We commence with explicit time marching schemes
because they do not involve matrix inversion or iterative solvers.
Before we write down the complete finite difference analog of
equations
(2)--(4),
we must give finite difference expressions to
the first and second spatial derivatives in the x- and
y-directions. We will use only centered differences unless otherwise
stated. Denoting 
, we have
Though we avoid having to use a matrix inverter to
solve for the values at 
, we pay the price by being restricted
in the time step we can take. The size of a time step one can take
in solving time-marching differential equations by explicit
methods is governed by the famous Courant--Friedrichs--Levy (CFL)
stability condition. For wave equations the time step is limited
by the wave speed, in this case the speed of the surface gravity
waves 
, and is given by
