A Second-Order JWKB Approximation with One Turning Point and Two Singular Points: Stability of an Accelerating Liquid Sphere
- 1 September 1971
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 12 (9) , 1955-1960
- https://doi.org/10.1063/1.1665829
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.Keywords
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