A Second-Order JWKB Approximation with One Turning Point and Two Singular Points: Stability of an Accelerating Liquid Sphere

Abstract

A formal higher-order matching procedure is employed to obtain a second-order asymptotic solution, of the JWKB type, to a Legendre-like differential equation with a large parameter. The equation has two second-order poles and a first-order turning point. In addition to the usual nonuniformity, the second-order JWKB approximation exhibits a divergent integral at these points. Eigenfunctions and eigenvalues, valid to the second order of approximation, are found by simultaneously matching the latter approximation to a turning-point expansion and two boundary-layer expansions. The solutions, which heretofore have not been described, are appropriate to the neutral stable surface waves manifest by an accelerating liquid sphere.