Civil Engineering Reference
In-Depth Information
Equation (10.75) can be rewritten (see Equation (3.54))
k
z
k
N
+
k
x
k
P
=
1
(10.82)
Let
θ
1
be the angle of the wave number vector
k
1
of components
k
x
and
k
z
with the
Z
axis. The angle
θ
1
and the modulus of the wave number vector satisfy the relation
k
1
cos
2
θ
1
k
N
k
1
sin
2
θ
1
k
P
+
=
1
(10.83)
Using the asymptotic limit of the effective density at high frequency gives the fol-
lowing angular dependence for the apparent tortuosity
α
∞
(θ
1
)
cos
2
θ
1
α
P
sin
2
θ
1
α
N
1
α
∞
(θ
1
)
+
=
(10.84)
∞
∞
where
α
P
∞
is the tortuosity in the plane
(X,Y)
and
α
N
∞
is the tortuosity in the direction
Z
. Tortuosity measurements at ultrasonic frequencies by Castagnede
et al
. (1998) show
a good agreement with this angular dependence. Using the asymptotic limit of the
effective density at low frequency gives the same relation for the apparent flow resistivity
σ(θ
1
)
cos
2
θ
1
σ
P
sin
2
θ
1
σ
N
1
σ(θ
1
)
+
=
(10.85)
10.6
Mechanical excitation at the surface of the porous
layer
The excitation is a unit stress field
τ
zz
applied at the free surface of the layer with a
space dependence exp
(
jωx/c)
. The boundary conditions at the free surface are given
by Equations (10.61) - (10.63) and Equation (10.64) is replaced by
−
+
τ
zz
=
σ
zz
−
p
e
(10.86)
The unit stress creates a plane wave in air with a spatial dependence exp[
−
j(ωx/c
−
k
0
z
cos
θ)
]where
k
0
is the wave number in the free air and cos
θ
is given by
(k
0
/ω)
2
−
(
1
/c)
2
cos
θ
=±
(10.87)
k
0
/ω
The choice in the physical Riemann sheet corresponds to Im cos
θ
≤
0. The pressure
p
e
and the velocity component
v
z
are related by
p
e
=−
v
z
Z
0
/
cos
θ
(10.88)
