Properties of Gases
Particle Velocity of a Gas
The
particle velocity of a gas flow at any point can be defined
as the average velocity (in
meters per second, m/s) of the air
molecules passing through a plane cutting
orthogonal to the flow. The
term ``velocity'' in this book, when referring to air, means
``particle velocity.''
It is common in acoustics to denote particle velocity by lowercase
.
The
volume velocity of a gas flow is defined as particle
velocity
times the
crosssectional area of the flow, or
where
denotes position along the flow, and
denotes time in
seconds. Volume velocity is thus in physical units of volume per
second (m
/s).
When a flow is confined within an enclosed channel, as it is in an
acoustic tube,
volume velocity is conserved when the tube
changes crosssectional area, assuming the density
remains
constant. This follows directly from conservation of
mass in a flow:
The total mass passing a given point
along the flow is given by
the mass density
times the integral of the volume volume
velocity at that point, or
As a simple example, consider a constant flow through two cylindrical
acoustic tube sections having crosssectional areas
and
,
respectively. If the particle velocity in cylinder 1 is
, then
the particle velocity in cylinder 2 may be found by solving
for
.
It is common in the field of acoustics to denote volume velocity by an
uppercase
. Thus, for the twocylinder acoustic tube example above,
we would define
and
, so that
would express the conservation of volume velocity from one tube
segment to the next.
According the
kinetic theory of ideal gases
[
180],
air pressure can be defined
as the
average momentum transfer per unit area per unit time
due to molecular collisions between a confined gas and its boundary.
Using Newton's second law, this pressure can be shown to be given by
one third of the average kinetic energy of molecules in the gas.
Here,
denotes the average squared particle
velocity in
the gas. (The constant
comes from the fact that we are
interested only in the kinetic energy directed along one dimension in
3D space.)
Proof: This is a classical result from the
kinetic theory of gases
[
180]. Let
be the total
mass of a gas
confined to a rectangular volume
, where
is the area of
one side and
the distance to the opposite side. Let
denote the average molecule velocity in the
direction. Then the
total net molecular momentum in the
direction is given by
. Suppose the momentum
is directed
against a face of area
. A rigidwall
elastic collision by a mass
traveling into the wall at velocity
imparts a momentum of
magnitude
to the wall (because the momentum of the mass is
changed from
to
, and momentum is conserved).
The average momentumtransfer per unit area is therefore
at any instant in time. To obtain the definition of pressure, we need
only multiply by the average collision rate, which is given by
. That is, the average
velocity divided by the
roundtrip distance along the
dimension gives the collision rate
at either wall bounding the
dimension. Thus, we obtain
where
is the density of the gas in mass per unit volume.
The quantity
is the average kinetic energy density of
molecules in the gas along the
dimension. The total kinetic
energy density is
, where
is the average molecular
velocity magnitude of the gas. Since the gas pressure must be the
same in all directions, by symmetry, we must have
, so that
Bernoulli Equation
In an ideal
inviscid, incompressible flow, we have, by
conservation of energy,
constant
where
This basic energy conservation law was published in 1738 by Daniel
Bernoulli in his classic work
Hydrodynamica.
From §
B.7.3, we have that the
pressure of a gas is
proportional to the average
kinetic energy of the molecules making up
the gas. Therefore, when a gas flows at a constant height
, some
of its ``pressure kinetic energy'' must be given to the kinetic energy
of the flow as a whole. If the mean height of the flow changes, then
kinetic energy trades with
potential energy as well.
Bernoulli Effect
The
Bernoulli effect provides that, when a gas such as air
flows, its
pressure drops. This is the basis for how aircraft wings
work: The crosssectional shape of the wing, called an
aerofoil
(or
airfoil),
forces air to follow a longer path over the top
of the wing, thereby speeding it up and creating a net upward force
called
lift.
Figure B.8:
Illustration of the Bernoulli effect in
an acoustic tube.

