BASIC EQUATIONS 



The mass of fluid entering the face AH ED in unit 

 time is clearly the rate at which mass is moving in the 

 X direction times the area of AH ED, or pUxlylc The 

 mass of fluid leaving the box through BCGF is a 



d{pUx) dipUy) d(pu^) 



Figure 2. Infinitesimal cube of fluid. 



similar expression, but with p and u^ measured at 

 (x + lx,y,z). The value of pu^ at (x + lx,y,z) is just 

 its value at {x,y,z) increased by lxd{pUx)/dx since h is 

 very small. That is, the mass leaving in one second 

 through face BCGF is 



\pUx + — (pMi)?xJZi 



= + — {pUx)lxUylz. 



Then the net increase per second in the mass inside 

 the box caused by the flow through the two faces 

 perpendicular to the x axis is 



d 



~ {pUx)lxl!/lz- 



dx 



Similarly, the net increase per second caused by the 

 flow through the two faces perpendicular to the y 

 axis is 



d 



dy 

 and through the two faces perpendicular to the z axis, 



d 



— —(pUz)lzkk- 



dz 



The total time rate of increase in the mass con- 

 tained in the box is simply the sum of these three 

 quantities, or 



+ 



+ 



\hkh 



(1) 



L dx dy ' dz 



Since no mass can be created or destroyed inside 

 the box, this rate of deposit of mass must result in a 

 corresponding change in the average density p inside 

 the box. That is, 



-£plxlylzj 



-[ 



d{pUx) d(pUy) d{pUz) 



+ 



+ 



Xlxf'yvz' 



dl ' " ''' L dx ' dy ' dz 



Canceling out the constant factor Ixlyk, we obtain the 

 general equation of continuity 



dp 

 dt 



= -[ 



d{pUx) , d{pUy) d{p 



+ 



+ 



OUz) l 



dz 1 



(2) 



dx ' dy ' dz 

 This equation can be simpUfied if it is assumed 

 that all displacements and changes of density are so 

 small that second-order and higher products of them 

 can be neglected. The actual density p, then, will not 

 be very different from the constant equilibrium den- 

 sity po. If 0- is defined by 



P — Po 



Po 



(3) 



then (T, the fractional change in density caused by the 

 displacement of the fluid from equilibrium, will be a 

 very small number. Henceforth a will be called the 

 fractional density change or condensation. 

 With this understanding, it is clear that 



d{pUx) 

 dx 



= ~[(Po + PoO-)Mx] 

 dx 



d dUx 



= —(PoMi) = POT— , 



dx dx 



since the second-order product aUx can be neglected. 

 By substituting this value of d(pUx)/dx and similar 

 expressions for d{pUy)/dy and d(pUz)/dz into equation 

 (2), the following simplified equation of continuity 

 results : 



^ _ { 

 dt~~\ 



dUx du 



+ T^ + 



(4) 



.dx dy 



Equation (4) is the form of the equation of con- 

 tinuity which will be used in the derivation of the 

 wave equation (27). 



For later reference, we shall note what happens to 

 the volume occupied by an infinitesimal mass of the 

 fluid when the fluid is given a small displacement 

 from equilibrium. If Vq is the volume occupied by the 

 small mass at equiUbrium, and v is the volume at 

 time t, then a fractional volume change co can be de- 

 fined by 



CO = —— • (5) 



