Civil Engineering Reference
In-Depth Information
1.5
1
0
0.5
−
0.5
0
−
1
−
0.5
−
1.5
−
1
−
2
−
1.5
−
2.5
−
2
−
3
0
5
10
15
20
25
30
0
5
10
15
20
25
30
(
m
−
1
)
(
m
−
1
)
k
0
sin
q
k
0
sin
q
(a)
(b)
Figure 7.10
The predicted reflection coefficient for a layer of material 2 of thickness
l
=
0
.
1 m at 0.3 kHz as a function of
k
0
sin
θ
.When
V
=−
1
,
sin
θ
=
1.
Taking into account the signs of the real and the imaginary part of cos
θ
p
and sin
θ
p
leads to 0
π/
2, as in Figure 7.8(b). The plane wave reflection coefficient can be
measured at sin
θ
real and larger than 1 with the Tamura method (Tamura 1990, Brouard
et al
. 1996). In the sin
θ
plane, sin
θ
p
is close to the real axis at sufficiently low frequencies
for a layer of finite thickness. The modulus of the reflection coefficient must present a
peak when sin
θ
is close to
|
sin
θ
p
|
. The predicted reflection coefficient for a layer of
thickness
l
=
0
.
1 m of material 2 is shown in Figure 7.10. The maximum of the modulus
of the reflection coefficient is larger than 1 for sin
θ
larger than 1. The reflected wave is
purely evanescent for sin
θ
real and larger than one, and does not carry energy, so there
is no power created. Measurements performed by Brouard
et al
. (1996) on a similar
material are in a good agreement with these predictions. Other experimental evidences of
this peak can be found in Brouard (1994), Brouard
et al
. (1996), and Allard
et al
. (2002).
The wave number vector in the free air above a thin layer, in the absence of damping,
is symbolically represented in Figure 7.8(a). This wave presents in air all the characteristic
properties of a surface wave, i.e. damping in the direction normal to the surface and
propagation without damping in the direction parallel to the surface with a velocity
c
0
/sin
θ
p
smaller than the sound speed
c
0
because sin
θ
p
>
1. Inside the porous layer, the
wave does not present the characteristic properties of a surface wave. The acoustic field
experiences total reflection on the rigid impervious backing and on the air porous layer
interface because the angle of refraction
θ
1
p
satisfies the following relations
≤
ϕ
≤
sin
θ
1
p
=
sin
θ
p
/n>
1
/n
(7.36)
The surface wave is the evanescent wave related to a trapped acoustic field in the
layer. This wave is very similar to the transverse magnetic (T.M.) electromagnetic surface
waves with the magnetic field perpendicular to the incidence plane above a grounded
dielectric described in Collin (1960), and Wait (1970). It can be called a surface wave,
depending on the restrictive conditions associated with this definition.
Semi-infinite layers
For a semi-infinite porous layer saturated by a perfect fluid, Equations (7.26) - (7.27) give
sin
θ
p
and cos
θ
p
real, and sin
θ
p
<
1, cos
θ
p
<
0. The wave number of the associated wave
