We numerically calculate cross sections for gravitational waves axially incident on a Kerr black hole. The resulting detailed cross sections display a wealth of structure which can be linked conceptually to phenomena familiar from other scattering problems. The limiting cross sections in chapter 6 assist analysis by allowing truncation of numerical calculations where the limiting results become applicable and by pointing to parameter ranges of particular interest.

We examine the angular and radial equations in detail, finding in each case a form of the solution which allows efficient numerical integration. We consider two methods of solution for the angular equation; a perturbation calculation for insight and its continuation for the actual numerical work. For the radial equation, considered in the remainder of the chapter, we use a slow stable solution for small values of r and apply a JWKB approximation to integrate rapidly to large r.

Angular equation

We first consider the perturbation of the angular equation about aω = 0, where as before, a is the specific angular momentum and ω is the scattered wave frequency.

One reason for considering the perturbation calculation is that it lays the foundation for the actual method used. In particular we follow a technique of Press & Teukolsky (1973) to expand the spin-weighted spheroidal harmonics conveniently in the spin weighted spherical harmonics in both cases. The perturbation calculation only obtains spheroidal values to first or second order in aω. The continuation method utilizes the same expansion of the spheroidal functions, but finds expressions for their derivatives with respect to aω. The numerical solution then integrates these derivatives.