Low order approximation of the spherical nonreflecting boundary kernel for the wave equation

We find low order approximations to the spherical nonreflecting boundary kernel for the wave equation in three dimensions. First we express the Laplace transform of the kernel as a rational function by solving for the zeros of a modified Bessel function. Then we formulate a linear time-invariant dynamical system whose transfer function is this rational function. Finally we use the Balanced Truncation method to generate low order approximations. We compare our approach with a direct L² minimization approach where a rational approximation is expressed as the ratio of two polynomials.