Figure
B.8 illustrates the Bernoulli effect for the case
of a reservoir at constant pressure
(``mouth pressure'') driving
an acoustic tube. Any flow inside the ``mouth'' is neglected. Within the
acoustic channel, there is a flow with constant particle
velocity .
To conserve energy, the pressure within the acoustic channel must drop
down to
. That is, the flow
kinetic energy subtracts
from the pressure kinetic energy within the channel.
For a more detailed derivation of the Bernoulli effect, see,
e.g.,
[
179]. Further discussion of its relevance
in musical acoustics is given in [
144,
197].
Referring again to Fig.
B.8, the gas flow exiting the
acoustic tube is shown as forming a
jet. The jet ``carries its
own
pressure'' until it dissipates in some form, such as any
combination of the following:
Pressure recovery refers to the
conversion of flow
kinetic energy back to pressure kinetic energy. In
situations such as the one shown in Fig.
B.8,
the flow itself is
driven by
the pressure drop between the confined
reservoir (pressure
) and the outside air (pressure
). Therefore, any pressure recovery would erode the
pressure drop and hence the flow
velocity .
For a summary of more advanced aeroacoustics, including consideration
of vortices, see [
196]. In addition, basic textbooks on
fluid mechanics are relevant [
171].
Acoustic intensity may be defined by
where
For a
plane traveling wave, we have
where
is called the
wave impedance of air, and
Therefore, in a
plane wave,
Acoustic Energy Density
The two forms of energy in a wave are
kinetic and
potential. Denoting them at a particular time
and position
by
and
, respectively, we can write them in
terms of
velocity and
wave impedance as follows:
More specifically,
and
may be called the
acoustic kinetic
energy density and the
acoustic potential energy density, respectively.
At each point in a
plane wave, we have
(
pressure equals wave
impedance times velocity), and so
where
denotes the acoustic
intensity (pressure times velocity) at time
and position
.
Thus, half of the acoustic intensity
in a plane wave is kinetic,
and the other half is potential:
^{B.30}
Note that acoustic intensity
has units of
energy per unit
area per unit time while the acoustic energy density
has
units of
energy per unit volume.
Energy Decay through Lossy Boundaries
Since the acoustic energy density
is the energy per unit
volume in a 3D sound field, it follows that the total energy of the
field is given by integrating over the volume:
In reverberant rooms and other acoustic systems, the field energy
decays over time due to losses. Assuming the losses occur only at the
boundary of the volume, we can equate the rate of totalenergy change
to the rate at which energy exits through the boundaries. In other
words, the energy lost by the volume
in time interval
must equal the acoustic
intensity
exiting the volume,
times
(approximating
as constant between times
and
):
The term
is the dotproduct of the (vector)
intensity
with a unitvector
chosen to be normal to the
surface at position
along the surface. Thus,
is
the component of the acoustic intensity
exiting the volume
normal to its surface. (The tangential component does not exit.)
Dividing through by
and taking a limit as
yields the following conservation law, originally published by
Kirchoff in 1867:
Thus, the rate of change of energy in an ideal acoustic volume
is
equal to the surface integral of the power crossing its boundary. A
more detailed derivation appears in [
349, p. 37].
Sabine's theory of acoustic energy decay in reverberant room
impulse
responses can be derived using this conservation relation as a
starting point.
The
ideal gas law can be written as

(B.45) 
where
The alternate form
comes from the
statistical
mechanics derivation in which
is the number of gas molecules in
the volume, and
is
Boltzmann's constant. In this
formulation (the
kinetic theory of ideal gases), the
average kinetic
energy of the gas molecules is given by
. Thus,
temperature is proportional to average kinetic energy of the
gas molecules, where the kinetic energy of a molecule
with
translational speed
is given by
.
In an ideal gas, the molecules are like little rubber balls (or
rubbery assemblies of rubber balls) in a weightless vacuum, colliding
with each other and the walls elastically and losslessly (an ``ideal
rubber''). Electromagnetic
forces among the molecules are neglected,
other than the electronorbital repulsion producing the
elastic
collisions; in other words, the molecules are treated as electrically
neutral far away. (Gases of ionized molecules are called
plasmas.)
