Civil Engineering Reference
In-Depth Information
K
and
ρ
are the bulk modulus and the density of the fluid, respectively. The quantity
A
is the amplitude of the acoustic pressure. From Equations (1.63) and (1.64), it follows that
the acoustic pressure
p
and the components of the displacement vector
u
are respectively
p(x,t)
=
A
exp[
jω(t
−
kx)
]
(2.3)
and
=
−
jAk
ρω
2
u
y
=
u
z
=
0
,
x
(x,t)
exp[
jω(t
−
kx)
]
(2.4)
Only the
x
component
υ
x
of the velocity vector does not vanish:
kA
ρω
exp[
jω(t
υ
x
(x,t)
=
−
x/c)
]
(2.5)
Equations (2.3) and (2.5) describe a travelling harmonic plane wave propagating along
the
x
direction. Pressure and velocity are related by
1
Z
c
p(x,t)
υ
x
(x,t)
=
(2.6)
with
(ρK)
1
/
2
Z
c
=
(2.7)
The quantity
Z
c
is the characteristic impedance of the fluid.
2.2.2 Example
As an example, for air at the normal conditions of temperature
T
and pressure
p
(18
◦
C
and 1 033
10
5
Pa), the density
ρ
0
, the adiabatic bulk modulus
K
0
, the characteristic
impedance
Z
0
, and the speed of sound
c
0
are as follows (Gray 1957):
×
ρ
0
=
1
·
213 kg m
−
3
K
0
=
1
·
42
×
10
5
Pa
Z
0
=
415
·
1Pam
−
1
s
c
0
=
342 m s
−
1
2.2.3 Attenuation
In a free field in air at acoustical frequencies, the damping can be neglected to a first
approximation when the order of magnitude of the propagation length is 10 m or less.
In the previous example, the effects of viscosity, heat conduction, and other dissipative
processes have been neglected. The phenomena of viscosity and thermal conduction
in fluids are a consequence of their molecular constitution. The description of sound
propagation in viscothermal fluids can be found in the literature (Pierce 1981, Morse and
Ingard 1986). The effects of viscosity and heat conduction on sound propagation in tubes
