Application of the Sonin-Pólya oscillation theorem
Authors:
E. V. Laitone and Wen Fan Lin
Journal:
Quart. Appl. Math. 32 (1974), 285-291
MSC:
Primary 33A30
DOI:
https://doi.org/10.1090/qam/430343
MathSciNet review:
430343
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Abstract: The necessary conditions for the Sonin-Pólya theorem, which predicts the variation of the successive relative maxima of the oscillating solutions of a second-order ordinary differential equation, are evaluated for the confluent hypergeometric functions to prove that the criteria for their decreasing sequence, as given in [1], [2] and [3], are incorrect. The Sonin-Pólya theorem is then applied to the differential equation for the linearized shallow-water edge wave that is produced by a beach of constant slope. The final application is to Allen’s differential equation for the angle of attack oscillations of a re-entry ballistic missile, and new stability criteria are obtained for both ascending and descending missiles that are coasting through the atmosphere.
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, AMS 55, National Bureau of Standards, 1964, pp. 504–535
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- L. J. Slater, Confluent hypergeometric functions, Cambridge University Press, New York, 1960. MR 0107026
G. Szegö, Orthogonal polynomials, Vol. 23, Amer. Math. Soc. Colloquium Publications, New York, 1939, p. 161
H. Lamb, Hydrodynamics, Dover, New York, 1945, pp. 276, 291 and 447
J. N. Hunt and A. M. O. Hamzah, Tidal waves in a canal of non-uniform depth, Pure Appl. Geophys. 67, 133–142 (1967)
- F. Ursell, Edge waves on a sloping beach, Proc. Roy. Soc. London Ser. A 214 (1952), 79–97. MR 50420, DOI https://doi.org/10.1098/rspa.1952.0152
H. J. Allen, Motion of a ballistic missile angularly misaligned with the flight path upon entering the atmosphere and its effects upon aerodynamic heating, aerodynamic loads, and miss distance, NACA TN 4048 (1957)
N. X. Vinh and E. V. Laitone, Longitudinal dynamic stability of a shuttle vehicle, J. Astronaut. Sci 19, 337–363 (1972)
M. Tobak and H. J. Allen, Dynamic stability of vehicles traversing ascending or descending paths through the atmosphere, NACA TN 4275 (1958)
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, AMS 55, National Bureau of Standards, 1964, pp. 504–535
H. Bateman, Higher transcendental functions, Vol. 1, McGraw-Hill, Book Co., New York, 1953, p. 291
L. J. Slater, Confluent hypergeometric functions, University Press, Cambridge, 1069, p. 119
G. Szegö, Orthogonal polynomials, Vol. 23, Amer. Math. Soc. Colloquium Publications, New York, 1939, p. 161
H. Lamb, Hydrodynamics, Dover, New York, 1945, pp. 276, 291 and 447
J. N. Hunt and A. M. O. Hamzah, Tidal waves in a canal of non-uniform depth, Pure Appl. Geophys. 67, 133–142 (1967)
F. Ursell, Edge waves on a sloping beach, Proc. Roy. Soc. London, A214, 79–97 (1952)
H. J. Allen, Motion of a ballistic missile angularly misaligned with the flight path upon entering the atmosphere and its effects upon aerodynamic heating, aerodynamic loads, and miss distance, NACA TN 4048 (1957)
N. X. Vinh and E. V. Laitone, Longitudinal dynamic stability of a shuttle vehicle, J. Astronaut. Sci 19, 337–363 (1972)
M. Tobak and H. J. Allen, Dynamic stability of vehicles traversing ascending or descending paths through the atmosphere, NACA TN 4275 (1958)
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Article copyright:
© Copyright 1974
American Mathematical Society
