Geometrical room acoustics
4.4 Enclosures with curved walls
In this section we consider enclosures, the boundaries of which contain curved walls or wall sections. Practical examples are domed ceilings, as are encountered in many theatres or other performance halls, or the curved rear walls of many lecture theatres. Concavely curved surfaces in rooms are generally considered as critical or even dangerous in that they concentrate the sound energy in certain areas and thus impede its uniform distribution throughout the room.
Formally, the law of specular reflection as expressed by eqn (4.1) is valid for curved surfaces as well as for plane ones, since each curved surface can be approximated by many small plane sections. Keeping in mind the wave nature of sound, however, one should not apply this law to a surface the radius of curvature of which is not very large compared to the acoustical wavelength. Whenever the radius of curvature is comparable or even smaller
Geometrical room acoustics 117 than the wavelength the surface will scatter an impinging sound wave rather than reflect it specularly, as described in Section 2.7.
Very often, curved walls in rooms or halls are spherical or cylindrical segments, or they can be approximated by such surfaces. Then we can apply the laws of concave or convex mirrors, known from optics. In the following it should be noted that the direction of the ray paths can be inverted.
In Fig. 4.11a the section of a concave, spherical or cylindrical mirror is depicted. Its radius of curvature isR. A bundle of rays originating from a point S is reflected at the mirror and is focused into the point P from which it diverges. Focusing of this kind occurs when the distance of the source from the mirror is larger thanR/2; if the incident bundle is parallel the focus is at distanceR/2. The source distancea, the distance of the focusb and the radius of the mirror are related by
1 a+1
b =2
R (4.17)
Figure 4.11 Reflection of a ray bundle from a concave and a convex mirror.
If the source is closer to the mirror thanR/2 (see Fig. 4.11b), the reflected ray bundle is divergent (although less divergent than the incident one) and seems to originate from a point beyond the mirror. Equation (4.17) is still valid and leads to a negative value ofb.
Finally, we consider the reflection at a convex mirror as depicted in Fig. 4.11c. In this case the divergence of any incident ray bundle is increased by the mirror. Again, eqn (4.17) can be applied to find the position of the
‘virtual’ focus after replacingRwith−R. As before, the distancebis negative.
The effect of curved surfaces can be studied more quantitatively by com- paring the intensity of the reflected ray bundle with that of a bundle reflected at a plane mirror. The latter is given by
I0= A
|a+x|n (4.18a)
while the intensity of the bundle reflected from the mirror is Ir= B
b−xn (4.18b)
In both formulaeAandBare constants;x is the distance from the centre of the mirror, and the exponentn is 1 for a cylindrical mirror and 2 for a spherical one. (It should be kept in mind that the distancebhas a negative sign for concave mirrors witha<R/2 and for convex mirrors.) Atx=0 both intensities must be equal, which yieldsA/B= |a/b|n. Thus, the ratio of both intensities is
Ir I0 =
1+x/a 1−x/b
n (4.19)
Figure 4.12 plots the level Lr=10·log10(Ir/I0) derived from this ratio for the cases depicted in Fig. 4.11 withn=2 (spherical mirror). The focus occurring for a>R/2 (curve a) is clearly seen as a pole. Apart from this, there is a range of increased intensity in whichLr>0. From eqns (4.17) and (4.19) it can be concluded that this range is given by
x<2 1
b−1 a
−1
= 1
R−1 a
−1
(4.20) Outside that range, the levelLris negative, indicating that the reflected bun- dle is more divergent than it would be when reflected from a plane mirror.
If a<R/2 (curve b), the intensity is increased at all distances x. Finally, the convex mirror (curve c) reduces the intensity of the bundle everywhere.
Geometrical room acoustics 119
Figure 4.12 Level difference in ray bundles reflected from a spherical and a plane reflector at distancex: (a) concave mirror,a=2R; (b) concave mirror,a=R/3; (c) convex mirror,a=2R.
According to eqn (4.19) the limit ofLrfor very large distances is Lr→10n·log10
b a
forx→ ∞ (4.21)
From these findings a few practical conclusions may be drawn. A concave mirror may concentrate the impinging sound energy in certain regions, but it may also be an effective scatterer which distributes the energy over a wide angular range. Whether the one or the other effect dominates depends on the positions of the source and the observer. Generally, the following rule3can be derived from eqn (4.20). Suppose the mirror in Fig. 4.11a is completed to a full circle with radiusR. Then, if both the sound source and the receiver are outside this volume, the undesirable effects mentioned at the beginning of this section are not to be expected.
The laws outlined above are valid only for narrow ray bundles, i.e. as long as the inclination of the rays against the axis is sufficiently small, or, what amounts to the same thing, as long as the aperture of the mirror is not too large. Whenever this condition is not met, the construction of reflected rays becomes more difficult. Either the surface has to be approximated piecewise by circular or spherical sections, or the reflected bundle must be constructed ray by ray. As an example, the reflection of a parallel bundle of rays from a concave mirror of large aperture is shown in Fig. 4.13. Obviously, the reflected rays are not collected just in one point; instead, they form an enve- lope which is known as a caustic. Next to the central ray, the caustic reaches the focal point in the distanceb=R/2 in accordance with eqn (4.17) with a→ ∞. Another interesting shape is the ellipse or the ellipsoid, which has
Figure 4.13 Reflection of a parallel ray bundle from a spherical mirror of large aperture.
Figure 4.14 Collection of sound rays in an elliptical enclosure.
two foci F1 and F2, as shown in Fig. 4.14. If a sound source S is placed in one of them, all the rays emitted by it are collected in the other one. For this reason, enclosures with elliptical floor plan are plagued by quite unequal sound distribution even if neither the sound source nor the listener are in a geometrical focus. The same holds, of course, for halls with a circular floor plan since the circle is a limiting case of the ellipse.
Geometrical room acoustics 121
Figure 4.15 Whispering gallery.
A striking experience can be made in such halls if a speaker is close to its wall. A listener who is also next to the wall, although distant from the sound source (see Fig. 4.15), can hear the speaker quite clearly even if the latter speaks in a very low voice or whispers. An enclosure of this kind is said to form a ‘whispering gallery’. The explanation for this phenomenon is simple. If the speaker’s head is more or less parallel to the wall, most of the sound rays hit the wall at grazing incidence and are repeatedly reflected from it. If the wall is smooth and uninterrupted by pillars, niches, etc., the rays remain confined within a narrow band: in other words, the wall conducts the sound along its perimeter. Probably the best known example is in St.
Paul’s Cathedral in London, which has a gallery at the circular base of the dome. Generally, a whispering gallery is an interesting curiosity, but if the hall is used for performances, the acoustical effects caused by it are rather disturbing.
4.5 Enclosures with diffusely reflecting walls,