In this paper, an investigation of the behavior of a finite difference scheme for solving initial value problems for the wave equation is reported. The scheme is a Lax-Wendroff method, which is second-order accurate. The objective of this work is to assess its accuracy in comparison to the well-known leapfrog method. To this end, spectral analysis is performed, which has a natural interpretation in terms of numerical dispersion relation. It is found that the scheme is highly dispersive, dissipative, and anisotropic. Numerical experiments were carried out to compare the performance of this scheme with the leapfrog scheme. It is apparent from our numerical tests and analysis that the standard leapfrog scheme outperforms a Lax-Wendroff scheme in solving linear wave equations.