Civil Engineering Reference
In-Depth Information
the acoustic fluctuating parts are relatively small; we may assume the phenomena to be
linear. Also note, as pointed out in the introduction, that we will assume
V
0
to be zero. It
should also be observed that this quantity as well as the particle velocity
v
is a vector
quantity.
Given these approximations we can write down the linear or the so-called acoustic
approximations for the governing fluid equations. These equations concern the
conservation of mass, the fluid forces and the relationship between changes in pressure
and density. We get
∂
ρ
= −∇⋅
ρ
v
,
(3.2)
0
∂
t
ρ
∂
v
∇=−
p
(3.3)
0
∂
t
2
and
pc
ρ
=
,
(3.4)
where
c
is the sound speed (phase speed) in the actual medium. Why it is termed phase
speed is due to the fact that travelling along with the wave at the same speed
c
one
always sees the same pattern; there is no change of phase. Eliminating the variables
v
and
ρ from these equations we get the
wave equation
2
1
∂
p
2
∇ −
p
=
0.
(3.5)
2
2
ct
∂
We now assume that the sound wave is split into partial waves having a harmonic time
dependency. This means that it is sufficient to observe just one frequency component at
the time. The sound pressure
p
and the particle velocity
v
at an arbitrary point in the
sound field may then be expressed as
{
}
j
ω
t
ˆ
pt
(,) Re
r
=
p
()e
r
⋅
(3.6)
{
}
j
ω
t
and
vr
( , )
t
=
Re
vr
ˆ
( ) e
⋅
,
where Re{…} signify that we shall use the real part of the expression, further that the
amplitudes of
p
and
v
, indicated by the “hats”, are in general complex values. As a rule
one implicitly uses the real value to obtain the real physical quantity. It is therefore
common practice to leave out the Re{…} in the expressions. When introducing the
harmonic time dependency Equation
(3.5)
will transform into the
Helmholtz equation
2
ω
∇+ =∇+ =
2
2
2
p
p
p
k
p
0,
(3.7)
2
c
where
k
(m
-1
) is denoted
wave number
.
The pressure and the particle velocity in a sound field may vary in a very complex
manner but still obeying the wave equation. Plane wave and spherical wave fields are
examples of idealized types of wave field that are not only important theoretically but
also in practical measurement situations.
