We present a study of elastic wave propagation in isotropic media. The Discontinuous Galerkin Method is applied to solve the elastodynamic equations which represent elastic wave propagation . The elastodynamic equations are transformed into a stress-velocity formulation. The Discontinuous Galerkin Method is a finite element that allows a discontinuity of the numerical solution at element interface. Through a proper choice of the flux computation points, the method only requires communication between elements that have common faces. The utilization of high-order Legendre polynomials as basisfun ctions has been shown to be more efficient in reducing the numerical dispersion and numerical dissipation. Discontinuous Galerkin Method is a compact method, high-order basis functions can be used easily without any essentially difficulty and even spectral accuracy becomes obtainable. Temporal discretization utilized explicit staggered leapfrog method. We compare the numerical results to the exact solutions and the comparison shows a good agreement.