J. W. Davis — Motion of Compressible Fluids. Ill 



dq _ ±—= — I dp I do- f/fl ) 



ds ~~ q* j- ( dp a- ds ds \ ' ^ ' 



dp 



Let q f be the lesser of the alternative velocities, and q" the 

 greater. It has already appeared, as illustrated in figure 1 

 that q' is less than <\/dp' ' ~dp\ and that q n is greater than 

 V dp"-r-dp n . \/dp~dp is the velocity of sound in the fluid 

 where the density is p. It will be simpler and more intelligible 

 to consider separately the effects of changes in a and 12 ; that 

 is, to treat one as a constant while the other varies. Thus we 

 learn from (13) that where the tube is narrowing, or the poten- 

 tial increasing, q f increases, and q" diminishes ; that where the 

 tube is widening or the potential diminishing, q' decreases and 

 q" increases; and that where the tube is uniform in section, or 

 the potential constant, q' and q" do not change. Comparison 

 of this statement with figure 1 shows, moreover, that the curve 

 (4) approaches the intersection of the axes while 12 increases, 

 and recedes from this point while 12 diminishes. 



When q', q ;/ , become equal, each satisfies equation (7). If 

 this happens where the tube is narrowing, or the potential 

 increasing, dq~ds == db oo , the upper sign pertaining to q' and 

 the lower to q". Represented graphically, with distances laid off 

 as abscissse and velocities as ordinates, the meeting of q', q ,r 

 occurs at a vertex having its convexity on the positive side. 

 Beyond the vertex extend imaginary values to the section 

 where appears a vertex with its convexity on the negative side. 

 This must be where the tube is widening, or potential decreas- 

 ing, and the velocity of the fluid is equal to the velocity of 

 sound. 



Equation (5) shows that wherever a has the same value in 

 the tube, extraneous forces not acting, or wherever 12 is the 

 same in a tube of uniform cross-section, the values of q are 

 the same. Hence the alternative velocities become equal, in 

 the one case, wherever in the tube <j has a particular value, 

 and, in the other case, wherever 12 has a particular value ; and, 

 wherever a is less, or 12 is greater, than its particular value, the 

 curves (4), (2), have receded from each other, and the velocities 

 are imaginary ; and, wherever a is greater, or 12 is less, than 

 the particular value, the curves (4), (2), have advanced upon 

 each other, and the velocities are unequal, and become more 

 and more separated as a or 12 departs from its particular value. 



While we confine our attention to a particular cross-section 

 of the tube, suppose the quantity of matter, M, that in unit 

 time flows across any section, retaining the constant energy C 

 per unit mass, continuously increases from zero, as in an 



