Wave Theory and Normal Modes

Wave theory is based on the study of wave motion within three-dimensional enclosures. It requires the establishment of boundary conditions which describe mathematically the acoustical properties of the walls, ceiling and other surfaces in the room. The difficulty involved in determining these boundary conditions for irregular shape and rooms containing furniture means that true analysis is possible for only a small number of idealised situations. Although the practical application of wave theory is quite limited, an understanding of its basis is essential in order to appreciate many of the problems which arise in room acoustics.

When employing wave theory, a room is considered as a complex resonator possessing many normal modes of vibration which are excited when a sound source is introduced to the room. The acoustic energy generated by the source acts to excite these room modes with the resulting sound energy residing in the standing waves established in the room. The characteristic frequencies of these vibrations depend on the room size and shape whereas the damping (or absorption) of the resulting waves depends upon the boundary conditions. Thus, every room imposes its own characteristics on to the sound source present.

Geometric Acoustics

If one assumes that the dimensions of a room are large compared to the wavelength, then sound waves may be considered in much the same way as light rays are treated in optics. This situation frequently occurs in architectural acoustics, especially in large auditoria. To continue the light analogy, sound rays are reflected from hard planar walls in accordance with the laws of reflection, i.e.: the incident ray, the reflected ray and the normal to the surface all lie on the same plane and the angle of incidence is equal to the angle of reflection. In the same way, sound rays incident on a curved surface will be either focussed (for concave) or dispersed (for convex).

Figure 1 - The effect of concave and convex shape on geometric sound rays.

The concept of a sound ray and the geometrical study of sound ray paths play an important role in the design of large rooms and auditorium, enabling troublesome echoes and flutter effects to be detected and dealt with at the design stage. A limitation of the geometrical approach is that usually only primary and possible secondary reflections can be studied before the sound ray being followed becomes 'lost' in the reverberant sound field and, in most enclosures, it is restricted to frequencies of 500 Hz and above.

Using Geometric Acoustics

Statistical methods are useful at the earliest stages of design, however, as more and more geometric information becomes available, why not use it. As architects, we need to be able to determine not only how much absorber to use, but what type of absorber and where to best put it. This is where the consideration of reflected sound rays can be quite useful.

The Placement of Reflectors and Absorbers

By analysing the paths of sound rays, it is easy to determine which areas require reinforcement (in the form of a reflector) and which require damping (in the form of absorber). Whilst designing for speech will be covered in more detail later, consider someone speaking at the rate of up to 8 syllables per second. Each syllable takes about 125 ms. Therefore, if clear reflections of the first syllable arrive midway through the second (or even the third) the speech may not be easily discernible by the listener.

In fact, work by Haas shows that the majority of reflected sound should arrive within 50 ms in order for it to reinforce the speech. This means that a path difference of 17 metres (340 x 0.05) between the direct and indirect sound is allowed for reinforcement and up to 43 metres (340 x 0.125) before the reflections become truly detrimental.

In this way, it is possible to design reflectors to minimise reflection paths to the entire audience area and place absorber to minimise the effect of late reflections.

Faults Attributable to Geometry

Statistical methods are quite limited in their capacity to predict acoustic faults within an enclosure. This is because most faults result from the geometry of the enclosure. A simple geometric analysis done on the drawing board can easily correct most of these at the design stage. The following are a few of the more common faults attributable to room geometry.

Spurious Echoes:

Occur when a strong reflection of the original signal can be clearly discerned by the listener. This is simply a matter of looking at the internal envelope of the enclosure and checking for possible sound paths which reflect off a sequence of large, highly reflective surfaces.

Figure 2 - The effect of a strong reflection arriving very late compared to the direct sound can be perceived as a distinct echo.

Picket Fence Echo:

This results from evenly spaced reflection paths, such as the rows of raised seating in amphitheatres and the evenly spaced curves of compressed fibre fencing. Depending on the number of steps and the path difference, such surfaces can produce a definite pinging sound when stuck by an impulsive sound source. The frequency of this ping is given by:

F = c / (2d)

where:
c = the speed of sound in air (m/s), and
d = the distance between successive steps (m).

Figure 3 - The reflected sound waves from repetitive elements can sometimes be detected as an audible 'pinging' sound.

Flutter Echo:

Occurs when both the source and receiver are between a pair of parallel, hard, surfaces. Some portion of the sound emitted by the source will get 'trapped' between the two reflective surfaces and will oscillate back and forth, being quite slow to decay. The listener will perceive this as a 'fluttering' noise. If the walls are a distance d apart, then the frequency of this flutter can be found in the same way as picket fence echo above.

Figure 4 - When sound energy gets trapped between two parallel reflective surfaces, a kind of 'flutter' can be detected by listeners.

Dead spots:

These can occur at positions which are far from reflecting surfaces and which receive sound only after it has passed over an absorbent surface. For example, at the rear of a gently raked theatre or cinema where the sound must pass over the audience and ceiling reflections are blocked by a balcony.

Figure 5 - A acoustic dead spot can occur where the direct sound is not clearly audible and no secondary reflections are heard to reinforce it.