Figure 15. Computation of pressure near 

 the free surface. 



where the j 3 k subscripts refer to positions in the plane of the still- 

 water surface as shown in Figure 10, and u and v the velocities 

 satisfying the linear long-wave equations. Where the initial velocity 

 field is known, u and v can be computed at time t Q + At using values 

 of u and v at time t and amplitudes at time t + At/2. This gives 

 the equations 



- - At , . 



3 ,k 3,k 2Ax 3 +1 >k 3~ l ,k J 



At 



Cn. 



- n. 



3 ,k 3 ,k 2Ay 3,k+l 'j\fc 



• 1> 



(141) 

 (142) 



At time t Q + At, the velocities u and v are given by 



At 



u . , = u . , - u . , (u . , , - u . , , ) 



3,k 3,k 2Ax 3 ,k 3+1, k 3-l,k J 



_ At_ 

 2 Ay 



" V 3,k (U J,^1 " U 3,k-J ~ ^t Cn J*l.* " W (143) 



V 3,k- V 3,k ~lt U 3,k Cv j + l,fe" V j-l,P 



" ^3,k ^3 Ml ~ \f.*-l } " ^y" %\k+l ~ "i.k-J (144) 



where the values on the right side of the equations are at previous 

 time steps as indicated. 



64 



