stability of and Waves in Stratified Flows 



p Is the deviation of the pressure from the hydrostatic pressure in 

 the primary flow, p is the density perturbation » g is the gravi- 

 tational acceleration, and 



dy 



The equation of continuity 



u + V =0 

 X y 



permits the use of a stream function \\> ^ in terms of which the velocity 

 components can be expressed: 



u = ijjy, V = - ij;^. (3) 



The linearized form of the equation of incompressibility is 



p^ +Up, +v'^' = 0, (4) 



in which 



-\ - dp 



^ " dy • 



If r\ is the vertical displacement of a line of constant density 

 from its mean position, the kinematic relationship 



Tl^ +UT1, = V = - 4j, (5) 



holds. All perturbation quantities will be assumed to be periodic in 

 X and have the exponential factor exp ik(x - ct) , so that from (5) and 

 (3) we have 



i|j = - (U-c)Ti, u= - [(U-c)7i]', v=ik(U-c)r|, (6) 



Then (1) and (4) give 



p = p(U-c)^Ti' and p = -p'ti. (7) 



Writing 



il(x,y,t) = F(y)e"^(''-'=*^ (8) 



221 



