The bottom side boundary equation is

The finite
difference equations shown as Fortran code are
Code Fragments
I, II, III, and IV,
respectively.
These difference equations are all second order in space and second
order in time.
The difference approximations for the boundary conditions must
match or nearly match those of the wave equation [17].
If the degree of approximation for the wave equation is **k**, the boundary
condition approximation must be **k** or **k-1**.