ii 2F i.i- 5F i^.i tl|F i>2.i- F i,3.i 



? ~ 2 



(10) 



2 F - 2F + F 

 3y 2 (Ay) 2 



(11) 



1£ „ " 3F i,,1 + 4F if1..1 " F i+2,,1 

 3x 2Ax 



3F _ F i,/j+1 ~ F i,,1-1 



3y 



2Ay 



(12) 



(13) 



Equations 11 and 13 are central differences, and Equations 10 and 12 are back- 

 ward differences. All four expressions have the same order of accuracy. 



19. The magnitude of the wave phase function gradient at any point 

 (i,j) is computed from the following expression, 



Vs| 2 , = k 2 1+ i- 

 'i,J i.J a lfJ ^ 



'2a. . - 5a. , . + 4a. _ . - a. ., ,' 

 (Ax) 2 



a i,1+1 - 2a i,.1 + a i,.1-1 

 (Ay) 2 



'i,J 



(14) 



- „ \/-3cc + 4cc -cc > 



- 3a i,.1 + 4a i + 1,.1- a i,2,1 «i„1 8iHKl„1 !i±2U 



2Ax A 2AX 



+ ( a i,>i - a i,.i-i Y CCg i,>i CCg i t , i-i 



\ 2Ay A 2Ay I 



This equation was derived by approximating the partial derivatives in Equa- 

 tion 4 using the finite difference operators given in Equations 10 through 

 13. The reason for selecting backward finite differences to approximate the 

 x-derivatives, specifically the curvature, will be discussed later in this 

 section. 



20. Equations 8 and 9 can both be expressed in the following general 

 form: 



12 



