2.1. D EFIN ITIO N OF ROOM ACOUSTICS
Consider a sound source w hich is situated in a room. Sound waves w ill propagate away from the source until they encounter one of the room's boundaries w here, in general, some of the sound energy w ill be reflected back into the room, some w ill be absorbed and some w ill be transmitted through the boundary. The complex sound fie ld produced by th e multitude of reflections and the behaviour of this sound fie ld as the sound energy in the room is allowed to build up and decay constitutes the acoustics of the room.
2.2. G EOMETRICAL ROOM ACOUSTICS
If one can assume that the dimensions of a room are large compared to the wavelength of sound then one may treat th e sound waves in the room in much the same way as light rays are treated in geometrical optics. This situa tion frequently occurs in architectural acoustics. In analogy w ith light rays, sound rays are reflected from hard plane w a lls in accordance w ith the laws of reflection i.e. the incident ray, the reflected ray and the normal to the sur face at the point of incidence all lie in the same plane; the angle of incidence is equal to the angle of reflection (Fig. 2.1). Therefore sound rays incident on
a curved surface w ill either be focused or dispersed depending on w hether the surface is concave or convex (Fig.2.2). Diffraction of sound rays can and does occur but the effect is more noticeable for low frequency, long w ave length sounds than w ith high frequency sounds of short wavelength.
Fig. 2.2. Reflections o f sound rays
Fig. 2.3. G raphical construction of the firs t reflections of the sound waves in a concert hall
The concept of a sound ray and the geometrical study of sound ray paths plays an im portant role in the design of large rooms and auditoria, enabling troublesome echoes and flu tte r effects to be detected and dealt with at the
stage of designing the building. Fig.2.3 shows how geometrical constructions can be used to position sound reflectors on the ceiling of a concert hall in o r der to improve the distribution of sound. A lim itation of the geometrical ap proach is that usually only the primary and possibly the secondary reflections can be studied before the sound ray being follow ed becomes "lo s t" in the re- verberent sound field.
2.3. GROW TH AND DECAY OF SO UND IN A ROOM
W hen a sound source is placed in a room, the sound intensity as measured at a particular point w ill increase in a series of small increm ents, due to the reflections arriving from the w alls, floor and ceiling, until an equilibrium posi tion is attained where the energy absorbed by the room is equal to energy ra diated by the sound source. When the sound source is abruptly switched o ff the sound intensity in the room w ill not suddenly disappear but w ill fade away gradually, the rate of decay being prescribed by the am ount and posi tion of the absorbing material in the room. This lingering of the sound is known as reverberation. The rate of absorption of sound energy in the room w ill be, in the main, proportional to the sound intensity so th at the growth and decay of sound pressure in the room is an exponential function of tim e (Fig.2.4.).
If one measures the sound pressure levels in dB in a decaying reverberant field as a function of tim e then one obtains a reverberation curve which is us ually a fairly straight line although the exact form depends on many factors in cluding the frequency spectrum of the sound source and the shape of the room.
2.4. REVERBERATION TIM E
At the beginning of this century W.C. Sabine carried out a considerable amount of research on the acoustics of auditoria and arrived at an empirical relationship between the volume of the auditorium , the am ount of absorptive material w ith in the auditorium and a quantity which he called the reverbera tion tim e. This relationship is now known as the Sabine form ula:
RT =
0,161 y
A (2.1)
w here RT = the reverberation time defined as the tim e taken for a
sound to decay by 60 dB after the sound source is abruptly switched off
V = the volume of the auditorium in m3
A = the total absorption of the auditorium in m2 -sabins.
The absorption unit of 1m2 -sabin represents a surface capable of absorbing sound at the same rate as 1m2 of a perfectly absorbing surface e.g. an open window.
