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
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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.
W. Thomson, Extraits de deux lettres adressées à M. Liouville, J. Math. Pures Appl. 12, 256–264 (1847)
N. M. Ferrers, On the distribution of electricity on a bowl, Quart. J. Math. 18, 97–109 (1882)
E. G. Gallop, The distribution of electricity on the circular disc and spherical bowl, Quart. J. Math 21, 229–256 (1886)
- 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. Lond. Math. Soc. 16 (1884/85), 286–306. MR 1575774, DOI https://doi.org/10.1112/plms/s1-16.1.286
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
- 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 127158, DOI https://doi.org/10.1017/s0305004100035325
Lord Rayleigh, The theory of the Helmholtz resonator, Proc. Roy. Soc. London Ser. A 92, 265–75 (1915); Sci. Papers 6, 365–375
- Arnold Sommerfeld, Partial Differential Equations in Physics, Academic Press, Inc., New York, N. Y., 1949. Translated by Ernst G. Straus. MR 0029463
- Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059774
- W. D. Collins, Some scalar diffraction problems for a spherical cap, Arch. Rational Mech. Anal. 10 (1962), 249–266. MR 147089, DOI https://doi.org/10.1007/BF00281192
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)
E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Cambridge Univ. Press, New York 1931
C. Neumann, Die Vertheilung der Elektrictät auf einer Kugelcalotte, Abh. Math. Phys. Cl. Königl. Sächs. Ges. Wiss. 12, 401–456 (1880)
- 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
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
- Horace Lamb, The dynamical theory of sound, 2nd ed. Dover Publications, Inc., New York, 1960. MR 0127692
W. Thomson, Extraits de deux lettres adressées à M. Liouville, J. Math. Pures Appl. 12, 256–264 (1847)
N. M. Ferrers, On the distribution of electricity on a bowl, Quart. J. Math. 18, 97–109 (1882)
E. G. Gallop, The distribution of electricity on the circular disc and spherical bowl, Quart. J. Math 21, 229–256 (1886)
A. B. Basset, On the potential of an electrified spherical bowl, and on the velocity due to the motion of an infinite liquid about such a bowl, Proc. London Math. Soc. 16, 286–306 (1885); A treatise on hydrodynamics, Vol. 1, Dover, New York, 1961 (reprint of 1888 edition), §§152–159
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
W. D. Collins, On some dual series equations and their application to electrostatic problems for spheroidal caps, Proc. Cambridge Philos. Soc. 57, 367–384 (1961)
Lord Rayleigh, The theory of the Helmholtz resonator, Proc. Roy. Soc. London Ser. A 92, 265–75 (1915); Sci. Papers 6, 365–375
A. Sommerfeld, Partial differential equations in physics, Academic Press, New York, 1949, pp. 162–164
P. M. Morse and H. Feshbach, Methods of theoretical physics, Vol. 2, McGraw-Hill, New York, 1953, pp. 1490–1494
W. D. Collins, Some scalar diffraction problems for a spherical cap, Arch. Rational Mech. Anal. 10, 249–266 (1962)
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)
E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Cambridge Univ. Press, New York 1931
C. Neumann, Die Vertheilung der Elektrictät auf einer Kugelcalotte, Abh. Math. Phys. Cl. Königl. Sächs. Ges. Wiss. 12, 401–456 (1880)
H. Lamb, Hydrodynamics, Cambridge Univ. Press, New York, 1932
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
H. Lamb, The dynamical theory of sound, 2nd ed., Dover, New York, 1960
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Article copyright:
© Copyright 1971
American Mathematical Society
