REFLECTION OF DIVERGING WAVES 373 



where the characteristic resistance is pc. Here 



2 _ dp 

 dp 



He assumes adiabatic expansion, so that p and c are functions of p only. If 

 a second incremental wave of pressure dp^ also traveling in the positive 

 direction, be added, its velocity increment, being relative to the medium, 

 will add to that already present. Its value will be related to dp through a 

 new characteristic resistance corresponding to the modified density result- 

 ing from the previous increment. Hence the velocity u resulting from a 

 large number of such waves will be 



Jo DC 



pc 



where w is the quantity represented by co in Lamb's version. If, then, all 

 of the wave propagation is in the positive direction 



u — w. 



Similarly, if an incremental wave is traveling in the negative direction, 



du = - — ~ , 

 pc 



and the condition for all the propagation to be in that direction is 



11 = —w. 



Obviously, then, if u has some other value than one of these it results from 

 the addition of increments some of which propagate in each direction. 

 Riemann deduces from the aerodynamic equations that 



I + (« + c) f) (to + u) = 0, (8) 



i + (" - c) A) (. _ „) = 0, (9) 



That is, the value of iv + ii is propagated in the positive direction with a 

 velocity of c + « and that of w — u, in the negative direction with a velocity 

 c — u. If, over a finite range of x, a disturbance be set up such that neither 

 of these quantities is zero, it must be made up of incremental waves in both 

 directions. However, as w + « propagates positively it will be accompanied 

 at any instant by a value of w — u which has been propagated from the other 

 direction. But, since the value of this was initially finite over a limited dis- 

 tance only, when all of this finite range is passed, a' — u will be zero, u will 



