### 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.