  We propose a new method for investigating the global properties of the
retarded Green's function $G_R(x',x)$ for fields propagating on an arbitrary
globally hyperbolic spacetime. Our method combines the Hadamard form for $G_R$
(this form is only valid within a normal neighbourhood of $x$) together with
Kirchhoff's integral representation for the field in order to calculate $G_R$
outside the maximal normal neighbourhood of $x$. As an example, we apply this
method to the case of a scalar field on a black hole toy-model spacetime, the
Pleba{\'n}ski-Hacyan spacetime, $\mathbb{M}_2\times\mathbb{S}^2$. The method
allows us to determine in an exact manner that the singularity structure of the
`direct' term in the Hadamard form for $G_R(x',x)$ changes from a form
$\delta(\sigma)$ to `$1/\pi\sigma$' after the null geodesic joining $x$ and
$x'$ has crossed a caustic point, where $\sigma$ is the world function.
Furthermore, there is a change of form from a $\theta(-\sigma)$ to a
`$-\ln|\sigma|/\pi$' in the `tail' term, which has not been explicitly noted
before in the literature. We complement the results from the Kirchhoff integral
method with an analysis for large-$\ell$ of the Green function modes. This
analysis allows us to determine the singularity structure after null geodesics
have crossed an arbitrary number of caustics, although it raises a causality
issue which the Kirchhoff integral method resolves. Because of the similarity
in the caustic structure of the spacetimes, we expect our main results for wave
propagation to also be valid on Schwarzschild spacetime.
