An asymptotic fundamental solution of the reduced wave equation on a surface

The physical problem of steady-state heat conduction in a thin shell is described by the “reduced wave equation” in which the differential operator is the (generally noneuclidean) Laplacian for the surface. A similar equation gives the approximation for steady-state waves in a prestressed curved membrane. A modification of the “geometric optics” asymptotic expansion, involving a Bessel function, is given for the fundamental point source solution. This is proven to be uniformly valid in the large, until a “caustic” is reached. Various features of the solution for a surface, which do not occur for the plane, are discussed.