At the boundary z = -d, 



9d 3d 



— + v — + w = 



3x 3y 



The variables n, p, q, and Q are expanded in the form 



(83) 



(84) 



Q = % + e Qj + e 2 Q 2 + . . . (85) 



where e is the ratio of wave amplitude to depth H/d. The variable 

 w is expanded as 



w = o(w + ew 1 + e 2 w 2 + . . .) 



where a is the ratio of water depth to wavelength d/L. The zero- 

 order solution gives 



(86) 



P = -z 

 o 



The first-order solution gives 



3t, 3x, 



3/i 



8n 1 3(du x ) 3(dv ) 



3t 



3x, 



3/i 







1 ""l "■' l 



a(du a ) 3(dv ) /3u 3v 1 



l 



3x n 



3/i 



3x, 3y 



The second-order solution gives 



|"3(d Ul ) 3(d Vl )-| 



3 racduj) 3(dv x ) 



z ^7 L 8x i + 3y i 



2 Z 3y x |_3x 1 + 3y x 



2 Z 3Xj |_3x 1 + 9y 1 J 



(87) 



(88) 



(89) 



(90) 



52 



