The integral relations equivalent to Helmholtz’s equation with closed‐boundary conditions possess nonunique exterior solutions at eigenwavenumbers of the corresponding interior problems. If the Green’s function for Helmholtz’s equation can be expanded into a series of orthogonal functions, then an infinite set of integral equations at the boundary results. These ‘‘null‐field’’ equations are shown to possess unique solutions for all wavenumbers. Furthermore, these solutions can be used to represent the solutions in the exterior domain as algebraic series.