Room acoustics - Building Acoustics - page 104

having walls of infinite stiffness. Solutions for such special cases may, however, give
some general information on sound fields in rooms. It is therefore useful to discuss some
of these cases, which we shall return to in section 4.4.2.
The development of numerical techniques in recent time has been formidable,
which include FEM (finite element methods), BEM (boundary element methods) and
various other numerical methods for predicting sound propagation in bounded spaces.
Using these, accurate solutions may be obtained for complex room shapes and boundary
conditions. First and foremost, these techniques are suitable in the lower frequency
ranges, i.e. when the ratio between a typical room dimension and the wavelength is not
too large. When using a FEM technique a reasonable number of elements per wavelength
are of the order three to four. If the typical room dimension is 10 metres one may at 100
Hz perhaps use 1000 elements. However, to calculate with the same accuracy at 1000 Hz
one needs 1000000 elements. Depending on the specific computer FEM software,
different types of elements are implemented, having some 8 to 20 nodes. At each of these
nodes we shall then calculate the sound field quantity in question. In spite of the large
capacity of modern computers, the limitations imposed on these calculations should be
obvious. It should, however, be stressed that FEM calculations have become very
important tools in the area of sound radiation and sound transmission, in particular where
a strong coupling between a vibrating structure and the surrounding medium is expected.
A number of other approximate methods have a long history in room acoustics. The
reason is that one normally is not interested in a detailed description frequency by
frequency. The average value in frequency bands, being either octave or one-third-octave
bands, has been more relevant. In the literature one will therefore find methods
characterized under headings such as statistical room acoustics and geometrical room
acoustics. The first term implies treating the sound pressure in a room as a stochastic
quantity with a certain space variance. The classical diffuse field model, also called the
Sabine model, is an extreme case in this respect. The latter name is a recognition of the
American scientist Wallace Clement Sabine (1868-1919) who published his famous
article “Reverberation” in the year 1900 containing a formula for the reverberation time
in rooms, a formula still being the most used. In a diffuse field model, the space variance
of the sound pressure is zero, the energy density is everywhere the same in the room.
Such a model may be seen as the acoustic analogue of the classical kinetic gas model.
There is also a long tradition for using geometrical models in acoustics, see e.g.
Pierce (1989). For geometrical acoustics in general, also denoted ray acoustics, the
concept of wave front is central. At a given frequency, a wave front is a surface where
the sound pressure everywhere is in phase. As the wave front moves in time, the line
described in space by a given point on the surface is called the ray path. Generally, it is
not necessary to assume that the amplitude is constant over the wave front or that the
wave front is a plane surface but in room acoustics this is assumed. Curved paths have no
place in geometrical room acoustics; the sound energy propagates along straight ray
paths just like light. Inherent in these geometrical models there is no frequency
information and the validity of the calculated results is in principle limited to a frequency
range where we may assume specular reflections and where diffraction phenomena may
be neglected. Such phenomena may, however, be included in these models by certain
artifices. We shall deal with them by giving an overview of the principles.