62
It is important that the boundary conditions 4.46 and 4.47 be used as prescribed
above. The parabolic equation gives N equations containing the N+2 unknowns A(I+1,J)
for J=0,N+1. The two boundary conditions are thus used to reduce the number of un
knowns to a number equal to the number of equations. If equation 4.47 is written in terms
of A(I+1,1) and A(I+1,2) at the lower boundary and in terms of A(I+1,N-1) and A(I+1,N)
at the upper boundary and the governing wave equation is written for J=2,N-1, then errors
may be introduced at the boundaries. This is especially true for the case of a reflective
boundary. Figure 4.3 shows waves freely passing through the lateral boundaries without
any noticeable reflection for the case of a plane wave with a period of 10 seconds propagating
over constant depth at an angle of 10 degrees to the x axis.