One-dimensional flow: weak waves

To a certain extent the results of this section have already been assumed in that certain expressions for the speed of sound propagation have been used. Pressure disturbances in gaseous and other media are propagated in longitudinal waves and appeal is made to elementary physics for an understanding of the phenomenon.

Consider the air in a stream tube to be initially at rest and, as a simplification, divided into layers 1, 2, 3, etc., normal to the possible direction of motion. A small pressure impulse felt on the face of the first layer moves the layer towards the right and it acquires a kinetic energy of uniform motion in so doing. At the same time, since layers 1, 2, 3 have inertia, layer 1 converts some kinetic energy of translational motion into molecular kinetic energy associated with heat, i. e. it becomes com­pressed. Eventually all the relative motion between layers 1 and 2 is absorbed in the pressure inequality between them and, in order to ease the pressure difference, the first layer acquires motion in the reverse direction. At the same time the second layer acquires kinetic energy due to motion from left to right and proceeds to react on layer 3 in a like manner. In the expansive condition, again due to its inertia, it moves beyond the position it previously occupied. The necessary kinetic energy is acquired from internal conditions so that its pressure falls below the original. Reversion to the status quo demands that the kinetic energy of motion to the left be transferred back to the conditions of pressure and temperature obtaining before the impulse was felt, with the fluid at rest and not displaced relative to its surroundings.

A first observation of this sequence of events is that the gas has no resultant mean displacement velocity or pressure different from that of the initial conditions, and it serves only to transmit the pressure pulse throughout its length. Secondly, the

4

Direction of

wave propagation

— at instant / 4- 8/

I

Fig. 6.5 displacement, and hence velocity, pressure, etc., of an individual particle of gas is changing continuously while it is under the influence of the passing impulse.

A more graphic way of expressing the gas conditions in the tube is to plot those of successive particles in the direction of movement of the impulse, at a given instant of time while the impulse is passing. Another curve of the particles’ velocities at an instant later shows how individual particles behave.

Fig. 6.5 shows a typical set of curves for the passage of small pressure impulses, and a matter of immediate interest is that an individual particle moves in the direction of the wave propagation when its pressure is above the mean, and in the reverse direction in the expansive phase.