at a will now be at a', and that originally at b will be at b', if bb' be made equal to aa'.

Fig.16.

B

By performing the same process for different instants, we can find as many corresponding positions of the longitudinally-vibrating particle as we please. It is true that we learn nothing new by this, since we cannot construct the wave-curve without knowing beforehand the mode of the particle's vibration [§ 12]. Still, when we are dealing with longitudinal particle-vibrations, and require to know the law of the variation of condensation and rarefaction at different points of a single wave, it is convenient to have a picture of the mode of vibration by which, as we know [§ 16], that law is determined. Such a picture we have in the form of the wave produced by the same mode of vibration when executed transversely.

Let us call the wave so

related to a given wave of condensation and rarefaction its associated wave.

19. Before leaving this portion of the subject, it will be advisable to draw the associated wave for

that particular mode of longitudinal vibration in which each particle moves as if it were the extremity of a pendulum traversing a path which is very short compared to the pendulum's length. The meaning of this limitation will be easily seen from Fig. 17.

Fig.17.

B

A

D

Let O be the fixed point of suspension; OA the pendulum in its vertical position; AB a portion of a circle with centre O and radius OA; a, b, c, d, points on this circle; AD a horizontal straight line through A; aa', bb', cc', dd' verticals though a, b, c, d, respectively. If the pendulum is placed in the position Oɑ, and left to itself, it will reach an equally inclined position on the other side of OA, i.e. will swing through twice the angle aOA before it turns back again. Similarly, if started at Ob it will swing through twice the angle b0A; if at Oc, through

Now the extre

twice the angle cOA; and so on. mity of the pendulum, when at a, is further from the horizontal line, AD, than when it is at b, since aa' is greater than bb'; and when at b further than when at c. If we make the pendulum vibrate through only a small angle, by starting it, say, in the position Od, its extremity will throughout its motion be very near to the horizontal straight line AD. If we make the angle small enough, or, which is the same thing, take Ad sufficiently small compared with OA, we may without any perceptible error suppose the end of the pendulum to move in a horizontal straight line instead of in a circular arc, i.e. along d'A instead of dA. To take an actual case, let us suppose the pendulum to be 10 ft. long, and its extent of swing 1 inch on either side of its vertical position. An easy geometrical calculation shows that the end of the pendulum will never be as much as

1

200

th of an inch out of the horizontal straight line drawn through it in its lowest position. This is a vanishing quantity compared to the length of the pendulum; we may, therefore, safely regard the vibration as performed along d'A instead of dA. Such a vibration, though executed in a straight line instead of in the arc of a circle, may be properly called a pendulum-vibration, as expressing the

law according to which it takes place. This law admits of simple geometrical illustration as follows. Let a ball, or other small object, be attached to some part of a wheel revolving uniformly about a fixed horizontal axis, so that the ball goes round and round in the same vertical circle with constant velocity. If the sun is in the zenith, ¿.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 bis.

Fig. 17 bis.

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 simultaneously transmitted along all the constituent filaments, 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, and 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 the 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.

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 about 30 inches high, as we know by the common barometer. Fig. 18 represents a section of a tube closed at one end, with a