Question 2 of 6: Room Acoustics — Reverberation Time and Hearing

Question 2: Room Acoustics — Reverberation Time and Hearing (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(1) The definition of reverberation time

When a steady sound source in an enclosed space is switched off, the sound does not stop; it continues to arrive at the listener as an ever-denser train of reflections from the room surfaces, each one weaker than the last because every reflection absorbs a fraction of the incident energy. The reverberation time, written T₆₀ or simply RT, is the time in seconds required for the sound pressure level in the room to fall by 60 dB below its steady-state value after the source has stopped. Sixty decibels is a factor of one million in sound intensity, and it was chosen by Wallace Sabine because it is roughly the range between the level of an orchestral forte and the background level of a quiet hall — that is, the point at which the tail of one note stops interfering with the next.

In practice a 60 dB decay is rarely measurable above the background noise of a real room, so the decay is measured over the first 20 or 30 dB and extrapolated; these are reported as T₂₀ and T₃₀, and ISO 3382 defines the measurement procedure. Reverberation time is not a single number for a room but a spectrum: it is quoted in octave or one-third-octave bands from 125 Hz to 4 kHz, because the absorption of every real surface varies strongly with frequency.

For design purposes the classical estimate is Sabine's equation, in SI units

where $V$ is the room volume in m³, $S_i$ the area of each surface in m², $\alpha_i$ its absorption coefficient at the frequency of interest, and $4mV$ the (small, high-frequency) contribution of absorption in the air itself. The units of $A$ are m² of absorption, historically called metric sabins. Applying it to the room in Figure 2.1: $T_{60} = 0.161\times 320/42.9 = 1.20$ s. Because $T_{60}$ is inversely proportional to $A$, doubling the absorption halves the reverberation time — 85.8 m² of absorption in the same room would give 0.60 s. Sabine's form assumes the sound field is diffuse and the absorption is low and evenly distributed; in a room with $\alpha$ above about 0.3, or with all the absorption on one surface, Norris–Eyring's form $T_{60} = 0.161V/[-S\ln(1-\bar\alpha)]$ is the better estimate.

(2) What too much and too little reverberation actually mean

A reverberation time that is too high means the room stores acoustic energy for too long. Every syllable of speech, or every note of music, is still audible while the next one arrives, so consecutive sounds overlap and smear into each other. The practical consequences are a measurable loss of speech intelligibility — consonants, which carry most of the information in speech and are short and quiet, are masked by the reverberant tails of the preceding vowels — and a build-up of steady background level, because noise from every source in the room decays slowly and accumulates. This is the familiar experience of a swimming pool, an atrium or a hard-surfaced cafeteria: people raise their voices to be heard over the reverberant field, which raises the field further. It also degrades the performance of public-address and fire-alarm voice systems, which is why NBCC and CAN/ULC-S1001 commissioning of voice communication systems is difficult in highly reverberant spaces.

A reverberation time that is too low means the room absorbs energy almost as fast as the source can supply it. The space sounds "dead" or "dry". Speech is perfectly intelligible but weak and effortful to produce, because the speaker gets no reinforcement from the room and must supply all the level directly; a lecturer in an over-absorbent room tires quickly and needs amplification for quite modest distances. For music the effect is worse: without reverberant support the sound loses fullness, blend and apparent loudness, individual instruments cease to fuse into an ensemble, and performers lose the acoustic feedback they rely on to hear one another. An anechoic chamber, at the limit, is actively unpleasant to occupy.

Good design therefore aims at a target that depends on use and on volume: roughly 0.4–0.6 s for a classroom or a small meeting room (ANSI/ASA S12.60 sets 0.6 s for classrooms up to 283 m³), 0.8–1.1 s for a lecture theatre or a drama theatre, 1.4–1.8 s for a symphony hall, and 2–3 s or more for organ and choral music in a church, where the long tail is part of the intended sound. Bringing the 1.20 s room of Figure 2.1 down to a classroom-suitable 0.80 s requires $A = 0.161\times320/0.80 = 64.4$ m², that is, an extra 21.5 m² of absorption — achievable with roughly 25 m² of high-performance acoustic ceiling tile.

