same vertical circle with constant velocity. If the sun is in the zenith, i. e. in such a position that the shadows of all objects are thrown vertically, the shadow of the ball on any horizontal plane below it will move exactly as does the bob of a pendulum.

The form of the associated wave for longitudinal pendulum-vibrations is shown in Fig. 17 (a).

Fig.17(a)

Retaining the form of the curve, we may make its amplitude and wave-length as large or as small as we please, as in the case of the waves in Fig. 4, (1) and (2), p. 13.

20. We have examined the transmission of waves due to longitudinal vibrations along a single very slender filament. Suppose that a great number of such filaments are placed side by side in contact with each other, so as to form a uniform material column. If, now, precisely equal waves are transmitted along all the constituent filaments simulta

neously, successive pulses of condensation and rarefaction will pass along the column. The parts in any assigned transverse section of the column will, obviously, at any given moment of time, all have exactly the same degree of compression or dilatation. When a pulse of condensation is traversing the section, its parts will be more dense, when a pulse of rarefaction is traversing it, less dense, than they would be, were the column transmitting no waves at all, and its separate particles, therefore, absolutely at rest. Let the column with which we have been dealing be the portion of atmospheric air enclosed within a tube of uniform bore. The phenomena just described will then be exactly those which accompany passage of a sound from one end of the tube to the other. It remains to examine the mechanical cause to which these phenomena are due.

the

Atmospheric air, in its ordinary condition, exerts a certain pressure on all objects in contact with it. This pressure is adequate to support a vertical column of mercury 30 inches high, as we know by the common barometer. In Fig. 18 is shown a section of a tube closed at one end, with a moveable piston fitting into the other. In (1) the air on both sides of the piston is in the ordinary atmospheric condition, so that the pressure on the right face of

the piston is counteracted by an exactly equal and opposite pressure on its left face.

(1)

(2)

Fig.18.

In (2) the piston has been moved inwards, so as to compress the air on the right of it. That on its left, being in free communication with the external air, is not permanently affected by the motion of the piston. In order to retain the piston in its forward position, it is necessary to exert a force upon it, in the direction of the arrow. If this force is relaxed, the piston is driven back. Since the pressure of the air on the left of the piston is just what it was before, that on its right must necessarily have increased. But this increase of pressure is accompanied by an increase of density, due to the compression of the air on the right of the piston. Hence increase of pressure accompanies increase of density.

If, as in (3), we reverse the process, by moving the piston outwards, the extraneous force must be exerted in the opposite direction, as shown by the arrow. The pressure on the right of the piston is therefore less than the normal atmospheric pressure on its left, i. e. diminution of pressure accompanies diminution of density. By experiments such as the above, it was shown, by the French philosopher Mariotte, that the pressure of air varies as its density.

21. Next, let us take a cylindrical tube open at one end and having a moveable piston fitting into the other, as in Fig. 19.

In (1) the piston is at rest at A, and the air in its ordinary atmospheric condition of density and

pressure. In (2) the piston is pushed inwards as far as C. While it is moving up to this position, the

air-particles in front of it are thrown into motion. Suppose that, at the moment when the piston reaches C, the particles at D are just beginning to be disturbed. The air which, in (1), occupied AB, is now crowded into CD, and is, therefore, denser than that further on in the tube. Now, let the piston be drawn back to E, (3), as much to the left of its original position, A, (1), as, in (2), it was to the right of it. The air in CD, (2), will, while this is taking place, expand into EF; for, being denser, it will also be at a greater pressure, than the air to the right of it. It will, therefore, act on the air in advance of it in the same way as the piston did on the air in contact with it when moving from A, (1) to C, (2). Hence the air in FG will be condensed, G being the point where the air particles are just beginning to be disturbed when the piston reaches the position E. Thus the air at D advances to F. Further, in consequence of the backward motion of the piston, the air in the neighbourhood of C, (2), has to move to E, (3). Thus the air originally in AB now occupies EF, which is greater than AB. It is therefore less dense than in (1), i.e. is in a state of rarefaction. Now, let the piston again advance to H, (4). The air in FG being at a greater pressure than that in its front, and still more so than that in its rear, will expand in both directions, causing a