Civil Engineering Reference
In-Depth Information
It should be noted that using the particle velocity as a variable, which we have done
in Equation
(3.94)
and also further on, is just a choice. A corresponding equation for e.g.
the displacement could be used as well. The phase speed of the longitudinal wave,
according to the equations above, will be given by
E
c
=
.
(3.96)
( )
L
2
ρυ
1
−
Examples of data for common materials are given in
Table 3.1
, which we may use to
calculate the wave speed for longitudinal waves. It should be noted that wave speed
normally found in tabled data applies to pure longitudinal waves, i.e. calculated from the
formula (
E
/ρ)
1/2
. The loss factor given in the table applies to the internal energy losses in
the material.
Table 3.1
Examples of material properties.
E-modulus
1
10
9
Pa
Density
kg/m
3
Poisson's
ratio
Loss factor
η
int
⋅10
-3
Material
Steel
7700-7800
190-210
0.28-0.31
~ 0.1
Aluminium
2700
66-72
0.33-034
~ 0.1
Glass
2500
60
-
0.6-2.0
Concrete
2300
32-40
0.15-0.2
4-8
Concrete
(lightweight aggregate)
400-600
1.0-2.5
~ 0.2
10-20
3.8
2
Concrete
(autoclaved aerated)
1300
~ 0.2
10-20
Gypsum plate (plasterboard)
800-900
4.1
~ 0.3
10-15
Chipboard
650-800
3.8
~ 0.2
10-30
Fir, spruce
400-700
7-12
~ 0.4
8-10
1
Dynamic E-modulus.
2
E-modulus for static pressure.
3.7.2
Shear waves
In a pure shear wave, also referred to as a transverse wave, we only get shear
deformations and no change of volume (see b) in
Figure 3.18
). The particle movements
are normal to the direction of wave propagation, and the wave equation for free wave
motion will be analogous to Equation
(3.94)
, i.e.
2
2
∂
v
∂
v
y
y
G
−
ρ
=
0,
(3.97)
2
2
∂
x
∂
t
where
v
y
represents the particle velocity normal to the direction of propagation. The shear
modulus is given by
E
G
=
,
(3.98)
2(1
+
υ
)
and for the phase speed
c
S
we get
