258 Building acoustics
6.6.1 Vibroacoustic reciprocity, background and applications
Generally, presenting the reciprocity theorem one uses the purely acoustical case as
depicted in Figure 6.30. The sound pressure at a given frequency measured at a certain
point in a fluid, this caused by an acoustic monopole situated at another position is
independent of an interchange of source and receiver. This is true even if the
transmission path comprises different media, boundary surfaces giving diffraction etc.
The only basic requirement is that the boundary surfaces react linearly. Questions related
to the influence of the dynamic behaviour of boundaries have engaged many scientists,
e.g. will the principle work when including porous materials in the system? Does it
require the boundaries to be locally reacting?
In fact, Rayleigh’s general principle of reciprocity implicitly implies that all types
of component may take part in the dynamic process, provided that their kinetic, potential
or dissipative energy is finite and positive functions of the velocity. Vibrating structures
may then take part without invalidating the principle, this being formally proved by
Lyamshev (1957). From this it follows that the transfer function between a mechanical
point force applied to a structure (e.g. a plate or a shell) and the sound pressure in a point
(see Figure 6.31) may be determined by exciting the structure by sound emitted by an
acoustic monopole.
Figure 6.31 Application of the principle of reciprocity. Mechanical forces and sound radiation.An extension of this point-to-point connection was to prove that a similar reciprocal
relationship exists between the sound radiation from a mechanical structure vibrating in a
given mode and the response of this mode for incident sound. This leads to the question
we shall be concerned with; the relationship between the point response of a structure to
diffuse field incidence and the radiated power from the structure excited in the same
point. The derivation is given in Cremer et al. (1988), where the following thought
experiment is presented:
A plate, comprising a part of the wall in a room, is driven by a point force F (see
Figure 6.32). We shall assume that the sound field set up in the room is diffuse and that
the sound power emitted may be expressed by
WaF= ⋅^2 , (6.114)
where a is a factor of proportionality. The power sets up an acoustic field in the room,
with a resulting sound pressure
p
F
u
Q
p
F
u
Q
=