Building Acoustics

194 Building acoustics

the former, for which the attenuation is large due to the viscous coupling between frame
and air, is traditionally denoted the slow wave.
The effect of an elastic frame may be quite pronounced for certain types of
absorber, especially for plastic foam materials. A simulated result is shown in Figure
5.29, where the shear modulus G of the frame is reduced from a maximum value (G1) of
2.0⋅ 107 (1+j⋅0.1) Pa in steps of 10. The other parameters for the material are identical to
the ones used in Figure 5.28 having Λ equal to 20μ. The calculations are performed
using a full Biot-model following the procedure and formulae given by Brouard et al.
(1995). To compare, we have also plotted the data, shown by the circular points, from the
latter figure that assumes an infinitely stiff frame according to the Allard/Johnson model.
Apparently, the results using the maximum value of the shear modulus gives nearly
identical result as a calculation assuming an infinitely stiff frame.

Figure 5.29 Normal incidence absorption factor of a material with an elastic frame modelled using Biot theory.
The shear modulus is reduced in steps with a factor of 10 from G1 = 2.0⋅ 107 (1+j⋅0.1) Pa. Other data are the
same as used in Figure 5.28 (Λ = 20μ). Symbols indicate data calculated by the Allard/Johnson model given by
Equation (5.65).

100 1000

200 400 600 800 2000 4000

Frequency (Hz)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

A

bso

rption

facto

r

Biot - G1

Biot - G2

Biot - G3

Allard-Johnson

Another example is given in Figure 5.30, which shows a comparison between
measured and calculated results, again using a full Biot model. The measurements are
conducted in a free field environment using a two-microphone technique. The
calculations are performed using two different methods, an analytical one and one using
a finite element method (FEM). The analytical method is the same as used in Figure 5.29.
Apparently, the two calculation methods are consistent with each other and the general
appearance is validated by the measurement results. It should be noted that a linear
frequency axis is used in this case.
The absorption has a maximum value in the frequency range 700–800 Hz, which is
caused by a rather strong movement of the frame at these frequencies. Calculating the
displacement at the surface of the foam material, a calculation possible by using FEM
technique, is shown in Figure 5.31. The displacement is calculated for a normal