254 



NATURE 



[7uly ^, 1878 



indirect way of accounting for a fact, and (as we have 

 seen) according to the kinetic theory it cannot hold, since 

 according to this theory, detuity and pressure can have 

 no influence on the velocity of the wave, and on the other 

 hand it is a known fact that the velocity of the molecules 

 in their exchange of motion (by which means alone they 

 can propagate the wave) is increased by the heat — indeed 

 this augmentation of velocity itself represents the added 

 "heat" This explanation of the increased velocity of a 

 sound-wave in a heated gas commends itself therefore 

 not only by its simplicity, but as a matter of scientific 

 truth. 



7. This serves also to explain in a direct and simple 

 manner the relation the velocity of sound in a gas bears 

 to the temperature. The absolute " temperature " of the 

 gas represents (as is known) simply the energy of the 

 molecules. The velocity of the molecules (as of any 

 moving system of bodies) is proportional to the square 

 root of their energy, and therefore proportional to the 

 square root of the absolute temperature (since the " tem- 

 perature " represents the energy). The velocity of sound, 

 therefore (which is proportional to the velocity of the 

 molecules), is thereby proportional to the square root of 

 the absolute temperature of the gas. 



8. To afford a more distinct idea of the mode of pro- 

 pagation of the wave and the physical effect (condensa- 

 tion and rarefaction) produced on the gas by its passage, 

 the following considerations may serve. It is an important 

 fact to keep in view that a system of bodies in free col- 

 lision, such as the molecules of a gas, do not move in a 

 mere chance or perfectly irregular manner, but a certain 

 regularity exists. It has been mathematically proved 

 that a forcible self-acting adjustment goes on among the 

 colliding molecules of a gas so as to cause them to move 

 in a special tmiwaeT, viz., so that an equal number of mole- 

 cules are moving in all directions, or as many molecules 

 are moving in any one given direction as in the opposite. 

 This mode of motion, if artificially disturbed, will correct 

 itself. It is this special mode of motion (or movement of 

 the molecules equally towards all directions) that pro- 

 duces the perfect equilibrium of pressure in all directions, 

 observed in a gas. 



9. From the fact that as many molecules are moving in 

 any one direction as in the opposite ; it follows that if an 

 imaginary plane be placed in any position outside a vessel 

 •containing gas, the number of molecules (in the vessel) 

 which at any instant are approaching the plane, is equal to 

 the number which at the same instant are receding from it. 

 Or otherwise, if we suppose any imaginary straight line in 

 a gas, and visualise the molecules upon this line, then, as 

 many molecules are moving in one direction as in the 

 opposite. In the case of those molecules which are 

 moving obliquely to the line, the resolved component of 

 the motion in the direction of the line can be taken. 

 This consideration enables the mode of motion of the 

 molecules of a gas in its normal state, and the manner of 

 propagation of waves through that mode of motion, to 

 be illustrated in a very simple manner. 



10. In the annexed diagram, let i, 2, 3, &c., represent a 

 line of spheres moving in such a way that as many spheres 

 are moving in one direction as in the opposite. A^ the 

 spheres marked with the odd numbers may be supposed 



to move in one direction, while those marked with the 

 even numbers move simultaneously in the reverse direc- 

 tion, the vis viva in the one direction balancing that in 

 the opposite direction (as is the case with a gas). Each 

 alternate sphere thus simply oscillates backwards and 

 forwards in opposite directions within the limits repre- 

 sented by the dotted lines in the diagram, the spheres 

 continually rebounding from each other, and the line of 



spheres tending to open out or expand and separate the 

 final controlling surfaces A and B (like the expansive 

 action of a gas). It will be observed that this is in prin- 

 ciple the only mode of motion possible by which the 

 spheres can be in equilibrium ; or half move in one direc- 

 tion and half in the opposite, so that the centre of gravity 

 of the whole is at rest, in analogy with the centre of gravity 

 of a portion of gas (the vis viva being at the same time 

 balanced). There are only minute differences of detail 

 as regards the comparison with a gas, none of principle. 

 One detail is that every alternate molecule (in a line of 

 molecules taken in a gas) does not necessarily move in an 

 opposite direction, but it is rigidly true (on account of the 

 vast multiplicity of molecules) that in any appreciable por- 

 tion of a line taken in a gas, as many molecules are moving 

 in one direction as in the opposite ; for if not, the gas could 

 not be in equilibrium in the direction of this line, whereas 

 it is known to be in equilibrium in every direction. Another 

 detail is that some of the molecules of a gas are moving 

 obliquely to such an imaginary line, so that the mean 

 path of the molecules is generally greater than that repre- 

 sented by the spheres. These details cannot however in 

 the least affect the principle, and therefore the above 

 method of illustration will serve (keeping in view the small 

 differences mentioned) to convey a perfectly just idea of 

 the character of the motion of the molecules of a gas in 

 its normal state, and the way in which through that mode 

 of motion "waves" are propagated through the gas. It 

 is evident that an illustration is desirable in order to 

 visualise clearly the facts. ^ 



11. Suppose, now, a slow oscillatory motion in the form 

 of a movement of vibration to be communicated to the 

 plane A. The plane B may be supposed removed and the 

 line of spheres extended indefinitely from the plane A. 

 Then at the first forward swing of the plane A, the sphere 

 I will receive an increment of velocity which it will 

 transfer by collision to sphere 2, the sphere i returning 

 with its n<Jrmal velocity to the plane, and receiving from 

 it a second increment, &c. By the forward swing of the 

 plane, a succession of small increments of velocity will 

 thus be propagated in the form of a pulse or semi-wave 

 along the line of spheres, the velocity of propagation of 

 the pulse being that of the spheres themselves. By the 

 backward swing of the plane (to finish one complete 

 vibration) a series of small decrements of velocity form- 

 ing the second half of the wave will be propagated in 

 precisely the same manner along the line of spheres. 

 Owing to the succession of increments of velocity received 

 by the spheres in the first half of the wave, these spheres 

 will be shifted bodily forwards (to a slight extent), and 

 owing to the succession of decrements of velocity sus- 

 tained by the spheres in the second half of the wave, 

 these spheres will be shifted (to a slight extent) bodily 

 backwards, an alternate closing and opening out of 

 the line of spheres corresponding to " condensations " and 

 "rarefactions" being the result. There is only one 

 slight (xjuantitative) difference in the case of an actual 

 gas. Owing to the fact that some of the molecules 

 in the case of a gas are moving (at the instant of passage 

 of the wave) obliquely to the line of propagation of the 

 wave, the rate of advance of the wave along the line of 

 propagation will necessarily be somewhat slower than the 

 velocity of the molecules which propagate it. It is (to 

 take a homely illustration) as if some couriers were trans- 

 mitting a message, and some of them were moving 

 obliquely to the line of transmission of the message, whea 

 evidently the rate of transmission would be less than the 

 velocity of the couriers. In order to obtain the true rate 

 of propagation of the wave in the gas, the oblique motions 

 of the molecules must be taken into account. 



12. In connection with a former paper * bearing on this 



' The mere fact of molecules, in the case of a gas, shifting their positions 

 (through diffusion) can of course nialce no difference, sines the same 

 character of motion is rigidly kept up. 



" Phil. Mag., June, 1877. 



