226 Mr. W. J. M. Rankine on Laplace's Theory of Sound. 



developed in the sudden condensation of air was sufficient to 

 account for it." [The excess of the actual velocity of sound 

 above its value as calculated by Newton.] " The explanation of 

 Laplace has been held to be untenable by many eminent mathe- 

 maticians and physicists ; for it is correctly argued, that a wave 

 of sound will commence with a rarefaction as often as with a 

 condensation, and that on one side of a nodal line on a bell or 

 vibrating plate, the wave commences with a rarefaction, while 

 on the other side it commences with a condensation; so that La- 

 place's reasoning would have been equally available if the velocity 

 of sound in air had proved one-sixth less than the theoretical velocity, 

 instead of one-sixth more, by considering the cold produced by rare- 

 faction." 



I have caused a portion of the above quotation to be printed 

 in italics, in order to indicate it as involving a representation of 

 the theory of Laplace, to which every one must object who has 

 read and understood his investigation of the question, or that of 

 Poisson, which is substantially the same. 



The impression natm'ally produced by the statement of Pro- 

 fessor Potter is, that, according to the theoiy of Laplace, the 

 heat developed by a wave of compression in air accelerates its 

 transmission, in the same manner as if the whole mass of air 

 were heated by some external cause ; so that a wave of dilatation, 

 cooling the air in a similar manner, must, according to the theory, 

 be retarded. 



Were this the nature of Laplace's theory, it could not stand 

 for a moment ; for as eveiy wave must consist of a compressed 

 and a dilated part, the different parts of a wave would travel with 

 different velocities. 



But there can scarcely be a greater misconception of Laplace's 

 views. His investigation starts from the principle, known to be 

 a fact, that when the density of a gas is changed, whether by 

 compression or dilatation, its temperature changes also, and it 

 does not assume a pressure proportional to the new density until 

 it has had time to recover its original temperature. The mo- 

 mentary variation of temperature being in the same direction 

 with the variation of density, the momentary variation of pres- 

 sure, whether positive or negative, is larger as compared with 

 the original pressure than the variation of density as compared 

 with the original density ; and this in a ratio which we may call 

 7, and whose value for atmospheric air is about 1"4. 



Now the velocity with which a disturbance of density is pro- 

 pagated is proportional to the square root, not of the total pres- 

 sure divided by the total density, but of the variation of pressure 

 divided by the variation of density. It is therefore greater than 

 the result of Newton's calculation in the ratio v^y : 1 ; and this 



