Wave propagation in the vicinity of a cylindrical object

Wave propagation in the vicinity of cylindrical objects is a problem which appears in many branches of physics and engineering. In particular, in the area of Geophysics there are many problems which include the presence of a cylindrical borehole. Unfortunatly, closed form solutions for such problems exist only for the most simple configurations. Therefore, presence of material variability as in the earth, one needs to resort to numerical solution techniques. However, use of methods such as finite differences, finite elements or pseudospectral methods in cartessian geometry will usually not be successful due to the small size of the borehole (on the order of 20 cm) compared to the distance which the waves propagate (often on the order of few kilometers). In addition the representation of a cylindrical object on a rectangular mesh causes spurrious diffractions. In this work we present a spectral technique for solving the two dimensional acoustic and elastic wave propagation problems in the presence of cylindrical objects. The method is based on spatial discretization of the solution in a cylindrical (r, θ) coordinate system where r is the distance from the center of the numerical mesh and θ is the radial angle. The solution is approximated by a Chebychev expansion in the r direction which results in non-uniform grid spacing (Figure 1). The fine grid spacing in the center of the mesh has the advantage of allowing many grid points to cover a very small physical space in that region. The Chebychev expansion also allows to treat the absorbing boundary conditions at the edges of the grid. For the azimuthal angle coordinate, we use the periodic Fourier expansion.