226 Building acoustics
In this case, we envisage that the wavelength in the surrounding medium is larger than
both partial wave numbers of the bending wave in the plate. The various parts of the
plate are vibrating in opposite phase separated by small distances, i.e. small when
compared with the wavelength in the medium around. The plate becomes a multipole
source; the movements of the various parts are not correlated or coordinated to be an
effective sound source. The only effective radiating areas are the areas situated near to
the corners, these are sufficiently far apart so as not to be mutually destructive. For the
edge mode, one of the partial wavelengths is larger then the wavelength in the medium
around, which results in a larger effective radiation area.
Both types of mode are also called slow acoustic modes because the phase speed of
the wave is smaller than the same in the surrounding media. As for the last mentioned
type of mode, the surface mode, the partial wavelengths and the speed is larger than in
the media around. We get fast acoustic modes where the whole surface is an efficient
radiator bringing the radiation factor towards the value of one.
Figure 6.13 Examples of modal pattern for a rectangular plate. Left figure – “corner mode”; both partial
wavelengths are smaller than the wavelength in the surrounding medium. Right figure – “edge mode”; one of
the partial wavelengths is larger than the wavelength in the surrounding medium.
6.3.4.1 Radiation factor for a plate vibrating in a given mode
Wallace (1972), based on the Rayleigh integral introduced in Chapter 3, calculated the
sound pressure and thereby the intensity in the far field from a plate where the velocity is
given by Equation (6.43). When integrating the intensity over a hemisphere over the
plate we get the radiated power and thereby the radiation factor by using Equation (6.25).
We shall not give the details here but for completeness we shall give the end result,
which is
++++++++++----------+++++--λnx λnx
λnzλnzλ
λ λ
λ
Quadrupole-
cancellingDipole-
cancelling++++++++++----------+++++--λnx λnx
λnzλnzλ
λ λ
λ
Quadrupole-
cancellingDipole-
cancelling