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 | 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.

**[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**

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Additional Information

DOI:
https://doi.org/10.1090/qam/284080

Article copyright:
© Copyright 1971
American Mathematical Society