The Art of the Helicopter John Watkinson - Karatunov.net

34 The Art of the Helicopter
Figure 2.12(b) shows that a solid object that moves against gas pressure increases
the velocity of the rebounding molecules, whereas in (c) one moving with gas pressure
reduces that velocity. The average velocity and the displacement of all the molecules
in a layer of air near to a moving body is the same as the velocity and displacement
of the body. Movement of a body results in a local increase, or decrease, in pressure.
Thus sound is both a pressure and a velocity disturbance. Integration of the velocity
disturbance gives the displacement.
Despite the fact that a gas contains endlessly colliding molecules, a small mass or
particle of gas can have stable characteristics if molecules leave and are replaced by
others having identical properties. As a result aerodynamics and acoustics seldom need
to consider the molecular structure of air and the constant motion can be neglected.
Thus when particle velocity and displacement is considered in aerodynamics or acoustics,
this refers to the average values of a large number of molecules. In an undisturbed
container of gas the particle velocity and displacement will both be zero everywhere.
When the volume of a fixed mass of gas is reduced, the pressure rises. The gas acts
like a spring; it is compliant. However, a gas also has mass. Sound travels through air
by an interaction between the mass and the compliance. Imagine pushing a mass via
a spring. It would not move immediately because the spring would have to be compressed
in order to transmit a force. If a second mass were to be connected to the first
by another spring, it would start to move even later. Thus the speed of a disturbance
in a mass/spring system depends on the mass and the stiffness. Sound travels through
air without a net movement of the air.
After the disturbance had propagated the masses would return to their rest position.
The mass/spring analogy is helpful for a basic understanding, but is too simple to
account for commonly encountered acoustic phenomena such as spherically expanding
waves.
Unlike solids, the elasticity of gas is a complicated process. If a fixed mass of gas is
compressed, work has to be done on it. This will generate heat in the gas. If the heat
is allowed to escape and the compression does not change the temperature, the process is
said to be isothermal. However, if the heat cannot escape the temperature will rise and
give a disproportionate increase in pressure. This process is said to be adiabatic and
the Diesel engine depends upon it. In most acoustic cases there is insufficient time for
much heat transfer and so air is considered to act adiabatically. Figure 2.13 shows how
the speed of sound c in air can be derived by calculating its elasticity under adiabatic
conditions.
Fig. 2.13 Calculating the speed of sound from the elasticity of air.