Building Acoustics

Statistical energy analysis (SEA) 267

Energy will be lost by radiation to the surroundings and by the exchange of energy
by a thermal coupling. We shall look at the possible combinations when the radiation
losses and the conductivity in the coupling, respectively, either has a low or high value.
The number of squares in Table 7.1 represents the energy (temperature) inside the
subsystems for the different combinations, certainly giving just a qualitative picture. A
situation which we shall look into further on is when the radiation losses are small
combined with a high conductivity, i.e. strong coupling. In this case the subsystems are
“sharing” the energy; there will be a so-called equipartition of energy.

Table 7.1 A qualitative picture of the energy (temperature) in the thermal system shown in Figure 7.1.

Radiation loss High Low

Conductivity System 1 System 2 System 1 System 2

High „„„ „„ „„„„„„„„ „„„„„„

Low „„„„ „ „„„„„„„„„„ „„„„„

Transferring this model into a vibroacoustic one having resonant acoustic volumes
(rooms etc.) and resonant solid structures, we may establish the following analogous
quantities:

• Thermal radiation losses Ù Losses due to absorption, internal losses in
  materials characterized by the reverberation time T 60 or loss factor η.


• Conductivity Ù Measure of the coupling strength, coupling loss factor ηij
  (may also be given by an impedance).


• Temperature (energy) Ù Sound pressure level in room, vibration level
  (velocity etc.) of solid structures.


• Thermal capacity Ù Modal density.

The last item is not self-evident but it is connected to one of the most important

assumptions for SEA.

7.2.2 Basic assumptions

The most important assumptions behind the method are the following:

1. The loss of energy within a subsystem is proportional to the total energy of the
   subsystem.


2. The energy transmitted from one subsystem to another is proportional to the
   modal energy difference.


3. The forces driving the different subsystems are independent, statistically
   speaking. We may add the energy response resulting from these forces to
   arrive at the total (modal) energy of each subsystem.
   To give an illustration of these assumptions and express them in a mathematical form,
   we shall again start with a system having two components or subsystems, marked 1 and
   2 in Figure 7.2. The total energy is E 1 and E 2 , respectively, of the subsystems having
   modal densities n 1 and n 2. The symbol W represents the power; Win for input power, W ́
   for transmitted power between subsystems and Wdiss for the energy dissipated or lost