Chapter 16: Acoustics

The study of what happens to sound in an enclosed space or as the result of interactions with large objects such as buildings is called acoustics. Humans have been trying to improve the acoustics of auditoriums and other public spaces since the time of the ancient Romans. Marcus Vitruvius Pollio worked as an architect and engineer in the first century BCE and is credited with installing resonating cavities called Echea in Roman amphitheaters. These were made of brass or clay and placed under the seats of the auditorium to enhance the voices of actors on stage. None of these devices still exist and it is not clear if they actually worked or not.

The rules presented previously (Chapter 7) of how waves interact with objects, such as reflection, refraction, path difference, diffraction and interference, govern how sound will behave inside rooms and auditoriums as well as outdoor concert pavilions. Although much is known about the topic, it is still difficult to know exactly what a given band or orchestra will sound like in a newly designed auditorium.

Key Terms:

Acoustics, reverberation, comb effect, echo, direct sound, reverberation time, absorptivity, standing waves, dead spot, feedback, binaural hearing, reflection (specular and diffuse), exponential decay, attenuation, infrasound, matching room acoustics to type of music.

16a: Acoustic Qualities

There are several general properties that are desirable in an auditorium or concert space. We would like every listener to have exactly the same sound experience. Any noise external to the room should be excluded or minimized. The sounds coming from the performers should be heard clearly and distinctly. If there are any echoes they should be below some threshold relative to the total sound so as not to interfere with the sound being listened to and should not change the sensation of where the music is coming from. The performers should not hear distracting echoes. One of the largest factors impacting how the shape, size and construction of an auditorium affect the sound of a musical performance is reverberation.

16b: Reverberation

Reverberation is the combined effects of multiple echoes in a room or concert hall as heard by a listener. Nearly all indoor spaces have reverberation but it is often missing at outdoor venues. If the walls of the room are hard there will be repeated echoes which may begin to overlap and cause the musical sound to be mushy or muddled. If the walls absorb too much of the sound the room will sound dead. We rely on reverberation to give us a sense of how large the room is; large concert halls have a different reverberation time than a small room.

Video/audio examples:

Here is a web site with sounds of reverberation in an empty room versus one with carpet.
Before and after example of unwanted reverberation.

In most of the simulations, animations and drawings of waves in this book (and others) it is assumed that the amplitude (and hence the loudness) of a sound wave doesn't change much as it travels. If the wave front is from a distant source it will be approximately flat (a so called plane wave) and the amplitude does not change as the wave propagates. However for a local source the wave front spreads out in a spherical shape and the further away you are from the sound source the softer it sounds. In other words, the amplitude of a small sound source obeys an inverse square law as it spreads out into open space (Chapter 8). This means the sound intensity decreases as 1/r² where r is the distance from the source. As a result of the inverse square law the direct sound arriving at the listener will be slightly less loud than the sound at the source and all reflected sound will be even softer. Doubling the distance from the sound will cause the amplitude to be 1/2² = 1/4 as intense. Likewise tripling the distance means the intensity will be 1/9^th as much.

In the room without carpet in the demo (above) a sudden sound (a clap) was made in one location. It takes sound, traveling at 345 m/s a few milliseconds to reach the receiver (a video recorder in this case). This is the direct sound and is slightly less loud than the original sound at the source. Split seconds later echoes start to arrive at the receiver. Each echo arrives later and with lower volume than the previous one because it travels a bit further. For example the first echo might come off the ceiling, the next from the nearest wall, the next from the wall behind the receiver, etc. A graph is shown below of the initial sound (blue at t₁) followed by the direct sound (green at t₂) and then the echoes (red, t_reflected and afterward). Only the first few echos are shown.

