Computation of the Motion of Long Water Waves 



focuses his attention on the modelling of long waves such as tsunamis 

 in areas with irregular bottom topography and complicated ocean 

 boundaries. His computation uses a space-staggered scheme (where 

 velocities , water levels , and depth are described at different grid 

 points) and a double time step operation in which the time integral 

 is considered over two successive operations in a manner designed 

 to make effective use of the space-staggered scheme. Among the 

 papers mentioned in this Introduction, only Leendertse [ 1967] , 

 Welch, et al. [ 1966] and Chan and Street used computer graphic 

 display for output of results. The value of graphic display is illus- 

 trated in the results presented in the remainder of this work. 



II. THE PRESENT WORK 



Street, et al. [ 1969] gave a progress report on the develop- 

 ment of computer programs for two numerical, finite -difference 

 models for the study of long water waves. These models and their 

 accompanying programs were, as noted above, given the acronyms 

 APPSIM and SUMMAC. Both were based on representation of the 

 motion of inviscid, incompressible fluids in terms of the Euler 

 equations of motion in Eulerian coordinates. Flow boundary con- 

 ditions were derived fronn physiccd requirements and the governing 

 equations at the boundaries. The mathematical models thus obtained 

 were then transformed to numerical, finite difference models for the 

 purposes of computation. In 1969 the study had been confined to 

 plane flows, but the numerical results had been verified by compari- 

 son with experiments and the work of others. The models were to 

 provide detailed flow field data in the portion of the wave shoaling 

 process where nonlinear effects are significant, but breaking has not 

 occurred. 



Our approximate simulation (APPSIM) is based on the method 

 of Peregrine [ 1967] and supplemented by the work of Madsen and Mei 

 [1969a, i969b] . APPSIM was implemented for quasi-two-dimensional, 

 plane flows (vertical motion integrated out). For the purpose of 

 implementing, testing, and verifying the program and method, we 

 simulated the propagation of solitary waves on a stepped slope which 

 represents the configuration of the continental slope and shelf, i.e. , 

 we examined long waves in moderately shallow water. The key 

 criteria to be satisfied were 



a. Solitary waves propagate stably on a horizontal bottom. 



b. Solitary waves decompose into undular bores when the 

 waves propagate onto a stepped slope [Street, et al» , 

 1968] . 



c. Wave heights must be in good quantitative agreement with 

 available experimental data. 



As reported by Street, et al. [ 1969] APPSIM met these criteria. 



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