Kirchhoff–Helmholtz representation

With the normal directed out of an obstacle $D$ into the fluid, a field solving $(\Delta +k^2)\psi =-Q$ outside $D$ has the representationHere $G_k$ is the outgoing Green function for the three-dimensional Helmholtz equation and the observation point lies outside the obstacle. The formula follows from Green second identity; the normal to the exterior fluid on its inner boundary is the negative of the stated obstacle normal. The surface term represents the scattered field.

For a scattered field satisfying ${\psi }_s=e^{ikr}{f}_{\infty }(\hat{\mathbf{r}})/r+O(r^{-2})$, the Kirchhoff–Helmholtz representation givesThe sign assumes the normal points out of the obstacle.