street^ Chan and Fromm 



The successful application of APPSIM to examples of plane 

 motion of long waves indicated that the method could be applied to a 

 quasi-three dimensional simulation (two horizontal space dimensions 

 with vertical effects represented in once integrated equations) of the 

 motion of waves over arbitrary bottom topography. We have ex- 

 tended APPSIM to handle general bottom topography and both solitary 

 and oscillatory wave inputs. The new method is called APPSIM2 and 

 presented in Section III below. 



An objective of our exact simulation was to provide detailed 

 information about wave processes near the shore and at the ocean- 

 structures interface. The Stanford- University- Modified Marker- and- 

 Cell (SUMMAC) method computes time -dependent, inviscid, incom- 

 pressible fluid flows with a free surface; the method is suitable for 

 analyzing two-dimensional flows. Initially, we simulated the propa- 

 gation of solitary waves in a horizontal channel filled with fluid to 

 unit depth and with vertical end walls. The solitary wave propagation 

 problem possessed several key features: 



a. The theories for the wave motion against the wall were 

 not in agreement with experiments. 



b. The solitary wave should propagate stably (without change 

 of form) in zones not near the channel walls. 



c. Perfect reflection from the walls should occur. 



We undertook a significant modification of the MAC method to create 

 a numerical scheme suitable for water wave simulation. As reported 

 in 1969, the resulting SUMMAC simulation met the criteria of stable 

 propagation and perfect reflection of solitary waves and resolved the 

 disagreement between theory and experiment for motion against a 

 vertical wall. 



The successful application of SUMlvLAC to the initial example 

 indicated the possibility of employing a modification of the same 

 technique to attack a variety of other problems. We have subsequently 

 studied the generation of water waves by a periodic pressure pulse 

 and the shoaling and run-up of solitary waves on a stepped slope and 

 on plane beaches. A summary of the presently implennented SUMMAC 

 method and results for the periodic pressure pulse problem are pre- 

 sented in Section IV of this paper. -An evaluation of the numerical 

 qualities of SUMMAC and a report of the shoaling and run-up studies 

 are given in Chan, et al. [ 1970] and Chan and Street [ i970b] . 



152 



