Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Potential and Rayleigh-scattering theory for a spherical cap

Author: John W. Miles
Journal: Quart. Appl. Math. 29 (1971), 109-123
MSC: Primary 78.31
DOI: https://doi.org/10.1090/qam/284080
MathSciNet review: 284080
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Harmonic functions are constructed for spherical-harmonic prescriptions of either a potential or its normal derivative on a spherical cap. The dipole-moment tensor and the Rayleigh-scattering properties of a spherical bowl, including the limiting case of a Helmholtz resonator, are determined. The results are uniformly valid with respect to the polar angle of the cap and resolve certain discrepancies in the existing literature.

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

  • [1] W. Thomson, Extraits de deux lettres adressées à M. Liouville, J. Math. Pures Appl. 12, 256-264 (1847)
  • [2] N. M. Ferrers, On the distribution of electricity on a bowl, Quart. J. Math. 18, 97-109 (1882)
  • [3] E. G. Gallop, The distribution of electricity on the circular disc and spherical bowl, Quart. J. Math 21, 229-256 (1886)
  • [4] A. B. Basset, On the Potential of an Electrified Spherical Bowl, and on the Velocity Potential due to the Motion of an Infinite Liquid about such a Bowl, Proc. London Math. Soc. S1-16, no. 1, 286. MR 1575774, https://doi.org/10.1112/plms/s1-16.1.286
  • [5] Lord Rayleigh, On the passage of waves through apertures in plane screens, and allied problems. Philos. Mag. 43, 259-272 (1897); Sci. Papers 4, 283-296
  • [6] W. D. Collins, On some dual series equations and their application to electrostatic problems for spheroidal caps, Proc. Cambridge Philos. Soc. 57 (1961), 367–384. MR 0127158
  • [7] Lord Rayleigh, The theory of the Helmholtz resonator, Proc. Roy. Soc. London Ser. A 92, 265-75 (1915); Sci. Papers 6, 365-375
  • [8] Arnold Sommerfeld, Partial Differential Equations in Physics, Academic Press, Inc., New York, N. Y., 1949. Translated by Ernst G. Straus. MR 0029463
  • [9] Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059774
  • [10] W. D. Collins, Some scalar diffraction problems for a spherical cap, Arch. Rational Mech. Anal. 10 (1962), 249–266. MR 0147089, https://doi.org/10.1007/BF00281192
  • [11] G. I. Taylor, The energy of a body moving in an infinite fluid, with an application to airships, Proc. Roy. Soc. London Ser. A 120, 13-21 (1928)
  • [12] E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Cambridge Univ. Press, New York 1931
  • [13] C. Neumann, Die Vertheilung der Elektrictät auf einer Kugelcalotte, Abh. Math. Phys. Cl. Königl. Sächs. Ges. Wiss. 12, 401-456 (1880)
  • [14] Horace Lamb, Hydrodynamics, 6th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. With a foreword by R. A. Caflisch [Russel E. Caflisch]. MR 1317348
  • [15] Lord Rayleigh, On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen, Philos. Mag. 44, 28-52 (1897); Sci. Papers 4, 305-326
  • [16] Horace Lamb, The dynamical theory of sound, 2nd ed. Dover Publications, Inc., New York, 1960. MR 0127692

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 78.31

Retrieve articles in all journals with MSC: 78.31

Additional Information

DOI: https://doi.org/10.1090/qam/284080
Article copyright: © Copyright 1971 American Mathematical Society

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website