Convergent low-frequency expansions for penetrable scatterers

We develop convergent series solutions in powers of the wavenumber (k=2π/λ) for the field (ψ) and the normalized (dimensionless) scattering amplitude (g) for scattering by lossless penetrable obstacles whose physical properties are specified by two real parameters. The first two terms of ψ are solutions of Laplace’s equation and the term of order kn, n⩾2, satisfies a two-parameter Poisson equation whose inhomogeneous term is proportional to the kn−2 term. The leading term of g is of order k3 (as obtained originally, by Rayleigh); the k4 term is zero for shapes that have inversion symmetry, and vanishes in the forward direction for all shapes; the kn terms, n⩾3, are expressed as volume integrals of functions involving the terms of ψ up to order kn−2. Equivalent expressions in terms of surface integrals are included. For a plane wave of arbitrary direction of incidence and a triaxial ellipsoid, we obtain explicitly the first four nonvanishing terms of ψ (to order k3) and the first two nonvanishing terms of Img (to order k5) and Reg (to order k8). Corresponding results for spheroids, needle, disc, sphere, and for the one-parameter problems are obtained as special cases. The necessary transformation of the ellipsoidal harmonics are also provided.