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 .
Volume Velocity of a Gas
The volume velocity of a gas flow is defined as particle velocity times the crosssectional area of the flow, or
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
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
Pressure is Confined Kinetic Energy
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.
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
Bernoulli Equation
In an ideal inviscid, incompressible flow, we have, by conservation of energy,
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 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].
Air Jets
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:
 heat (now allowing for ``friction'' in the flow),
 vortices (angular momentum),
 radiation (sound waves), or
 pressure recovery.
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
Acoustic intensity may be defined by
For a plane traveling wave, we have
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 waveimpedance 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}
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:
Sabine's theory of acoustic energy decay in reverberant room impulse responses can be derived using this conservation relation as a starting point.
Ideal Gas Law
The ideal gas law can be written as
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
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 pressurewave 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].
Isothermal versus Isentropic
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.
Adiabatic Gas Constant
The relative amount of compression/expansion energy that goes into temperature versus pressure can be characterized by the heat capacity ratio
In terms of , we have
where for dry air at normal temperatures. Thus, if a volume of ideal gas is changed from to , the pressure change is given by
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.
Heat Capacity of Ideal Gases
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
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
Speed of Sound in Air
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}
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
Tables B.1 and B.2 (adapted from [314]) give some typical values for air.


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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Wave Equation in Higher Dimensions
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Wave Equation for the Vibrating String