Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 335 / 1 / i / Solution

Use the outgoing Green function for the three-dimensional Helmholtz equationLet ${\mathbf{n}}^{\prime }$ point out of the obstacle $D$ into the exterior. Apply Green second identity to the field and $G_k$ in a large exterior region. Their Sommerfeld radiation conditions make the outer-sphere contribution vanish. The normal on the exterior region's inner boundary is $-{\mathbf{n}}^{\prime }$, so the Kirchhoff–Helmholtz representation isThe first integral is the incident source field ${\psi }_i$. Therefore the scattered field is the surface term. Since ${\psi }_i$ solves the homogeneous Helmholtz equation within the obstacle, its surface term is zero at exterior observation points; the same representation can be written using only scattered-field traces:The unknown boundary condition determines these traces, but the representation itself does not require choosing that condition.