Building Acoustics

Waves in fluid and solid media 91

The quantity I is the cross sectional area moment of inertia of the plate per unit width.
As mentioned above, the expressions given for the wave number and phase speed
presupposes that the plate is thin, i.e. the wavelength should be larger than six times the
plate thickness. Another way of expressing this is by demanding that cB should be less
than 0.3⋅cL. (How can you show this?) If this condition is fulfilled the error should be
less than 10%.
We may if need be, by using results from Mindlin (1951), calculate a corrected
phase speed c'B if the condition above is not fulfilled. The corrected phase speed is given
by

(^3333)
B BG

11 1

,

c' ccγ

=+

⋅

(3.106)

where γ is a factor depending on Poisson’s ratio υ according to Table 3.2.

Table 3.2 Correction table for Equation (3.106).

υ 0.2 0.3 0.4 0.5

γ 0.689 0.841 0.919 0.955

Examples of calculated phase speed for concrete plates, in the thickness range of
50–200 mm, are shown in Figure 3.19. Corresponding data for steel plates, having
thickness covering the range of 1–10 mm, are shown in Figure 3.20. Calculated results in
both diagrams are performed using thin plate theory as well as thick plate theory (see
Equations (3.103) and (3.106)). For the chosen range of plate thickness and frequency
range there is practically no difference when it comes to the steel plates. The limit on the
thin plate theory will in this case correspond to a phase speed of approximately 1500 m/s.
As for the concrete, however, the corresponding limit will approximately be 1000 m/s,
which may be seen clearly from the two sets of curves.
Shown in both figures is also the phase speed in air. The point of intersection
between this line and the corresponding curves for the different plate thickness, i.e.
where cair is equal to cB, is defining the so-called critical frequency fc. This quantity is of
fundamental importance when it comes to sound radiation from plates in bending
vibrations (see section 6.3.3).

3.7.3.2 Eigenfunctions and eigenfrequencies (natural frequencies) of plates

In accordance with the general observations in section 2.5.3, we are in a position to
describe the vibrations in structural elements, such as beams, plates and shells, by
eigenfunctions and corresponding eigenfrequencies. Starting out from these functions we
may, in an analogous way as in section 3.6 above, e.g. calculate transfer functions
between an input force and a chosen velocity component. This will be a continuance of
the calculations on discrete (lumped) mechanical systems given in sections 2.5.1 and
2.5.2.