! 



180 Lord Rayleigh on the Discharge of 



waves in the jet is the same for the same nozzle and the 

 same pressure. 



2. The pressure at which the stationary sound waves begin 

 to develop is the same in air, carbonic acid, and hydrogen, 

 and is equal to *9 atmosphere. 



This is the ipressure-excess behind the nozzle, so that the 

 whole pressure there is 1*9 atmosphere. The environment 

 of the jet is at one atmosphere pressure. 



Emden, comparing his observations with the theory of 

 Saint- Venant and Wantzel, then enunciates the following con- 

 clusion : — The critical pressure, in escaping from which into 

 the atmosphere the gas at the nozzle's mouth moves with the 

 velocity of sound, is equal to the pressure at which stationary 

 sound waves begin to form in the jet. So far, I think, 

 Emden makes out his case ; but he appears to over-shoot 

 the mark when he goes on to maintain that after the critical 

 pressure-ratio is exceeded, the escaping jet moves everywhere 

 with the same velocity, viz. the sound-velocity ; and that 

 everywhere within it the free atmospheric pressure prevails. 

 He argues from what happens when the motion is strictly in 

 one dimension. It is true that then a wave can be stationary 

 in space only when the stream moves with the velocity of 

 sound ; but here the motion is not limited to one dimension, 

 as is shown by the swellings between the disks. Indeed the 

 propagation of any wave at all is inconsistent with uniformity 

 of pressure within the jet. At the surface of the jet, but not 

 within it, the condition is imposed that the pressure must be 

 that of the surrounding atmosphere. 



The problem of a jet in which the motion is completely 

 steady in the hydrodynamical sense and approximately 

 uniform was taken up by Prandtl *, both for the case of 

 symmetry round the axis (of z) and in two dimensions. In 

 the former, which is the more practical, the velocity com- 

 ponent iv is supposed to be nearly constant, say W, while 

 u and v are small. We may employ the usual Eulerian 

 equations. Of these the third, 



dw dw dw dw 1 dp 



at ax ay dz p dz 



reduces to 



w dw 1 dp 



W dz~ pdz~> 



.... (2) 



when we introduce the supposition of steady motion and 

 * Phys. Zeitschrift, 5 Jahrgang, p. 599 (1904). 



