T 

 9u 3u au dr\ _ Bx ,, rt „>. 



— r+u-— + v — • + -rr - f v + -5— — = (103) 



at at 9t dx a d + n 



T 



3v 3v 3v 3n r By n ,, n ,, 



— + u — + V — + g — - + f U + -= — a — = (104) 



8t 3x 3y s 3y a d + n l J 



and the continuity equation is 



It- + |- (du + nu) + |- (dv + nv) = (105) 



9t 3x v 3y 



where the bottom stress is given by 



gu 



and 



T _ = £J /u 2 + v 2 (106) 



Bx r 2 



gv 



/u 2 + v 2 (107) 



Chen, Divoky, and Hwang (1975) give numerical equations for the two- 

 dimensional case based on the Boussinesq equations (see Sec. IV, 5). 



5. Computer Models . 



Solutions of the equations for long water waves are obtained by 

 numerical means. Leendertse (1967) gives the following method for solv- 

 ing the linearized long-wave equations by using a space staggered scheme 

 as shown in Figure 10. Taking the subscript n to indicate the value 

 at time t, the subscript n + 1/2 to indicate the value at time t + At/ 2, 

 and the subscript n + 1 to indicate the value at time t + At, the compu- 

 tations use alternate sets of equations at alternate time steps At as 

 shown below. First u and n are calculated implicitly and v explic- 

 itly at time t + At/2, then v and n implicitly and u explicitly at 

 time t + At, then u and n implicitly and v explicitly at time 

 t + 3At/2, etc. Calculating u at point (j + 1/2, k) , r\ at (j ,k) , 

 and v at (j ,k + 1/2), as defined in Figure 10, the calculations are, 

 at times t + At/2, t + 3At/2, t + 5At/2, . . . 



u ; = u 4Atf^ ---g(^ (108) 



n+l/2 n 2 a v n 2 As & \dx) , , n ^ 



x 'n+l/2 



Via ■ \ - Tit'lk"* + ft* u W/ 2 + |y u* + ^ V U c 109 ) 



'n+l/2 - \ - 2 az T c u n+l/2 



1 a* c 1 At / 8n\ , 11ro 



t At f u . ,„ - «- — g { —-) (HO) 



2 As 6 \9y/ 



55 



