An Analytic Method for Computing the Infinite Sums Occurring in the Geoelastic Disk Load Problem

Abstract

AbstractThe elastic displacements caused by a point load imposed on the surface of a layered, spherical, and self‐gravitating Earth can be expressed in terms of infinite series of Legendre polynomials or their derivatives, multiplied by constants, called Love numbers, that depend on the summation index or degree. Truncating these infinite series causes oscillatory errors in the computed displacement field, particularly in the near field of the load, but Farrell (1972, https://doi.org/10.1029/RG010i003p00761) circumvented this problem using a Kummer transformation that relied on an asymptotic approximation for the Love number spectrum and three useful identities for series of Legendre polynomials. Unfortunately, nobody has determined a similar work‐around for the disk load problem since in this case the infinite expansions consist of products of Legendre polynomials or their derivatives, and no similarly useful identities have been identified to date. We present an alternative means to the same general goal by replacing the infinite series with closed‐form expressions involving elliptic integrals.