Gases under High Pressures. 181 



neglect the terms of the second order. In like manner the 

 other equations become 



wf =- 1 t- w l ;- = - ll! f. . . ( 3) 



dz p dx dz p dij K ' 



Further, the usual equation of continuity, viz. 



(4) 





J j 1 — l/« • . . 



a.i' rt</ etc 



here reduces to 



i*44v w j i =0 . ... (5) 



\a# </y as/ as 



If we introduce a velocity-potential (/>, we have with use 

 of (2) 



_,, V? dp W 2 dw W 2 <? 2 <4 ,„. 



y * = —fTz = cP ~dT-^-d?> ■ ■ (6) 



where a, = \Z(dpjdp), is the velocity of sound in the jet. In 

 the case we are now considering, where there is symmetry 

 round the axis, this becomes (r 2 = x 2 -\- y 2 ) 



^'^(1-^=0, . . (7) 

 dr- r dr \ tr J dz 2 v / 



and a similar equation holds for w, since w = d<j>ldz. 



If the periodic part of w is proportional to cos fiz, we have 

 for this part 



d 2 w , 1 die , /W 2 ,\ ~ g 

 dr' r dr \cr J 



and we may take as the solution 



w=W + H cos /3z . J { a/( W- -a 2 ) . fir/a}, . (9) 



since the BesseFs function of the second kind, infinite when 

 r = 0, cannot here appear. The condition to be satisfied at 

 the boundary (r = R) is that the pressure be constant, equal 

 to that of the surrounding quiescent air, and this requires 

 that the variable part of w vanish, since the pressure varies 

 with the total velocity. Accordingly 



Jo{V(W 2 -a 2 )./3R/a} = 0, . . . U0) 



which can be satisfied only when W> a, that is when the 

 mean velocity of the jet exceeds that of sound. The wave- 



M 



