Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A boundary layer approach to the whispering gallery phenomenon

Author: B. J. Matkowsky
Journal: Quart. Appl. Math.
MSC (2010): Primary 35B40, 35J05; Secondary 35C20
DOI: https://doi.org/10.1090/qam/1513
Published electronically: August 2, 2018
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Abstract: We consider the eigenvalue problem for the Laplace operator in a two dimensional domain, bounded by a smooth, closed convex curve $ C$, on which the eigenfunctions vanish. Waves whose wavelengths $ \lambda $ are very small may be viewed as particles traveling along specific paths termed rays, along which the waves propagate. Small wavelengths $ \lambda $ correspond to large wave numbers $ k=\frac {2 \pi }{\lambda }$, which are the eigenvalues. Therefore, we restrict attention to the consideration of large eigenvalues. Keller and Rubinow considered the problem from the point of view of geometrical optics. They constructed eigenvalues and eigenfunctions by means of rays for special geometries, and qualitatively discussed general geometries. They noted the existence of caustics, i.e., envelopes of rays in the domain, with the property that the amplitude of the eigenfunctions is appreciable only in the region between the boundary and the caustic. Inside the caustic the solution is negligibly small. The caustic is the locus of the turning points on the rays, at which the behavior changes from oscillatory (wavelike) to exponentially decaying. We are interested in those eigenfunctions for which the caustic is ``near'' the boundary. They correspond, in acoustics, to the whispering gallery modes, by means of which a person speaking near the wall of a convex room can be heard across the room, but not in the interior of the room. We consider domains bounded by a general smooth convex curve $ C$, and employ a boundary layer approach to the problem, to determine these eigenvalues and eigenfunctions. Our results provide improvements to those of Keller and Rubinow.

References [Enhancements On Off] (What's this?)

  • [1] A. Sommerfeld and I. Runge, Anwendung der Vektorrechnung auf die Grundlagen der geometrischen Optik, Ann. Phys. 35, 277-298 (1911).
  • [2] Lord Rayleigh, The Problem of the Whispering Gallery, Philosophical Magazine 20, 1001-1004 (1910).
  • [3] Lord Rayleigh, Further Application of Bessel's Function of High Order to the Whispering Gallery and Allied Problems, Philosophical Magazine 27, 100-109 (1914).
  • [4] L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Riebung, Verhandlungen der 3rd International Mathematiker Kongress in Heidelberg, 484-491 (1905).
  • [5] Harold Jeffreys, On Certain Approximate Solutions of Lineae Differential Equations of the Second Order, Proc. London Math. Soc. (2) 23 (1924), no. 6, 428–436. MR 1575202, https://doi.org/10.1112/plms/s2-23.1.428
  • [6] G. Wentzel, Eine Verallgemeinerung der für Quantenbedingungen fur die Zwecke der Wellenmechanik, Zeit. für Pysik 38, 518-529 (1926).
  • [7] H. Kramers, Wellenmechanik und halbzahlige Quantisierung, Zeit. fur Phys. 39, 818-840 (1926).
  • [8] L. Brillouin, La mécanique ondulatoire de Schrödinger; une méthode générale de resolution par approximations successives, Compt. Rend. Acad. Sci. 183, 24-26 (1926).
  • [9] Joseph B. Keller, A geometrical theory of diffraction, Calculus of variations and its applications. Proceedings of Symposia in Applied Mathematics, Vol. 8, For the American Mathematical Society: McGraw-Hill Book Co., Inc., New York-Toronto-London, 1958, pp. 27–52. MR 0094120
  • [10] J. B. Keller and S. Rubinow, Asymptotic solution of eigenvalue problems, Annals of Physics 9, 24-75 (1960).
  • [11] Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059774
  • [12] F. W. J. Olver, The asymptotic expansion of Bessel functions of large order, Philos. Trans. Roy. Soc. London. Ser. A. 247 (1954), 328–368. MR 0067250, https://doi.org/10.1098/rsta.1954.0021
  • [13] Bernard J. Matkowsky, ASYMPTOTIC SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS IN THIN DOMAINS, ProQuest LLC, Ann Arbor, MI, 1966. Thesis (Ph.D.)–New York University. MR 2616044

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Additional Information

B. J. Matkowsky
Affiliation: Department of Engineering Science and Applied Mathematics, Northwestern University, Evanston, Illinois 60208
Email: b-matkowsky@northwestern.edu

DOI: https://doi.org/10.1090/qam/1513
Received by editor(s): June 25, 2018
Received by editor(s) in revised form: July 3, 2018
Published electronically: August 2, 2018
Dedicated: This paper is based on a section of the author’s dissertation at the Courant Institute of Mathematical Sciences in 1966, with Joe Keller, teacher, colleague, and friend. The paper is dedicated to his memory.
Article copyright: © Copyright 2018 Brown University

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