GENEKAL THEOEY OF SOUND WAVES 205 



which is otherwise obvious from the meaning of "divergence." 

 In the case of irrotational motion, this becomes 



V 2 < = 0, ........................... (6) 



which is identical with "Laplace's equation" in the theory of 

 attractions. The same equation occurs in the theory of steady 

 electric (or thermal) conduction in metals. If, for example, $ 

 denote the electric potential, the formulae (3) of 69 give the 

 components of current, provided the specific resistance of the 

 substance be taken to be unity. This analogy will be found 

 useful in the sequel. 



The theory of the motion of incompressible fluids is capable 

 of throwing more light, occasionally, on acoustical phenomena 

 than might at first sight be anticipated. We are apt to forget 

 that the velocity with which changes of pressure are propagated 

 in water is after all only four or five times as great as in air, 

 and that the visible (or at all events easily imaginable) motions 

 of water, under circumstances where the compressibility has 

 obviously little influence, may supply a valuable hint as to the 

 behaviour of a gaseous substance under similar conditions. This 

 remark will have frequent illustration in the following chapters. 



The kinetic energy of a system of sound waves is 



. 



The potential energy, as given by the argument of 60, is 



...(8) 

 The integrations extend over the region affected. 



71. Spherical Waves. 



In the case of plane waves with fronts perpendicular to Ox 

 the equation (4) of 70 reduces to 



**-<-** m 



dt*~*dtf' 



whence <f> -f(ct x) + F(ct + x) ................ (2) 



This need not be further discussed. 



