200 DYNAMICAL THEOEY OF SOUND 



By dividing any finite region into rectangular elements we 

 see that the total flux outwards across the boundary must be 

 equal to the volume-integral of the divergence, or 



This can of course be proved mathematically without attributing 

 any kinematical meaning to the symbols. 



68. Equations of Motion. 



To form the dynamical equations, we fix our attention on 

 that portion of matter which at the instant t occupies the 

 rectangular space &xyz. On the hypothesis of infinitely 

 slow motion its acceleration of momentum parallel to x is 

 p Sx &y$z .du/dt, where p is the density. The mean pressures 

 on the respective faces may be taken to be the pressures at the 

 centres of those faces, and the total pressures on the two faces 

 perpendicular to x are therefore 



The difference gives a force dp/dx. Sx&y&z in the direction of 

 ^-positive. Equating this to the acceleration of momentum, we 

 obtain the first of the following system of equations : 



du _ dp dv _ dp dw _ _dp ,-. 



p dt~~d~x' p dt~~dy' p ~dt~~dz' 



Since the variations of p when multiplied by du/dt, ..., ... may 

 be neglected, we may replace p by its equilibrium value p , but 

 it will not always be necessary to preserve the suffix. 



As in 59 we write 



} ........................ (2) 



where s denotes the condensation (p po)/po, and K is the cubic 

 elasticity of the fluid. If we further write 



c 2 = */po, ........................ (3) 



as before, we obtain 



du_ 8s dv_ _ 8s <^__ c2 ds ( ft 



dt~ dx> dt~ d' dt~ dz ....... W 