(3) The variables that govern reverberation time

Sabine's equation names the two dominant variables explicitly and hides two more. In order of importance:

Room volume V. Reverberation time is directly proportional to volume. Sound travels further between reflections in a large room, so fewer absorption events occur per second and the decay is slower. Doubling the volume of a room while keeping the surface treatment the same lengthens the reverberation time by roughly the cube root of eight over four — in practice, large rooms are inherently more reverberant, which is why volume per seat is the first parameter an auditorium designer sets.

Total absorption A, that is, the areas of the surfaces multiplied by their absorption coefficients. This includes not only the fabric of the room — plaster, glass, carpet, acoustic tile, curtains — but the furnishings and, critically, the occupants. A seated audience is one of the most absorbent "materials" in an auditorium (of the order of 0.4–0.5 m² of absorption per person), so a hall's reverberation time can fall by a third between empty and full; upholstered seating is specified partly so that this difference is small and the rehearsal acoustic resembles the performance acoustic.

Frequency. Absorption coefficients are strongly frequency dependent, so reverberation time is a curve rather than a number. Porous absorbers (mineral fibre, carpet, curtains) work well above about 500 Hz and poorly at low frequency; panel and membrane absorbers, and structural elements such as gypsum board on studs, absorb at low frequency. Untreated rooms therefore tend to have a long low-frequency tail, and a "bass ratio" (the ratio of the 125–250 Hz reverberation time to the 500–1000 Hz value) slightly above one is desirable for music and undesirable for speech.

Air absorption, and hence temperature and relative humidity. The $4mV$ term matters only above about 2 kHz and only in large volumes, but in a concert hall it is what limits the high-frequency reverberation, and $m$ depends on humidity — which is why the relative humidity in a large hall is an acoustic as well as a comfort parameter.

Room shape and the distribution of the absorption. These do not appear in Sabine's equation at all, which is precisely its limitation. If all the absorption is concentrated on one surface, or if the room is long and narrow or strongly coupled to an adjoining volume such as a stage house or a balcony recess, the sound field is not diffuse, the decay is not a single straight line, and Sabine over-predicts the reverberation time. Diffusing surfaces, splayed walls and articulated ceilings exist to make the real room behave more like the idealisation.

(4) The smallest detectable change in sound pressure level

Under laboratory conditions, with a steady broadband signal presented to a trained listener, the just-noticeable difference in sound pressure level is about 1 dB, and this is the figure to quote. In the field, with real signals and ordinary listeners, a 1 dB change is essentially imperceptible; a change of about 3 dB is the smallest that is reliably noticed, 5 dB is clearly noticeable, and 10 dB is judged as roughly a doubling (or halving) of loudness. These numbers explain why decibel arithmetic feels counter-intuitive to non-specialists: a 1 dB change corresponds to a sound-pressure ratio of only $10^{1/20} = 1.122$, while the 10 dB change that sounds "twice as loud" is a factor of ten in intensity. It also explains why doubling the number of identical noise sources — which adds 3 dB — is barely audible, and why halving the sound power of a fan is a disappointing investment unless several such measures are combined.

(5) The audible frequency range

The nominal range of human hearing is 20 Hz to 20 kHz. This is the range used to define acoustic instrumentation and audio bandwidth, and it corresponds to a young, undamaged ear. Two qualifications belong in a complete answer. First, the upper limit falls steadily with age and noise exposure (presbycusis): a typical 60-year-old hears little above 8–12 kHz, and noise-induced loss characteristically begins as a notch near 4 kHz. Second, sensitivity is very far from uniform across the range — the ear is most sensitive between about 2 and 5 kHz, where the ear canal resonates, and needs far more sound pressure at 63 Hz than at 1 kHz to produce the same loudness. That frequency dependence, described by the equal-loudness contours of ISO 226, is exactly what the A-weighting network of a sound level meter approximates, which is why occupational noise limits under the provincial OH&S regulations and CSA Z107.56 are expressed in dBA. For building services work the octave bands from 63 Hz to 8 kHz cover everything of practical interest, and room criteria such as NC or RC curves are specified over that range.