PROBLEMS WITH POLAR COORDINATES

The major disadvantage of using a polar coordinate frame is that it is only suitable for calculating the Green's function for a single point. When Green's functions are to be combined, they must be interpolated to a common coordinate frame. In contrast, data from rectangular coordinate frames centered at different points can be combined by a simple shift along the coordinate axes.

All of the polar coordinate figures in this paper have been interpolated onto the same mesh as the rectangular coordinate frame data using a bilinear interpolator. In some figures, the artifacts produced by the interpolation step are clearly visible. The polar data has a ``rough'' quality that the rectangular coordinate data does not. The problem is most severe when the radius is large. In these cases the spacing between the different radii is much larger than the largest spacing on the rectangular grid. In these regions aliasing may become a problem if the waves propagate at an high angle to the polar grid.

Another problem is that the trial solution used in equation (2) is only an approximate solution. It is only valid in the far field and breaks down at low frequency and small radius. This can be seen from the formula for the radial wavenumber,

$$k_r = \frac{i}{2r} \pm \sqrt{\frac{\omega^2}{v^2}-\frac{k_\theta^2}{r^2}-\frac{1}{4r^2}}$$

The expression inside the square-root becomes negative for all values of $k_\theta$ when $\frac{1}{4r^2} \gt \frac{\omega^2}{v^2}$. This provides a lower limit to the radius at which to specify the initial condition $r \gt \lambda/2$. In practice it is advisable to make the initial radius a least two wavelengths of the lowest frequency.