Let us next consider the motion of air in a cylindrical tube, in the particular case in which the leg of a vibrating tuning-fork is applied at one end. This is a simple case of the propagation of sound-waves. We shall treat it by a synthetical process, some what like that given by Newton.
As we have already seen (see PENDULUM), a simple vibration such as that of a pendii lum or tuning fork is the resolved part, in a definite line, of the uniform motion of a point in the circumference of a circle. What we have now to show is, that such a motion of all the particles of air in the pipe, the phase of the vibration (or the position of the particle in its path at any instant) depending on its distance from the end of the tube. is consistent with mechanical principles. When this is done, it will be easy us to trace, in this particular example, the process by which the wave is propagated from one layer of the fluid to the next. We must now consider (a little more closely Sian in PENDULUM or SouNo) the nature of the simple vibration of each particle of the air.
Suppose P to move, with uniform velocity V, in the circle APB, and let PQ be drawn perpendicular to the fixed diameter, OA, then the acceleration of P's motion is Vi OA in the direction P0. Hence in the motion of Q, which is a simple vibration, we have, by lie rule for resolving velocities and accelerations (see VELOCITY), Velocity of Q— in the direction QO; OQ ' Acceleration of Q = OA OA in the direction QO.
Next cow! ler two particles of air near one another in the axis of the tube, or the masses of air in tie o contiguous cross-sections of the tube. If the phase of vibration were the same for outh they would be equally displaced from their original positions and the air between them would be neither compressed nor dilated.
Hence, that a wave may pass, the phases must be differ ent. Let, then, Q represent the position of the one par ticle, or layer, in its line of vibration at any instant; Q', the simultaneous position of the other. The first will be displaced through a space OQ from its position of rest; the second, through a space OQ'; and their distance will therefore be altered by the amount QQ', which may be taken to represent the compression or dilatation. But it is easy to see that, as P and P' move round, QQ' is always proportional to PQ. Hence the compression or dilatation of the air in any cross-section of the tube is proportional to the velocity with which it is moving. Hence the dif
ference of pressures before and behind any such section is proportional to the difference of velocities—i.e., to the acceleration of the motiott while the section passes over a space equal to its own thickness. And this is consistent with mechanical principles, for the mass of air in the section is constant, while the differ ence of pressures before and behind produces the acceleration, and should therefore be proportional to it. The particles of air in cross-sections of the tube therefore vibrate, each in the same period as does the tuning-fork, but the phase is later for each section in. 'proportion to its distance from the fork. Where the phase is one or more whole vibra tions later than that of the fork. the motion is exactly the same as that of the fork, and simultaneous with it. At all other points, it is the same as that of the fork, but. not siniultanegus. Thus the greatest displacement of the fork is immediately shared by the layer next it, later by the next layer, and so on. Thus, a ware of displacement trav els along the tube from one section to the next, while each particle merely oscillate& backward and forward through (in general) a very small space about its position of rest.
The reader who has followed the little geometrical investigation above will have no, difficulty in proving for himself that the velocity with which the wave travels is proportional to vq, where p is the pressure, and p the density of the air. The easiest mof doing this is to express, in terms of these and other quantities, the equation given us by the laws of motion, Mass X Acceleration = Difference of pressures, and to 4ssutne that Hooke's (q.v.) law holds, even during the sudden compression of air. This, we know, is not the case.; so that a correction has to be applied to the above expression; • depending on the heat developed by sudden compression or lost in sudden rarefaction, by each of which the elastic force of the air is increased. But this has been already discussed in SOUND.
The above formula shows us, however, that the velocity of sound is not affected by the pressure of the air—i.e., the height of the barometer—since, in still air, p is pro. portional to p. The velocity does depend on the temperature, being, in fact, proportional (ceteris paribus) to the square root of the temperature measured from absolute zero. See