The
mass of the gas in volume
is given by
, where
is
the
molar mass of the gass (about 29 g per mole for air). The
air density is thus
so that we can write
That is,
pressure is proportional to density
at constant
temperature
(with
being a constant).
We normally do not need to consider the (
nonlinear)
ideal gas law in
audio acoustics because it is usually
linearized about some
ambient pressure
. The physical pressure is then
, where
is the usual acoustic pressure
wave variable. That is, we are
only concerned with small pressure
perturbations in typical
audio acoustics situations, so that, for example, variations in volume
and density
can be neglected. Notable exceptions include
brass instruments which can achieve nonlinear
soundpressure regions,
especially near the mouthpiece [
198,
52].
Additionally, the aeroacoustics of air
jets is nonlinear
[
196,
530,
531,
532,
102,
101].
If air compression/expansion were
isothermal (constant
temperature ), then, according to the
ideal gas law , the
pressure would simply be proportional to density
. It turns
out, however, that
heat diffusion is much slower than audio acoustic
vibrations. As a result, air compression/expansion is much closer to
isentropic (constant
entropy ) in normal acoustic
situations. (An isentropic process is also called a
reversible
adiabatic process.) This means that when air is compressed by
shrinking its volume
, for example, not only does the pressure
increase (§
B.7.3), but the temperature
increases as
well (as quantified in the next section). In a constantentropy
compression/expansion, temperature changes are not given time to
diffuse away to thermal equilibrium. Instead, they remain largely
frozen in place. Compressing air heats it up, and relaxing the
compression cools it back down.
The relative amount of compression/expansion energy that goes into
temperature
versus
pressure
can be characterized by the
heat capacity ratio
where
is the
specific heat (also called
heat
capacity) at constant pressure, while
is the specific heat at
constant volume. The
specific heat, in turn, is the amount of
heat required to raise the temperature of the gas by one degree. It
is derived in
statistical thermodynamics [
138]
that, for an
ideal gas, we have
, where
is the
ideal
gas constant (introduced in Eq.
(
B.45)). Thus,
for any
ideal gas. The extra heat absorption that occurs when heating a gas
at constant pressure is associated with the
work (§
B.2)
performed on the volume boundary (fore times distance = pressure times
area times distance) as it expands to keep pressure constant. Heating
a gas at constant volume involves increasing the
kinetic energy of the
molecules, while heating a gas at constant pressure involves both that
and pushing the boundary of the volume out. The reason not all
gases have the same
is that they have different
internal degrees of freedom, such as those associated with
spinning and vibrating internally. Each degree of freedom can store
energy.
In terms of
, we have

(B.46) 
where
for dry air at normal temperatures. Thus,
if a volume of ideal gas is changed from
to
, the pressure
change is given by
and the temperature change is
These equations both follow from Eq.
(
B.46) and the
ideal gas law
Eq.
(
B.45).
The value
is typical for any
diatomic
gas.
^{B.31} Monatomic inert gases, on the other hand,
such as Helium, Neon, and Argon, have
.
Carbon
dioxide, which is
triatomic, has a
heat capacity ratio
. We see that more complex molecules have lower
values because they can store heat in more degrees of freedom.
In statistical thermodynamics [
175,
138],
it is derived that each molecular degree of freedom contributes
to the molar
heat capacity of an
ideal gas, where again
is the
ideal
gas constant.
An ideal
monatomic gas molecule (negligible spin) has only
three degrees of freedom: its
kinetic energy in the three spatial
dimensions. Therefore,
. This means we expect
a result that agrees well with experimental measurements
[
138].
For an ideal
diatomic gas molecule such as air, which can be
pictured as a ``bar bell'' configuration of two rubber balls, two
additional degrees of freedom are added, both associated with spinning
the molecule about an axis
orthogonal to the line connecting the
atoms, and piercing its
center of mass. There are two such
axes. Spinning about the connecting axis is neglected because the
moment of inertia is so much smaller in that case. Thus, for diatomic
gases such as dry air, we expect
as observed to a good degree of approximation at normal
temperatures.
