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

Abstract

This paper discusses a finite element method, discontinuous Galerkin time domain approach that solves the 2-D acoustic wave equation in cylindrical coordinates. The method is based on discretization of the wave field into a grid of r and θ where r is the distance from the centre of the domain and θ is the radial angle. The Galerkin formulation is used to approximate the solution of the acoustic wave equation for the r and θ derivatives. The boundary conditions applied at the boundaries of the numerical grid are the free surface boundary condition at r = 1 and the absorbing boundary condition applied at the edges of the grid at r = 2. The solution is based on considering wave motion in the direction normal to the boundary, which in this case is the radial direction over radial angle θ ∈ [0o , 30o ]. The exact solution is described in terms of Bessel function of the first kind, which forms the basis of the boundary conditions for the values of pressure and eventually sufficient accuracy of the numerical solution. The algorithm generated in Matlab is tested against the known analytical solution, which demonstrates that, pressure of the wave increases as the radius increases within the same radial angle. The domain was discretized using linear triangular elements. The main advantage of this method is the ability to accurately represent the wave propagation in the free surface boundary with absorbing boundary condition at the edges of the grid, hence the method can handle wave propagation on the surface of a cylindrical domain. The resulting numerical algorithm enables the evaluation of the effects of cavities on seismograms recorded in boreholes or in cylindrical shaped tunnels.