follow from this compression. First, its elasticity is augmented through the mere augmentation of its density. Secondly, its elasticity is augmented by the heat of compression. It was the change of elasticity which resulted from a change of density that Newton took into account, and he entirely overlooked the augmentation of elasticity due to the second cause just mentioned. Over and above, then, the elasticity involved in Newton’s calculation, we have an additional elasticity due to changes of temperature produced by the sound-wave itself. When both are taken into account, the calculated and the observed velocities agree perfectly.
But here, without due caution, we may fall into the gravest error. In fact, in dealing with Nature, the mind must be on the alert to seize all her conditions; otherwise we soon learn that our thoughts are not in accordance with her facts. It is to be particularly noted that the augmentation of velocity due to the changes of temperature produced by the sonorous wave itself is totally different from the augmentation arising from the heating of the general mass of the air. The average temperature of the air is unchanged by the waves of sound. We cannot have a condensed pulse without having a rarefied one associated with it. But in the rarefaction, the temperature of the air is as much lowered as it is raised in the condensation. Supposing, then, the atmosphere parcelled out into such condensations and rarefactions, with their respective temperatures, an extraneous sound passing through such an atmosphere would be as much retarded in the latter as accelerated in the former, and no variation of the average velocity could result from such a distribution of temperature.
Fig. 13.
Whence, then, does the augmentation pointed out by Laplace arise? I would ask your best attention while I endeavor to make this knotty point clear to you. If air be compressed it becomes smaller in volume; if the pressure be diminished, the volume expands. The force which resists compression, and which produces expansion, is the elastic force of the air. Thus an external pressure squeezes the air-particles together; their own elastic force holds them asunder, and the particles are in equilibrium when these two forces are in equilibrium. Hence it is that the external pressure is a measure of the elastic force. Let the middle row of dots, Fig. 13, represent a series of air-particles in a state of quiescence between the points a and x. Then, because of the elastic force exerted between the particles, if any one of them be moved from its position of rest, the motion will be transmitted through the entire series. Supposing the particle a to be driven by the prong of a tuning-fork, or some other vibrating body, toward x, so as to be caused finally to occupy the position a′ in the lowest row of particles: at the instant the excursion of a commences, its motion begins to be transmitted to b. In the next following moments b transmits the motion to c, c to d, d to e, and so on. So that by the time a has reached the position a′, the motion will have been propagated to some point o′ of the line of particles more or less distant from a′. The entire series of particles between a′ and o′ is then in a state of condensation. The distance a′ o′, over which the motion has travelled during the excursion of a to a′, will depend upon the elastic force exerted between the particles. Fix your attention on any two of the particles, say a and b. The elastic force between them may be figured as a spiral spring, and it is plain that the more flaccid this spring the more sluggish would be the communication of the motion from a to b; while the stiffer the spring the more prompt would be the communication of the motion. What is true of a and b is true for every other pair of particles between a and o. Now the spring between every pair of these particles is suddenly stiffened by the heat developed along the line of condensation, and hence the velocity of propagation is augmented by this heat. Reverting to our old experiment with the row of boys, it is as if, by the very act of pushing his neighbor, the muscular rigidity of each boy’s arm was increased, thus enabling him to deliver his push more promptly than he would have done without this increase of rigidity. The condensed portion of a sonorous wave is propagated in the manner here described, and it is plain that the velocity of propagation is augmented by the heat developed in the condensation.
Let us now turn our thoughts for a moment to the propagation of the rarefaction. Supposing, as before, the middle row a x to represent the particles of air in equilibrium under the pressure of the atmosphere, and suppose the particle a to be suddenly drawn to the right, so as to occupy the position a″ in the highest line of dots: a″ is immediately followed by b″, b″ by c″, c″ by d″, d″ by e″; and thus the rarefaction is propagated backward toward x″, reaching a point o″ in the line of particles by the time a has completed its motion to the right. Now, why does b″ follow a″ when a″ is drawn away from it? Manifestly because the elastic force exerted between b″ and a″ is less than that between b″ and c″. In fact, b″ will be driven after a″ by a force equal to the difference of the two elasticities between a″ and b″ and between b″ and c″. The same remark applies to the motion of c″ after b″, to that of d″ after c″, in fact, to the motion of each succeeding particle when it follows its predecessor. The greater the difference of elasticity on the two sides of any particle the more promptly will it follow its predecessor. And here observe what the cold of rarefaction accomplishes. In addition to the diminution of the elastic force between a″ and b″ by the withdrawal of a″ to a greater distance, there is a further diminution due to the lowering of the temperature. The cold developed augments the difference of elastic force on which the propagation of the rarefaction depends. Thus we see that because the heat developed in the condensation augments the rapidity of the condensation, and because the cold developed in the rarefaction augments the rapidity of the rarefaction, the sonorous wave, which consists of a condensation and a rarefaction, must have its velocity augmented by the heat and the cold which it develops during its own progress.
It is worth while fixing your attention here upon the fact that the distance a′ o′, to which the motion has been propagated while a is moving to the position a′, may be vastly greater than that passed over in the same time by the particle itself. The excursion of a′ may not be more than a small fraction of an inch, while the distance to which the motion is transferred during the time required by a′ to perform this small excursion may be many feet, or even many yards. If this point should not appear altogether plain to you now, it will appear so by and by.
§ 10. Ratio of Specific Heats of Air deduced from Velocity of Sound
Having grasped this, even partially, I will ask you to accompany me to a remote corner of the domain of physics, with the view, however, of showing that remoteness does not imply discontinuity. Let a certain quantity of air at a temperature of 0°, contained in a perfectly inexpansible vessel, have its temperature raised 1°. Let the same quantity of air, placed in a vessel which permits the air to expand when it is heated—the pressure on the air being kept constant during its expansion—also have its temperature raised 1°. The quantities of heat employed in the two cases are different. The one quantity expresses what is called the specific heat of air at constant volume; the other the specific heat of air at constant pressure.21 It is an instance of the manner in which apparently unrelated natural phenomena are bound together, that from the calculated and observed velocities of sound in air we can deduce the ratio of these two specific heats. Squaring Newton’s theoretic velocity and the observed velocity, and dividing the greater square by the less, we obtain the ratio referred to. Calling the specific heat at constant volume Cv, and that at constant pressure Cp; calling,