Suppose a continuous sound is made at the front of an auditorium for some length of time instead of a clap like in the demonstration video. The first graph below shows the sound at the source, starting at time t₁, ending at time t_f and lasting a length of time, T. A listener will experience a slight delay until the sound reaches them (t_direct in the second graph) and hear the sound as starting at t₂. A split second after the listener hears the sound directly from the source they will begin to hear echoes off walls, floor and ceiling, causing the sound to be slightly louder (the stair steps in SIL starting at t_reflected). The listener does not hear the sound start to decay until a time t_direct after the sound is turned off (sound is still traveling to the listener for a brief time after the source is turned off). After a length of time T the listener hears the direct sound stop (at time t_decay) and only the echoes remain and they start to decay. The time it takes the sound to drop in volume by 60 dB from when it starts to decay is called the reverberation time (T_r in the graph).

The decay of sound in the graph is fairly linear ; it pretty much drops off in a straight line (the SIL scale is logarithmic so an exponential decay in intensity appears as a straight line in an intensity level graph). Real decay curves can have this shape or may be curved or may be irregular shaped. In part this depends on the shape of the room and the materials present which we investigate next.

An approximate equation (due to Wallace Sabine) for calculating reverberation time in seconds is T_r = (0.16 s/m) V/S_e where T_r is the reverberation time, V is the volume of the room in m³ and S_e is the effective absorption area in m². The effective absorption is S_e = α₁S₁ + α₂S₂ + α₃S₃ + ... where S₁ is the surface area that has absorptivity α₁ etc. If the absorptivity or absorption coefficient is zero the surface is a perfect reflector of sound. Absorptivity is measured in units of sabin. A perfect absorber would have an absorptivity of 1.0 S. Below is a table of the absorptivity of various surfaces, in Sabins. Notice that even painting a concrete wall has a significant effect on the absorptivity. This is because unpainted concrete and brick are porous and as a result can absorb more sound.

As we saw in Chapter 7, a path difference can cause destructive or constructive interference due to a phase difference. If there is strong reflection it can be the case that reflected sound and sound coming directly from the stage may reach a listener out of phase and thus cancel destructively for a particular wavelength as shown in the diagram below.

The effect depends on the amount of delay and can result in peaks and dips in the frequency spectrum at the listener location. This is sometimes called the comb effect and can be produced artificially by adding a recording to itself but with a slight delay. This can also lead to locations in a concert hall called dead spots, where the range of frequencies in the original sound source is not heard correctly. As a result of interference between the direct sound and reflected sound, different kinds of music sound better with different reverberation times. Medieval music was often performed in cathedrals with very long reverberation times, sometimes as long as 13 seconds. The composers and musicians of the time gradually modified the music to fit the venue. This type of music is usually very simple and uses lots of harmonic notes so that the echoes would harmonize with and reinforce the notes being sung. Pipe organ music generally sounds better with more reverberation for the same reason. In general, symphony music sounds best with less reverberation than pipe organs but with more reverberation than chamber music. Symphony halls typically have reverberation times of two to five seconds; 2.5 seconds is considered optimal. Speech usually sounds best with very little reverberation; too many echoes makes the speaker difficult to understand. Opera houses and speech auditoriums are designed to have reverberation times of 0.8 to 1.5 seconds. A typical living room in a house has a reverberation time of 0.4 seconds.

Reverberation can be artificially added to recorded and live music using electronic equipment (sometimes referred to as comb filters). This may be done, for example to a voice to give it a more full bodied sound or to make an outdoor concert sound like it is indoors. Reverberation may also be added to give a recording a sound as if the listener was in a larger auditorium when in fact they are listening from headphones. The sound tracts of most modern recorded music, movies and TV programs are heavily processed to overcome deficiencies in the sound quality due to the acoustics of the recording location.

Video/audio examples:

A short YouTube about reverberation.
A short YouTube about Balloon Pop with and without reverberation.

16c: Other Acoustical Effects

As we saw in Chapter 11 on stringed instruments, the equation for the Helmholtz frequency modes inside a rectangular container with height H, length L and width W is f_n,l,m = v/2 ((n/L)² + (l/H)² + (m/W)²)^1/2. Here v is the speed of sound in air and n, l, and m are mode numbers. For a given set of mode numbers there will be standing waves in the container. You can hear standing waves when you sing in a rectangular shower stall. Stand in the shower stall and, starting with a low tone, gradually increase the pitch. You will find certain notes that sound louder; these are the standing waves for that enclosure. In a rectangular auditorium the possibility of standing waves results in two problems. For a piece of music certain notes will be enhanced while others will not. You can explore these modes in this box modes simulation Applet (notice you can grab and rotate the box to see different modes from different angles).

