Three-dimensional Green’s function for wave propagation in a linearly inhomogeneous medium—the exact analytic solution

A new exact solution of the Helmholtz equation is obtained in closed form for the case of a point source in a layered medium with refractive index variation of the form n = (1 + Az)^xn. The solution is obtained by starting with a onefold integral representation for the solution that was derived by Holford [J. Acoust. Soc. Am. 70, 1427–1436 (1981)] and Jeng and Liu [J. Acoust. Soc. Am. 81, 1732–1740 (1987)]. The solution is particularly interesting because the classical ray picture for the medium exhibits a caustic and therefore a shadow zone is formed in the unbounded medium. The power flux due to a point source in such a medium is computed and the flow of energy into the region beyond the caustic surface is illustrated. The effects of scattering by turbulence are studied and it is shown that scattering can drastically affect the field levels in the shadow zone. Also, the field patterns of a simple array are studied in such a medium and are compared to patterns in a homogeneous medium. PACS numbers: 43.20.Bi, 43.20.Fn.