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 lower-case .
Volume Velocity of a Gas
When a flow is confined within an enclosed channel, as it is in an acoustic tube, volume velocity is conserved when the tube changes cross-sectional 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 cross-sectional 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 upper-case . Thus, for the two-cylinder acoustic tube example above, we would define and , so that
Pressure is Confined Kinetic Energy
According the kinetic theory of ideal gases , 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 . 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 rigid-wall 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 momentum-transfer 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 round-trip distance along the dimension gives the collision rate at either wall bounding the dimension. Thus, we obtain
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.
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 cross-sectional 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.
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.
Acoustic intensity may be defined by
For a plane traveling wave, we have
Therefore, in a plane wave,
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:
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.
The ideal gas law can be written as
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 electron-orbital repulsion producing the elastic collisions; in other words, the molecules are treated as electrically neutral far away. (Gases of ionized molecules are called plasmas.)
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 sound-pressure 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 constant-entropy 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.
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.
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
There is also a (weaker) dependence of air absorption on temperature .
Theoretical models of energy loss in a gas are developed in Morse and Ingard [318, pp. 270-285]. 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 time-constant for vibration-relaxation. The relaxation time-constant 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 maximum-loss frequency, and somewhat higher above. Thus, the humidity level changes the dispersion cross-over frequency of the air in a resonant mode.
Wave Equation in Higher Dimensions
Wave Equation for the Vibrating String