Environmental Engineering Reference
In-Depth Information
Equations (5.49) and (5.50) connect the oscillation frequency
ω
and the wave
number by the relation
ω
D
c
s
k
,
(5.51)
where the speed of sound
c
s
is
r
p
0
mN
0
D
r
1
m
@
p
c
s
D
N
.
(5.52)
@
An equation of the type of (5.51) that connects the frequency
of the wave with its
wave number
k
is called a dispersion relation. We see that here the group velocity
of sound propagation
ω
@
ω
/
@
k
isthesameasthephasevelocity
ω
/
k
and does not
depend on the sound frequency.
To find the sound velocity, it is necessary to know the connection between vari-
ations of the gas pressure and density in the acoustic wave. For long waves, the
regions of compression and rarefaction do not exchange energy during wave prop-
agation. Hence, this process is adiabatic, and the parameters of the acoustic wave
satisfy the adiabatic equation
pN
γ
D
const ,
(5.53)
where
c
p
/
c
V
is the adiabatic exponent. That is,
c
p
is the specific heat capacity
at constant pressure, and
c
V
is the specific heat capacity at constant volume. On
the basis of expansion (5.48), the wave parameters are related by
γ
D
p
0
p
0
N
0
N
0
D
γ
.
Because of the state equation (4.11),
p
D
NT
,where
T
is the gas temperature, we
obtain
p
0
/
N
0
D @
p
/
@
N
D
γ
T
. Thus, the dispersion relation (5.51) yields
r
γ
T
m
k
.
ω
D
(5.54)
The sound velocity is seen to be of the order of the thermal velocity of gas particles.
Figure 5.8 demonstrates this dependence for acoustic oscillations in air.
The dispersion relation (5.54) is valid under adiabatic conditions in the wave if
atypicaltime
for heat transport in the wave is short compared with the period
of oscillations 1/
τ
. Assuming the heat transport in the wave to be due to thermal
conductivity, we have
ω
r
2
/
k
2
)
1
, where the distance
r
is of the order of
τ
(
the wavelength and
is the thermal diffusivity coefficient. From this we obtain the
adiabatic criterion
c
s
ω
(5.55)
for the wave, where
c
s
is the wave velocity. For example, in the case of air under
standard conditions (
p
D
1 atm), condition (5.55) has the form
ω
5
10
9
s
1
.
