WKB solution of the wave equation for a plane angular sector

Abstract

The two angular Lamedifferential equations that satisfy boundary conditions on a plane angular sector ~PAS! are solved by the Wentzel-Kramers-Brillouin ~WKB! method. The WKB phase constants are derived by matching the WKB solution with the asymptotic solution of the Weber equation. The WKB eigenvalues and eigenfunctions show excellent agreement with the exact eigenvalues and eigenfunctions. It is shown that those WKB eigenvalues and eigenfunctions that contribute substantially to the scattering amplitude from a PAS can be computed in a rather simple way. An approximate formula for the WKB normalization constant, which is consistent with the WKB assumptions, is derived and compared with the exact normalization constant. @S1063-651X~98!00507-8# PACS number~s!: 42.25.2p I. INTRODUCTION In a previous paper @1#, formulas for the Wentzel- Kramers-Brillouin ~WKB! eigenvalues satisfying Dirichlet or Neumann boundary condition on a plane angular sector ~PAS! were reported and the WKB eigenvalues, $n,m%, were compared with the exact eigenvalues for PAS's of different corner angles ~60°, 90°, and 120° !. A historical review of the solution of the wave equation for a PAS was also given in the above paper. It suffices to say that, to our knowledge, no approximate solution of this problem has been reported in the literature. In this paper, the two coupled Lameequations are solved by the WKB method. The WKB analysis in this paper is valid for large values of n. Depending on the sign of m, one of the two Lameequations has turning points. When umu is small the turning points occur where the angles are small. In this region it proves more accurate to obtain the WKB phase constants by matching it with the asymptotic solution of the Weber equation. For large values of umu, it is shown that this phase constant reduces to the phase constant obtained when the solution is matched with the asymptotic solution of the Airy's equation as is commonly done in quantum mechanics. This paper is organized in the following way: In the sec- ond section the WKB solution is formulated and the WKB phase constants are derived. In the third section formulas for the WKB eigenvalues for Dirichlet and Neumann boundary conditions are derived and a comparison between the WKB and exact eigenfunctions is presented. The fourth section contains a derivation of approximate WKB solutions which are valid for small values of m/n. Finally, an approximate formula for the WKB normalization constant consistent with WKB assumptions is derived in section five.