182 Lord Kayleigh on the Discharge of 



length (X) o£ the periodic features along the jet is given 

 by\ = 27r//3. 



The most important solution corresponds to the first root 

 of (10), viz. 2*405. In this case 



x= 27rRy / (W7a 2 -l) fn) 



2-405 • • v v 



The problem for the two-dimensional jet is even simpler. 

 If b be the width of the jet, the principal wave-length is 

 .given by 



\=2/,v'(W 2 /a 2 -l) (12) 



The above is substantially the investigation of Prandtl, who 

 iinds a sufficient agreement between (11) and Emden's 

 measurements *. 



It may be observed that the problem can equally well be 

 treated as one of the small vibrations of a stationary column 

 of gas as developed in ' Theory of Sound/ §§ 26$, 340 (1878). 

 If the velocity- potential, symmetrical about the axis of z, be 

 also proportional to e^ kat+ ^\ where k is such that the wave- 

 length of plane waves of the same period is 2tt//c, the equa- 

 tion is § 340 (3) 



^ + 1 # +( P_^ )( ^ = 0, . . . (13) 

 dv r dr . 



and if k> ft 



<i> = e« k « t+ MJ {\/{k 2 -/3 2 ) .r}. . . . (14) 



The condition of constant pressure when r = H gives as 

 before for the principal vibration 



V / (/r-.5 2 ).R=2-405 (15) 



The velocity of propagation of the waves is ka/fi. If w T e 

 equate this to W and suppose that a velocity W is superposed 

 upon the vibrations, the motion becomes steady. When we 

 substitute in (15) the value of k, viz. W/3/a, we recover (11). 

 It should perhaps be noticed that it is only after the vibrations 

 have been made stationary that the effect of the surrounding 

 ■air can be properly represented by the condition of uniformity 

 of pressure. To assume it generally would be tantamount 

 to neglecting the inertia of the outside air. 



* When W<r«, /3 must be imaginary. The jet no longer oscillates, 

 •but settles rapidly down into complete uniformity. This is of course the 

 usual ca^e of gas escaping* from small pressures. 



