Wu 



The differential equation for 41 (in a source free region) can be written 



pvV + p'(s^) 



-2 (V0)'+ gz 



H'(0) = 0, 



which is due to Dubreil-Jacotin (11) and Long (12). In terms of the pseudo- 

 stream-function 0(x,y) of Yih (10), such that 



Eq. (14) becomes, somewhat simpler in form, 



(14) 



(15) 



VV + gzp'(^) - H'(0) - 0. (16) 



These equations have been often used as a starting point for studying steady 

 flows of finite amplitude. 



INFINITESIMAL DISTURBANCES; LINEARIZED 

 FORMULATION 



If the disturbed flow differs only slightly from the basic flow, we may intro- 

 duce the perturbation quantities; for the general case of compressible flows of 

 an ideal gas, we write 



lo + fli 



P - PQ-^ Pi 



P = Pn + P: 



S = S„ + S, 



(17) 



in which qo = (U,v,0), the perturbation velocity q, has the components (ui,Vi,Wj), 

 Pq) Po > ^0 ^^® related to each other by 



Po(z) = -gPo 



(18) 



and 



-l/y Sq/c^ -l/y S/c, 



^oPo e " P = pp e r 



const , 



(19) 



where 7 = Cp/c^ is the ratio of the specific heats. The local speed of sound 

 relative to the medium of the basic flow is Cq , where 



(dpo/dpo) = yPo/Po 



(20) 



Substituting Eq. (17) into Eqs. (1-3), assuming qi, pj, Pj, Sj all small com- 

 pared with their appropriate counterparts of the basic flow, we obtain the lin- 

 earized equations as: 



Po "^iv qj + (Dpj + WjPo) = PqQ 



Dqi + Wi(U'e^+ V'e ) 



-VPi - gPiC^ , 



Dpi + WjPg = (Dpi - gPoWi)/cc 



(21) 

 (22) 

 (23) 



554 



