o dP 



icaW-gR+—+rP = 

 dz 



i(oR + rw= 



o o o 



du dv dw 

 — + — +— =0 

 dx dy dz 



(67) 

 (68) 



(69) 



Elimination of u, v, w, and R yields 



2ff' dP 



<o^'-fl dx co^-f^ dy 



gr 



where f^^L and p'^^ 

 ' dy dz 



,2t-' 



r P=0 (70) 



gr-oj^dz fgY^_^2) 



We now consider waves with infinitely long crests, — = 0, ai^d neglect the 



small terms 



gr-aj2 dz 



^«1. 



It follows that 



-f 



with a separation 



yields 



d^ 2ft d_ 



dy^ \^-f^ dy 



dx 



1 (9^P dP 



gr -oj^ I dz^ dz 



dz' 

 2i-' 



gr- 



P (y, z) = G (y) P (z) 

 d^G 2ff' dG 



(71) 



(72) 



+ f (a>^-r}G=Q 



dy^ oj^-f^ dy 



d^P ^ dP 



dz' 



dz 



(gr - co^k 



2r' 



gr-oj' 



p = o 



(73) 



(74) 



After differentiation of (74) under consideration of (67) , (68) , and 

 it follows that 



— + r — + fgr - <y^) e w = 



dz"^ dz 



«1, 



gF-oj^ dz 

 (75) 



Therefore, equation 73 can be interpreted as the y— dependency of W. 



26 



