224 Building acoustics
2
0
c 2.c m
f
π B= (6.41)
For homogeneous plates we may write
2
0
c
L,
1.8
c
f
ch≈
⋅ ⋅
(6.42)
where cL is the phase speed for longitudinal waves and h is the plate thickness. Figure
6.11 shows the critical frequency for plates of some typical building materials.
6.3.4 Sound radiation from a finite size plate
We showed in the previous section that for frequencies lower than the critical frequency
no radiation could occur from a plate of infinite size. This is certainly not the case for
real plate structures of finite size but how shall this be calculated, e.g. for a rectangular
plate having edges of length a and b? This will be rather more complicated than the
idealized example with the infinite plate. Assuming that the vibration of the plate is
determined by its natural modes, the radiation will depend on the actual modal pattern,
which in turn is determined by the modes taking part and their individual vibration
amplitudes. This implies that the mean surface velocity of the plate does not uniquely
determine the radiated power. In principle therefore, one cannot calculate the radiation
factor solely from the dimensions and material properties. The vibration generating
mechanism or the form of excitation must also be known. With the latter we have
knowledge of the actual source, what kind of source and how it actually is driving the
plate.
In most practical cases, having a stationary mechanical excitation, the structure will
be forced into vibration by a more or less broad banded source. This means that the
vibration pattern is a combination of the natural modes having eigenfrequencies inside
the actual frequency band being excited into resonance. The contribution from each of
these modes will depend on how the structure is driven by the source. In our case,
concerning the rectangular plate, we shall be quite pragmatic assuming that all modes
having their natural frequency within the actual frequency band have the same velocity
amplitude. Data given in standards, e.g. EN 12354–1 is calculated using this assumption
and we shall give some examples below.
However, it will be quite useful to calculate the radiation factor for a single mode to
see how critical the vibration pattern is concerning the radiated power. We shall therefore
calculate the radiation factor for a simply supported plate set in an infinite baffle. We
assume that the plate is vibrating in a simple harmonic way with a velocity given by
(^) y(,) ˆsin x sin z 0 ,0 ,
nx nz
uxz u x a z b
ab
⎛⎞π ⎛⎞π
=≤⎜⎟⎜⎟ ≤≤≤
⎝⎠⎝⎠
(6.43)
where nx and nz are the modal numbers in the x- and z-direction, respectively. This is
illustrated in Figure 6.12, where the plate vibrates in a (5, 4) mode. The corresponding
wave number are, as shown in Chapter 3, given by