7. The Numerical Scheme 



Frequently, it is of interest to know how much of the setup can be 

 attributed to the onshore effects and how much can be attributed to the 

 alongshore effects. To determine these two separate and distinct effects 

 in the computational scheme. Equation (11) can be written in the equiva- 

 lent forms as follows: 



X k W cos 6 .^_. 



8x " gD ^^^^ 



9x gD ^ -" 



where the total setup along the x-axis is simply 





9S 



as 



8S 



X 



+ y 



yx 



8x 



3x 



Equations (12) , (13) , and (14) can be solved by computing in incre- 

 ments of time and space provided that the increments taken are not too 

 large. Furthermore, the functional relationships as given by the above 

 equations are taken as continuous over the entire interval . In the 

 niomerical analog of the equations. Ax and At are taken as nonuniform 

 spacing and time steps. This will allow taking coarse spacing, Ax, where 

 the seabed is relatively flat, and fine spacing where the bed slope changes 

 more rapidly near shore. Nonuniform time steps. At, allow more frequent 

 storm-surge computations during the period when the maximum rise in water 

 level is anticipated which is sometimes useful in developing the storm 

 surge hydrograph. The discrete position, x, along the traverse line and 

 the time level, t, are defined as: 



(15) 



jc = X - y (Ax) . 



1=1 



t = t + y (At) 



o >, ^ -^n 

 1 + 1 



where x^ is the distance from the shoreline to the most seaward position 

 prescribed on the traverse line. The summation of Ax for all i's up to 

 and including IM, the maximum value of i at shore, is simply x ; thus at 

 the shore x = 0. Actually, according to the coordinate system taken, x 

 would be negative seaward of the shoreline; however, taking x positive 

 does not effect the computation scheme, and furthermore such a choice is 

 consistent with the computational schemes presently used. Normally, t^ 

 will be taken as zero and the time level is simply the summation of At 

 for all n's up to, and including, some specified value of w = tIM. 



16 



