A Crank-Nicolson finite difference method for incompressible Navier-Stokes equations is developed and is applied to find the numerical solution of wave elevation on the free surface in a two dimensional tank. The numerical simulation of wave elevation on the free surface is studied with different excited accelerations and Reynolds number. From the benchmarks, the proposed finite difference method agrees well with the previous published works. As shown in the numerical results, we draw a conclusion that the wave elevations decay gradually with time. The number of the beating periods diminish gradually when the Reynolds number decreases. Finally, we find that the wave elevations on the free surface keep a related stable height, when the Reynolds number decreases.