At high temperatures, new degrees of freedom appear associated with
vibrations in the molecular bonds. (For example, the ``bar bell'' can
vibrate longitudinally.) However, such vibrations are ``frozen out''
at normal room temperatures, meaning that their (quantized) energy
levels are too high and spaced too far apart to be excited by room
temperature collisions [
138, p. 147].
^{B.32}
The
speed of sound in a gas depends primarily on the
temperature, and can be estimated using the following formula
from the
kinetic theory of gases:
^{B.33}
where, as discussed in the previous section, the
adiabatic gas
constant is
for dry air,
is the
ideal gas
constant for air in
meterssquared per secondsquared per
degrees
Kelvinsquared, and
is
absolute temperature in degrees
Kelvin (which equals degrees
Celsius + 273.15). For example, at zero
degrees Celsius (32 degrees
Fahrenheit), the speed of sound is
calculated to be 1085.1 feet per second. At 20 degrees Celsius, we
get 1124.1 feet per second.
Air Absorption
This section provides some further details regarding acoustic air
absorption [
318]. For a
plane wave, the decline of
acoustic
intensity as a function of
propagation distance
is given
by
where
Tables
B.1 and
B.2 (adapted from
[
314]) give some typical values for air.
Table B.1:
Attenuation constant (in inverse
meters) at 20 degrees Celsius and standard atmospheric pressure
Relative 
Frequency in Hz 
Humidity 
1000 
2000 
3000 
4000 
40 
0.0013 
0.0037 
0.0069 
0.0242 
50 
0.0013 
0.0027 
0.0060 
0.0207 
60 
0.0013 
0.0027 
0.0055 
0.0169 
70 
0.0013 
0.0027 
0.0050 
0.0145 

Table B.2:
Attenuation in dB per kilometer at
20 degrees Celsius and standard atmospheric pressure.
Relative 
Frequency in Hz 
Humidity 
1000 
2000 
3000 
4000 
40 
5.6 
16 
30 
105 
50 
5.6 
12 
26 
90 
60 
5.6 
12 
24 
73 
70 
5.6 
12 
22 
63 

There is also a (weaker) dependence of air absorption on
temperature
[
183].
Theoretical models of energy loss in a gas are developed in Morse and
Ingard [
318, pp. 270285]. Energy loss is caused by
viscosity,
thermal diffusion,
rotational
relaxation,
vibration relaxation, and
boundary losses
(losses due to
heat conduction and viscosity at a wall or other
acoustic boundary). Boundary losses normally dominate by several
orders of magnitude, but in resonant modes, which have
nodes along the
boundaries, interior losses dominate, especially for polyatomic gases
such as air.
^{B.34} For air having moderate amounts of water
vapor (
) and/or
carbon dioxide (
), the loss and dispersion
due to
and
vibration relaxation hysteresis becomes the
largest factor [
318, p. 300]. The vibration here
is that of the molecule itself, accumulated over the course of many
collisions with other molecules. In this context, a diatomic molecule
may be modeled as two
masses connected by an ideal
spring. Energy
stored in molecular vibration typically dominates over that stored in
molecular rotation, for polyatomic gas molecules [
318, p.
300]. Thus, vibration relaxation hysteresis is a loss
mechanism that converts wave energy into heat.
In a resonant mode, the attenuation per
wavelength due to vibration
relaxation is greatest when the
sinusoidal period (of the resonance)
is equal to
times the
timeconstant for vibrationrelaxation.
The relaxation timeconstant for oxygen is on the order of one
millisecond. The presence of water vapor (or other impurities)
decreases the vibration relaxation time, yielding loss maxima at
frequencies above 1000 rad/sec. The energy loss approaches zero as
the frequency goes to infinity (wavelength to zero).
Under these conditions, the
speed of sound is approximately that of
dry air below the maximumloss frequency, and somewhat higher above.
Thus, the humidity level changes the dispersion crossover frequency
of the air in a resonant mode.
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