Sound transmission 257
Apart from the very low frequency range we get a very good fit between measured
and calculated results. The discrepancies at low frequencies may be caused partly by the
limited specimen size, partly by a certain stiffness making the mats act like a plate. A
more probable explanation is found in the measurement conditions; very low reduction
indexes imply feedback between the sending and receiving room. When deriving
Equation (6.6), which gives the sound reduction index, we implicitly assumed that there
was no coupling between these rooms. At the same time one may get problems
concerning the diffusivity when the partition is strongly absorbing.
6.6 A relation between airborne and impact sound insulation
We have in general treated our two cases, a construction’s behaviour by point impacts
versus a distributed excitation as a sound field, separately. Under certain conditions,
however, we are able to derive a direct relationship between these two ways to
characterize the sound insulation properties of a partition, i.e. a relation between the
sound reduction index and the impact sound pressure level. One then makes use of a very
important principle in vibroacoustics, the principle of reciprocity, a principle we have
referred to several times in the preceding chapters. Reciprocity implies a mutual
relationship and in its most general form tells us that the response at a certain point in a
linear elastic system that is caused by exciting another point in the system is invariant by
interchanging the source and receiver.
This was postulated by Lord Rayleigh as far back as 1873, but one had to wait
nearly 100 years before a formal proof was put forward by Lyamshev (1957). This laid
the foundation for many practical applications of the principle. Among these are many
connected to sound radiation from vibrating structures, response of structures excited by
sound fields and sound transmission. In all cases, reciprocal measurements of transfer
functions may often be simpler and less time consuming than the equivalent direct
measurements.
Figure 6.30 Acoustic reciprocity. Exchanging the position of a monopole source and the position of the sound
pressure measurement.
More recently Fahy (1995) has presented an extensive review of theory and
practical applications. We shall not repeat it here, only to give a short overview and
present an expression for the relationship between the radiated acoustic power from a
structure excited by a point force and, conversely, the velocity when the structure is
excited by a diffuse sound field. Next, we shall use this relationship to derive an
expression linking the sound reduction index and the impact sound pressure level.
p 2
p 1
Q 1
Q 2
p 1
Q 1 =p 2
Q 2