition, whence r_d = √(QR/16π)), positive closer to the source, falling by 6 dB per doubling of distance beyond. It is the steady-state cousin of the clarity C₈₀ of tab 1.
A steady source fills the room with two kinds of sound. The direct sound falls by 6 dB per doubling of distance, as in free space (dashed). The reverberant sound, the energy bounced around many times, is nearly the same everywhere (dotted): its level is set by the balance between the power the source pumps in and the power the walls absorb. Their sum (black) follows the 1/r² law near the source and flattens beyond the critical distance r_d, where the two are equal and the total is 3 dB above each. This is "reverberation evens out the sound". Add absorption and r_d grows, the plateau drops: a drier, clearer room, but the back rows get less sound. Most of an audience sits beyond r_d, which is why the reverberant field, and so T, matters so much.
f = (c/2) √[(n_x/L_x)² + (n_y/L_y)² + (n_z/L_z)²]
p ∝ cos(n_xπx/L_x) cos(n_yπy/L_y) cos(n_zπz/L_z), rigid walls.
Low modes (below ~60 Hz) are hard to hear on laptop speakers: use headphones or external speakers.
Sound: ready
Modes. The air in a closed box is the 3-D version of the string and the membrane: standing waves with a cosine in each direction (pressure is maximal at rigid walls). Axial modes (one index non-zero, orange) bounce between one pair of walls, tangential (two, blue) between four, oblique (three, green) between all six. Their number grows as f³ (Weyl's law, the curve on the right), so at low frequency they are few and far apart and the room colours the bass: the cube is the worst case, with many modes piled on the same frequencies. The Schroeder frequency f_S ≈ 2000√(T/V) (dashed) is where about three modes fall within one resonance width, Δf ≈ 2.2/T: above it the modes overlap and the statistical picture of tab 2 takes over, below it single modes dominate. A longer T means narrower resonances, so f_S rises with T (try the slider). The clap is heard at the opposite corner through the modes below 300 Hz only: the room's "boom". Flutter echo. Between two parallel reflecting walls a clap returns every 2d/c: an impulse train, whose spectrum is a comb with teeth at multiples of c/2d. The ear hears that spacing as a pitch, the buzzing "ring" of a corridor or a stairwell: compare with the same clap without walls, and try 1–2 m (pitch ≈ 85–170 Hz). With large walls the wave between them is nearly plane and only the absorption α makes it die out (−60 dB after about 6.9/α round trips); with small walls spherical spreading kills the echoes much sooner and the ring becomes a short buzz. Tilted or diffusing walls break the train.