GENERAL THEORY OF SOUND WAVES 199 



the ratio of the maximum value of udu/dx to du/dt is ka. The 

 restriction to " infinitely slow " motions therefore means that 

 the amplitude must be small compared with X/2-7T. 



If we fix our attention on any geometrical surface, open or 

 closed, drawn in the region occupied by the fluid, the expression 



(lu + mv + nw) BS . Bt, 



where (/, m, n) is the direction of the normal drawn from an 

 elementary area BS of the surface, towards one side, measures 

 the volume which in the infinitely short time Bt crosses BS. 

 The coefficient of Bt in this expression is called the "flux" 

 across BS, and its integral 



(lu + mv + nw) dS, . (8) 



taken over the surface, is called the total flux across the latter 

 towards the side on which the normals are supposed drawn. It 

 measures the rate at which fluid is being carried across the 

 surface, expressed in terms of volume per unit time. 



To calculate the flux outwards across the boundary of an 

 elementary rectangular region BxByBz having its centre P at 

 the point (x, y, z), we note that the average velocities parallel 

 to x, over the faces ByBz, being equal to the values of u at the 

 centres of these faces, will be 



respectively. The difference of the fluxes, from left to right, 

 across these faces is accordingly du/dx .BxByBz. Adding the 

 corresponding terms for the other pairs of faces, we obtain the 

 result 



C r 



(c 



ou dv dw\ ~ . 



~- + 5- + )Ba:ByBz (9) 



dy oz 



The expression in brackets gives a sort of measure of the rate 

 at which the substance in the neighbourhood of P is on the 

 whole flowing away from P. It is therefore called the 

 " divergence " of the vector (u, v, w), and is denoted by 

 div (u, v, w) ; thus 



. du dv dw 

 , W ) = + + (10) 



