Equations 9 and 10, and defining V = Ui + Vj yields 



ui = gk tanh (kd) 1/2 + Uk cos 9 + Vk sin 9 



where g is the acceleration due to gravity, and d is the local water 

 depth. 



15. Taking the differential of Equation 14 yields 



(14) 



1 JJk 

 k 3x 



_39 (U sin 9 - V cos 6) 

 3x A 



3U 3V . 

 -t— cos 9 + -r— sin 



dX 3x 



where 



i + 



2kd 



sinh (2kd) 



(cj - Uk cos 9 - Vk sin 9) ».. 

 o 3d 



A sinh (2kd) 3x 



- U cos 9 - V sin 9 |+ U cos 9 + V sin 



.Likewise 



1 _3k 

 k 3y 



_39 (U sin 9 - V cos 9) 

 8y A 



3U 3V . 

 -r— cos 9 + -— sin 

 dy dy 





(to - Uk cos 9 - Vk sin 9) 



o 



A sinh (2kd) 

 16. By substitution of Equations 15 and 16 into Equation 12 



89 



9y 



391 a sin 9 (U sin 9 - V cos 9) 



_| cos 9 + j 



3y 



sii 



cos 9 (U sin 9 - V cos 9) 



r su 



-r- COS 



3y 



9 + 



-r— sin 9 

 3y 



' 3U 



A 

 9 + 



3V . . 

 -r— sin 9 

 3x 





A 





a) - Uk cos 9 - Vk sin 9 . , 



o 3d 



A sinh (2kd) 3y 

 u - Uk cos 9 - Vk sin 9 „. 



A sinh (2kd) 



= 



(15) 



(16) 



(17) 



13 



