Building Acoustics

246 Building acoustics

calculatation assuming an ideal diffuse incident sound field; i.e. assuming sound
incidence evenly distributed over all angles and with random phase. We shall approach
the practical problem by again going back to our infinitely large plate.
In Chapter 5 (section 5.5.3.2) we calculated the statistical absorption factor for a
sound-absorbing surface. The same type of integral may be used to calculate a statistical
or diffuse field transmission factor τd. We shall write

2

d

0

2()sincosd.

π

τ = ∫τφφφφ (6.99)

Inserting the transmission factor from Equation (6.98) with the wall impedance
according to Equation (6.91) we shall not be able to give an analytical solution to the
integral. Limiting the solution to low frequencies by using the approximation given by
Equation (6.80), the result may be written

(^) d00[]
d

1

RRR10 lg 10 lg 0.23 (dB).

τ

=⋅ = −⋅ (6.100)

The expression is commonly referred to as the diffuse field mass law. Cremer presented a
similar expression as far back as 1942, valid for frequencies above the critical frequency,
which we shall write

(^) d c
0c

2

10 lg 10 lg 5 dB for.

fm f

Rff

Zf

⎡⎤ ⎛⎞πη

=+−⎢⎥⎜⎟ >>

⎣⎦ ⎝⎠

(6.101)

We shall return to these expressions below when we address the problem of transmission
through real walls or panels, taking the finite size into consideration.

6.5.2 Sound transmission through a homogeneous single wall

Starting out from the observations on sound transmitted through an infinitely large plate,
we shall move on to the practical case of sound transmission from one room to another
by way of a single homogeneous wall or floor. Several options are available for
calculations. One possibility is an analytical solution starting out from a description of
the sound field in the rooms coupled to the structural wave field in the wall, all of them
expressed by a sum of the natural modes. The task is then to calculate the coupling
between each of these modes, which is a relatively complex task (see e.g. Josse and
Lamure (1964) or Nilsson (1974)). Using finite element methods (FEM) may be seen as
a modern version of such procedures (see e.g. Pietrzyk (1997)). The power of such
methods lies in the ability to investigate specific situations, preferably in the lower
frequency range.
Statistical energy analysis (SEA) is also a powerful method under the condition that
the modal fields have a sufficient number of eigenfrequencies inside the actual frequency
band (see e.g. Craik (1996)). The strength of this method lies therefore in treating
problems in the mid and high frequency range. We shall therefore give an introduction,
backed up by some examples, for this method in the next chapter.