stability of and Waves in Stratified Flows 



III. EXTENSION OF MILES' THEOREM 

 Following Howard [ 1961] , we set 



G = W'/^F, 



where W = U - c. Then (9) can be written as 



(pWG')' - [(pU')72 + k^W +pW-'(U'V4 - gP] G = 0. (14) 



The boundary condition at the bottom is 



G(0) = 0. (15) 



The interfacial conditions (11) become 



p^(WG' - U'G/2)^ - P (WG' - U'G/2)^ = gApW"'G, (16) 



to be applied at the surfaces of density discontinuity, and in particu- 

 lar the upper-surface condition becomes 



WG' - U'G/2 = gW'G, or G(d) = , (17) 



depending on whether the upper surface is free or fixed. 



Multiplying (14) by G , where the asterisk indicates the com- 

 plex conjugate, and integrating fronn the bottom to the first surface 

 of density discontinuity and then from discontinuity to discontinuity 

 throughout the fluid domain, and utilizing (15), (16), and (17), we 

 have 



^pWLlG'l' +k'|G|'] +y(pU')'|G|V2 +yp[U'V4- gP]W*|G/w|' 



-^gAjpW*|G/w|^-^[(pU')^ - (pU')J |G|V2 = 0, (18) 



i i 



in which each of the integrals is over the entire fluid domain exclusive 

 of the surfaces of density discontinuity (i.e. , it is a summation of 

 integrals over the layers of continuous density distributions), and 

 the summation is over the discontinuities, including the free surface 

 if there is one. If the flow is unstable, c. > 0, and the imaginary 

 part of (18) is 



rp[|G'|'+k'|G|'] +rp[gP- U'V4] |G/W|2 +YgA p|G/w|2 =0, 

 ^ ^ V ' (19) 



223 



