Building Acoustics

304 Building acoustics

(^) ()

()

2

00

sa

00 00

sa

,cos

cos cos

c

ZZ

cc

ZZ

U

τMθ M

UU

MM

−

=

⎛⎞⎛

⎜⎟⎜++

⎝⎠⎝

.

⎞

⎟

⎠

(8.38)

a)

b)

Figure 8.23 a) Anti-symmetric and b) symmetric wave motion in a sandwich element with compressible core.

To calculate the transmission factor for diffuse field incidence, we shall as usual
integrate the expression; over a range 2π for the azimuth angle θ and up to an angle of
approximately 80° for M, the latter to simulate laboratory conditions.
The derivation of the expressions giving Zs and Za starts from the equations of
motion for the face sheets and the core. The equations for the face sheets allows for in-
plane movements and for bending, whereas the core is described as a homogeneous and
elastic material allowing for dilatational as well as shear wave motion. A 4x4 impedance
matrix is set up to represent the core, relating both the normal and shear stress amplitudes
to the velocity amplitudes of the face sheets, transverse as well as in-plane.
The next step is to link these impedance components, i.e. the matrix coefficients, to
the equations of motion of the face sheets to arrive at the sought-after velocity
amplitudes caused by the incident sound pressure. In this way, Moore and Lyon arrived
at, given identical face sheets, two uncoupled equations describing the symmetric and
nonsymmetric motion, respectively, and thereby to explicit expressions for Zs and Za for
direct input to Equation (8.38).
Measured and predicted results for a sandwich panel of 13 mm plasterboards and a
55 mm thick core of a polyurethane foam material (PUR) is shown in Figure 8.24.
Taking account of the uncertainty of the material properties, the calculations are
performed using two different values for the modulus of elasticity. The applied material
data are given in Table 8.2.
As apparent from the figure, the fit between predicted and measured results is very
good. The double wall resonance, caused by the symmetric or dilatational movement, is
evident around 800 Hz in the same way as the effect of coincidence shows up in the
frequency range 2500–3000 Hz. The frequency of the former one, the lowermost
dilatational resonance, is given by