2.5. ABSORPTIO N COEFFICIENT
Material Frequency, Hz
125 250 500 1000 2000 4000
Air, per cu. m. nil nil nil 0,003 0,007 0,02
Acoustic paneling 0,15 0,3 0,75 0,85 0,75 0,4 Plaster 0,03 0,03 0,02 0,03 0,04 0,05 Floor, concrete 0,02 0,02 0,02 0,04 0,05 0,05 Floor, wood 0,15 0,2 0,1 0,1 0,1 0,1 Floor, carpeted 0,1 0,15 0,25 0,3 0,3 0,3 Brickwall 0,05 0,04 0,02 0,04 0,05 0,05 Curtains 0,05 0,12 0,15 0,27 0,37 0,50 Total absorption of
one seated person 0,18 0,4 0,46 0,46 0,51 0,46
780120
The absorption coefficient of a material, as originally defined by Sabine, is the ratio of the sound absorbed by the m aterial to that absorbed by an equiva lent area of open w indow hence the absorption coefficient of a perfectly ab sorbing surface would be 1 .Providing one knows the superficial areas and the absorption coefficients of the various m aterials to be used, the reverbera tion tim e of an auditorium can be determ ined at the design stage. To fa c ili tate such calculations sets of tables have been published giving the absorp tion coefficients of the commonly used building materials as a function of fre quency. The variation in the reverberation tim e of specially designed reverber ant rooms (also known as live or hard rooms) as one introduces or removes absorptive material is a standard method for determining absorption coeffi cients (refer to the International Standard ISO 354). (See Chapter 6).
2.6. D ERIVATIO N OF FORMULAE FOR REVERBERATION TIM E
Theoretical derivations of Sabine's form ula are usually based on geometri cal acoustics utilising the assumptions that the sound in the enclosure is d if fuse and that all directions of propagation are equally probable. This is a gross sim plification of the actual behaviour of sound in enclosures because it neglects such important factors as room modes, the positioning of absorptive material, the influence of the shape of the room and others. For fairly reverb erant rooms w ith a uniform distribution of absorptive m aterial, Sabine's fo r mula gives a good indication of the expected behaviour of sound in the room. As a room becomes more and more "d ea d " i.e. the boundaries become more and more absorbent, so the results obtained from employing Sabine's fo r mula become more and more inaccurate. In the limiting case of a completely dead or anechoic room w here the absorption coefficient of the boundaries is unity then the reverberation tim e is obviously zero because a reverberant field cannot exist in these conditions. However, for this situation Sabln's fo r mula gives a fin ite value of 0,161 V /A for the reverberation tim e. Several d if ferent approaches have been used to derive equations w hich give values of reverberation time which are in better agreement w ith the measurement re sults for rooms containing little absorption. As examples, tw o such formulae are quoted here.
Eyring's formula for reverberation time is
RT — S ln(1 — a ) 0,161 1 / (2.2) where _ _ « tS -i + « 2 ^ 2 + • • • + y - n S n S i + s 2 + ■ • • + sn
+ sn the areas of the various materials
a-,, cx2 , • • • the respective absorption coefficients.
Eyring's formula gives results w hich are in much better agreement w ith the measured reverberation times for dead rooms than Sabine's form ula. Also, the Eyring form ula gives the correct value of RT = O for an anechoic room i.e. fo r a = 1. One drawback of this improved form ula is that it is only strictly valid for rooms w hich have the same value of a for all boundaries.
The theory of M illington and Sette leads to the form ula
RT = £ - s, ln (1 - x,)0 ' - r - 1(T - --- Y ( 2 3 )
w here Si = the area of the ith m aterial
«j = the absorption coefficient of the ith m aterial.
W hen the materials in a room have a wide variety of absorption coefficients then the best predictions of reverberation times are obtained by employing the M illington and Sette formula. This form ula can be obtained by s u b s titu t ing the effective sound absorption coefficient ae = — ln(1—«¡1 into Sabine's form ula.
M illington and Sette's formula indicates that highly absorbing m aterials are more effective than would be anticipated in influencing the reverberation
tim e. For example, w hen the absorption coefficient of a material is greater
than 0 ,6 3 then the effective absorption coefficient is seen to be greater than one.