Waves in fluid and solid media 89
(^) S.
G
c
ρ= (3.99)
Using Equation (3.96), (3.98) and (3.99) we get for a plate
S
L1
2
c
c−υ
= (3.100)Inserting a Poisson’s ratio of 0.3 as an example will give a ratio of these wave speeds of
approximately 0.59. The wave speed of a shear wave will then always be less than the
speed of a longitudinal wave. Lastly, we point to the fact that pure shear waves will, like
pure longitudinal waves, generally occur only in bodies where the dimensions are very
large compared with the wavelength.
3.7.3 Bending waves (flexural waves)
Bending waves are likely to be excited in bodies or structures where one or two
dimensions are becoming small compared to the wavelength at an actual frequency. This
implies that this wave type will be dominant in common construction elements, for
example beams and plates. This again means that it takes on a central position in building
acoustics, also due to these waves being easy to excite. Furthermore, the particle velocity
will be normal to the direction of propagation, which also means that it is normal to the
surface of a beam or plate (see c) in Figure 3.18). This again implies that there will be an
efficient coupling to the surrounding medium (air), which means that the plate or beam
potentially could be an efficient sound source. We may easily be aware of this fact by
knocking on a thin metal plate.
Our treatment of bending waves will mainly be concerned with simple thin plate
models, also called Bernoulli–Euler models. In these models, one presuppose that the
deformation of an element due to bending is much larger than the one caused by shear
and, furthermore, the rotation of the element is neglected. A limit for using thin plate
models, often referred to in the literature, is that the wavelength of the bending wave
must be larger than six times the thickness of the beam or plate. For quite common
thicknesses of concrete this may be a limitation and one should then apply thick plate
models (Reissner–Mindlin). We shall limit ourselves to giving some examples of the
differences one may encounter by using these models.
The treatment will also, if not pointed out otherwise, be limited to plates of
isotropic materials, which means that the material properties are independent of
direction. One then needs only two quantities, the modulus of elasticity and Poisson’s
ratio, to describe the linear relationship between forces and displacements.
Unfortunately, a large group of building materials exhibit anisotropy, the material
properties depend on direction. Wooden materials are typical examples where the
properties depend on the direction of the fibres. Other examples are composite materials
reinforced by fibres. A special type of anisotropy is denoted orthotropic. An orthotropic
plate is a plate where the material properties are symmetric about three mutually
perpendicular axes. Well-known examples are corrugated panels often used in industrial
buildings; having a waveform or a more sophisticated trapezoidal cross section, the latter
normally called cladding. It should be noted, however, that to apply the general theory of
orthotropic plates to corrugated panels one has to find the equivalent orthotropic