streets Chan and Fromm 



APPSIM2 is based on Peregrine's equations and is described in detail 

 in Section III below. 



Heitner [ 1969] presented a nonlinear method based on 

 Lagrangian coordinates and a finite element representation of the 

 fluids. His theory retains terms representing the kinetic energy of 

 the vertical motion; thus, like the methods described just above, 

 Heitner's approximate method permits permanent waves to propa- 

 gate. Unlike those methods, Heitner's formulation gives a repre- 

 sentation of wave breaking inception, bore formation and run-up 

 on a linear beach for plane flows. 



The exact, plane-flow models are represented by the work of 

 Brennen [ 1970] , Hirt, et al . [ 1970] , and Chan and Street [ 1970] . 

 All are based on the exact equations of motion; however, Chan and 

 Street [ 1970] work in Eulerian coordinates , Brennen [ 1970] uses 

 Lagrangian coordinates and equations, and Hirt, £t al. [ 1970] em- 

 ploy Lagrangian coordinates but retain the Eulerian form of the 

 governing equations. Based on the Marker-and-Cell (MAC) method 

 [Welch, et al. , 1966] , Chan and Street's model for water waves is 

 called SUMMAC and is discussed in Section IV of this paper, Brennen 

 has applied his technique to tsunami generation by ocean floor move- 

 ments and to run-up on a beach. Hirt, _et al. , made applications to 

 wave sloshing in a tank by way of verification of their method, called 

 LINC, which has wide application in ocean wave analyses as well. 

 All the exact, plane-flow methods mentioned employ some form of 

 finite -difference or cell representation of a discrete grid of points. 



Finally, the quasi-three-dimensional methods are represented 

 by the work of Pritchett [ 1970] and Leendertse [ 1967] and by the 

 extension to two horizontal dimensions of the APPSIM program of 

 Street, et al. [ 1969] discussed above. Pritchett presents a code 

 for solving incompressible, two-dimensional, axisymmetric , time- 

 dependent, viscous fluid flow problems involving up to two free 

 surfaces. The basic equations are exact; heuristic models for tur- 

 bulence simulation are used. Scalar quantities such as heat and 

 solute concentration can be traced, and the fluid may be slightly non- 

 homogeneous. Most of the variables, their placement in the compu- 

 tational mesh, and the free surface treatment are those used in MAC 

 [ Welch, et al. , 1966] . 



Leendertse [ 1967] developed a computational model for the 

 Ccdculation of long-period water waves in which the effects of bottom 

 topography, bottom roughness and the earth's rotation were included. 

 The equations of motion are vertically integrated so only the two 

 horizontal space-dimensions remain (much in the manner of Peregrine, 

 [1967] ), but while Peregrine [ 1967] , Madsen and Mei [ 1969a, 1969b] 

 and Street et al. [ 1969] retain terms to account for the vertical 

 ac c el e rat ions "of the fluid, Leendertse [1967] does not. His equations 

 become the usual nonlinear shallow water wave equations with added 

 terms to account for the bottom roughness and earth's rotation. He 



150 



