CHAPTER VII 



GENEKAL THEORY OF SOUND WAVES 



67. Definitions. Flux. Divergence. 



In respect of notation it is convenient now to take a point 

 of view somewhat different from that adopted in the preceding 

 chapter. We denote by u, v y w the component velocities, 

 parallel to rectangular axes, considered as functions of position 

 (x, y, z) and of time t. With each point of space there is 

 accordingly associated, at any given instant, a vector (u, v, w), 

 and the whole assemblage of such vectors gives an instantaneous 

 picture of the distribution of velocity*. On the other hand 

 the variations of u, v, w with the time, for given values of 

 x, y, z, give the history of what goes on at a particular place f, 

 but supply in the first instance no information as to the 

 careers of the various particles which (so to speak) successively 

 cross the scene. 



When we proceed to calculate the component accelerations 

 of the particle which at the instant t is in the position (x, y, z) 

 we have to take account of the fact that after the lapse of a 

 time however short its velocities u, v, w will be given by the 

 respective functions of the altered position as well as the altered 

 epoch. Suppose that at two successive instants ^, ^ a particle 

 occupies the positions P and P', respectively, and that the 

 corresponding values of the ^-component of the velocity are 



* M. Marey and others have taken photographs, of short exposure, of a two- 

 dimensional current of water carrying suspended motes. The image of each 

 mote is drawn out into a short line, which indicates the direction and magnitude 

 of the corresponding velocity. 



t As if we were to view the surface of a stream through a narrow tube, 



