Numerical investigations into two-dimensional hyperbolic and parabolic equations have garnered considerable interest across various domains. In this work, the Galerkin method is applied to construct the numerical solution of the two-dimensional telegraph equation with variable coefficients. The related equation, extensively used in engineering, was derived in 1876 by Heaviside, who addressed the charging and discharging of a finite length of cable under the effects of induction. This study demonstrates that the Galerkin method is a highly effective and powerful tool for numerically solving such problems. Several examples of numerical experiments are provided.