The Acoustic Wave Propagation Equation: Discontinuous Galerkin Time Domain Solution Approach

Abstract

This paper discusses a finite element method, discontinuous Galerkintime domain approach that solves the 2-D acoustic wave equation incylindrical coordinates. The method is based on discretization of thewave field into a grid ofrandθwhereris the distance from the centreof the domain andθis the radial angle. The Galerkin formulation isused to approximate the solution of the acoustic wave equation for therandθderivatives. The boundary conditions applied at the boundariesof the numerical grid are the free surface boundary condition atr= 1and the absorbing boundary condition applied at the edges of the grid atr= 2. The solution is based on considering wave motion in the directionnormal to the boundary, which in this case is the radial direction overradial angleθ∈[0o,30o]. The exact solution is described in terms ofBessel function of the first kind, which forms the basis of the boundaryconditions for the values of pressure and eventually sufficient accuracyof the numerical solution. The algorithm generated in Matlab is testedagainst the known analytical solution, which demonstrates that, pres-sure of the wave increases as the radius increases within the same radialangle. The domain was discretized using linear triangular elements. Themain advantage of this method is the ability to accurately represent the wave propagation in the free surface boundary with absorbing bound-ary condition at the edges of the grid, hence the method can handlewave propagation on the surface of a cylindrical domain. The resultingnumerical algorithm enables the evaluation of the effects of cavities onseismograms recorded in boreholes or in cylindrical shaped tunnels.