Brennen and Whitney 



where t is the time interval. Acceleration values, Z|| , needed in 

 the f^e surface condition (Section 4C) are approximated by 

 (Zf*'^ - Zf"'^)/T (error order T^Zfttt). Thus the main part of the 

 solution involves finding P| ' knowing Z , Z^ and their previous 



values . 



The first time step (fronn p = to p = 1) requires a little 

 special attention. Clearly Z (a,b) is chosen to fit the required 

 initial conditions. But further information is required on a free 

 surface which will enable the accelerations in that condition to be 

 found (see Section 4C) . 



B. Spatial Solution 



A nnethod of the present type is restricted to a finite body of 

 fluid, S. However, S, could be part of a larger or infinite mass of 

 fluid if an "outer" approximate solution of sufficient accuracy was 

 available to provide the necessary matching boundary conditions at 

 the interface. The region, S, need not be fixed Ln time. It would 

 Indeed be desirable, for example, to "follow" a bore. 



In a great number of cases of widely different physical ge- 

 ometry including all the examples of Section 6, it is convenient to 

 choose S to be rectangular in the (a,b) plane. This rectangle 

 (ABCD, Fig. 1) is then divided into a set of elem^ental rectangles. 

 The motion of each of these cells of fluid is to be followed by deter- 

 mining the Z values at all the nodes. 



Fig. 1. The Rectangular Lagranglan Space, S, Showing the 

 Numbering Conventions Used 



122 



