204 DYNAMICAL THEOKY OF SOUND 



constant round any circuit, provided we imagine the circuit to 

 move with the fluid. If initially zero for every circuit which 

 can be drawn in a finite portion of the fluid, it will remain zero 

 for every such circuit. 



70. General Equation of Sound Waves. 



We postulate henceforth the existence of a velocity potential, 

 at all events in the case of a uniform medium, to which we 

 confine ourselves for the present. We have then, from 

 68 (6) 



This symbol V 2 is called the "Laplacian operator," from its 

 constant occurrence in the analytical theory of attractions as 

 first developed by Laplace. Again, by differentiation of 69 (4) 

 with respect to t we get 



Finally, by elimination of s, 



w 



This may be regarded as the general differential equation of 

 sound waves in a uniform medium. If a solution can be 

 obtained which gives prescribed initial values to </> and s 

 (or dQ/dt), and satisfies the other conditions of the problem, the 

 subsequent value of s is given by (3), and the values of u, v, w 

 by 69 (3). 



We may stop for a moment to notice the form assumed by 

 the equations when the fluid is incompressible. This may be 

 regarded as an extreme case, in which c is made infinite, whilst 

 s is correspondingly diminished, in such a way that c 2 s, which 

 = (p p )/p , remains finite. The equation of continuity, 68 

 (6), takes the form 



(5) 