A second problem when there are standing waves is the occurrence of nodes and anti-nodes. You can often detect these nodes by moving your head slightly to the left or right in a room where there are parallel walls and some ambient noise; you will notice the sound changes a little bit as you move your head. A listener seated at a node hears less of that frequency while a listener at a anti-node hears more. You may have noticed that auditoriums are not perfectly square but have sloping floors and ceilings and walls at an angle to the stage as seen in the picture below of Ríos Reyna Hall in Caracas, Venezuela. This is in part to eliminate or reduce standing waves. However the effects of a non-rectangular room on reflected sound can be complicated and unpredictable. As a result, acoustical design is still somewhat of an art.

Newly constructed concert halls may be built with floating floors, mounted on springs and dampeners to isolate the space from external vibrations. The walls and ceilings may also be isolated from the surroundings using acoustically isolating connectors to attach them to external walls and ceilings. This 'box within a box' construction greatly reduces interference from external noise.

If the building is already in place it may be possible to improve the sound and remove at least some of the standing waves by using absorbing barriers of various sizes and shapes. Recall from Chapter 7 that, although sound obeys the law of reflection, if the surface has irregularities of the same size as the wavelength the reflection will be diffuse instead of specular. Obviously objects like chandeliers, seating, and lights will also interact with sound wavelengths of the same size causing diffuse reflection. Diffuse reflection tends to help prevent standing waves and dead spots.

Although it is not known if the Echea of Marcus Vitruvius Pollio actually reinforced the sound of the actor's voices onstage, a similar type of resonance cavity is used today to reduce unwanted sound from fans in air ducts. If a resonance cavity is attached to an air duct and has a resonance frequency just below the fan frequency the cavity will produce a sound wave exactly out of phase of the sound from the fan. This tends to cancel the unwanted noise from the fan.

As a final aid to getting sound from the stage of a concert hall to the listeners, a orchestral or band shell is often used. As we saw in Chapter 7, waves emitted from the focal point of a concave mirror are reflected outward in parallel beams. An orchestra shell is half a concave parabolic mirror. The orchestra or band performs at or near the focal point of the curved surface and the surface projects the sounds out in even parallel paths to the audience. These shells can be used indoors or, as in the picture below, outside.

16d: Sound Reinforcement

As if the problem of sound reflection wasn't complicated enough, most auditoriums today use amplified sound in the form of microphones and speakers. If presented with two different sources (for example a singer and the amplified singing from a speaker) our ear-brain hearing system may be confused about where the sound is coming from. If the sounds arrive more than 30 milliseconds apart (the case when the speaker and singer are more than 10 m apart) we will hear an echo. If the speakers are closer to the listener than the original source the sound will appear to come from the speaker, not the singer because our ear-brain system identifies whichever signal getting to the ear first as the source. This is called the precedence effect and can be manipulated to fool the listener into thinking the sound comes from the singer and not the speaker. This will work, even if the direct signal is much weaker than the sound from the speakers. To arrange this, a very slight delay is often introduced in the amplification system so that the direct sound arrives just ahead of the amplified sound of the speakers. If there are speakers at different locations in the auditorium this requires careful arranging of speakers and the amount of delay as shown in the following diagram.

A common problem with amplified sound is feedback. Feedback occurs when sound from the speaker is fed back into the microphone (usually accidentally). The amplifier system tries to amplify the sound again (and again and again). Once the amplifier reaches its maximum capacity to amplify, the sound is distorted and an unpleasant, high pitched squeal results (although a few musicians have actually used feedback as part of their performance).

Video/audio examples:

Wikipedia on acoustic feedback.
Jimi Hendrix was a rock star in the 1960s who helped make the use of feedback in music popular.
Two discussions of acoustics: serious, for the cinema.

A sound sample of feedback: