112 J. IF. Dams — Motion of Compressible Fluids. 



infinite number of consecutive cases of steady motion, a and 

 XI are now constant ; the curve (4) is fixed, and the curve (2) 

 recedes from coincidence with the axes to an indefinite dis- 

 tance. The alternative velocities approach each other, become 

 equal, and become imaginary. Each section has, therefore, a 

 limited capacity for the discharge of fluid. Evidently this 

 capacity varies as <r, and is greater as the curve (4) is farther 

 from the intersection of the axes, or as H is less, or as C is 

 greater, and is less as 12 is greater or C is less. If the energy 

 per unit mass should also increase with sufficient rapidity, the 

 capacity of each section would not be reached ; otherwise, it 

 is ultimately reached, and the section whose area is least, or at 

 which the potential is greatest, limits the capacity of the tube 

 as a conduit. 



When the capacity of the tube is not reached, the velocity 

 is either everywhere less than the velocity of sound, or every- 

 where greater. When the capacity of the tube is reached, the 

 velocity at the section of least area or where the potential is 

 greatest, Is equal to the velocity of sound at that point, and 

 there the first derivative of the velocity with respect to dis- 

 tance assumes the indeterminate form — . To resolve this, 







differentiate (2) with respect to s, and cancel the term contain- 

 ing dcr-^ds ; then 



d \ dp \ d 2 p dp 

 ds [dp f ~~ dp 2 ds 



__d 2 pp_dq^ 



dp 2 q ds' K V 



With the aid of (14) differentiate 



numerator and denominator 



of (13) once and solve for — : 





/ dp q d 2 a- 

 do / dp a ds 2 



ds ^ 2q + ^ 

 dp 2 q 



/ d 2 a 



dp 2 q 



TTT1 d 2 <r d 2 Q . dq 

 WbeD ds~* ' ° r S*' 1S zer0 > ds 



= ±0. 



When the capacity of the tube is reached, the conjugate 

 flows meet at the critical section with equal velocities, densi- 

 ties, and pressures, and are, consequently, there indistinguish- 

 able. On either side of this section, which of the two possible 

 flows is actual, is determined by conditions existing on that 

 side. Hence it is possible for the actual flow to have a veloc- 

 ity less than sound on one side and greater on the other. That 

 this is not a natural case appears as follows : On either side of 



