Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



An asymptotic fundamental solution of the reduced wave equation on a surface

Author: C. R. Steele
Journal: Quart. Appl. Math. 29 (1972), 509-524
MSC: Primary 73.35; Secondary 80.35
DOI: https://doi.org/10.1090/qam/408397
MathSciNet review: 408397
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The physical problem of steady-state heat conduction in a thin shell is described by the ``reduced wave equation'' in which the differential operator is the (generally noneuclidean) Laplacian for the surface. A similar equation gives the approximation for steady-state waves in a prestressed curved membrane. A modification of the ``geometric optics'' asymptotic expansion, involving a Bessel function, is given for the fundamental point source solution. This is proven to be uniformly valid in the large, until a ``caustic'' is reached. Various features of the solution for a surface, which do not occur for the plane, are discussed.

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

  • [1] G. S. S. Avila and J. B. Keller, The high-frequency asymptotic field of a point source in an inhomogeneous medium, Comm. Pure Appl. Math. 16, 363-381 (1963) MR 0164133
  • [2] V. V. Bolotin, Equations for the non-stationary temperature fields in thin shells in the presence of sources of heat, Prikl. Math. Meh. 24, 361-363 (1960) = J. Appl. Math. Mech. 24, 515-519 (1960) MR 0118102
  • [3] L. M. Brekhovskikh, Waves in layered media, Izdat. Akad. Nauk SSSR, Moscow, 1957; English transl., Academic Press, New York, 1960 MR 0112392
  • [4] P. R. Garabedian, Partial differential equations, Wiley, New York, 1964 MR 0162045
  • [5] Yu. A. Kravtsov, Two new asymptotic methods in the theory of wave propagation in inhomogeneous media (review), Soviet Physics-Acoustics 14, 1-17 (1968)
  • [6] R. E. Langer, On the asymptotic solutions of ordinary differential equations, with reference to the Stokes' phenomenon about a singular point, Trans. Amer. Math. Soc. 37, 397-416 (1935) MR 1501793
  • [7] R. M. Lewis, N. Bleistein and D. Ludwig, Uniform asymptotic theory of creeping waves, Comm. Pure Appl. Math. 20, 295-328 (1967) MR 0213101
  • [8] R. P. Nordgren and P. M. Naghdi, Propagation of thermoelastic waves in an unlimited shallow spherical shell under heating, Proc. Fourth U. S. Nat. Congr. Appl. Mech. (Univ. of California, Berkeley, Calif., 1962), vol. 1, Amer. Soc. Mech. Engrs., New York, 1962, pp. 311-324 MR 0159475
  • [9] R. P. Nordgren, On the method of Green's function in the thermoelastic theory of shallow shells, Internat. J. Engrg. Sci. 1, 279-308 (1963) MR 0153168
  • [10] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, New York, 1944 MR 0010746
  • [11] T. J. Willmore, Differential geometry, Oxford Univ. Press, London, 1959
  • [12] J. -S. Yang, Thermal stresses in thin shells, Ph.D. Thesis, Stanford University, 1969; see also Stanford University Report, SUDAAR No. 371, Stanford Calif., 1969

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73.35, 80.35

Retrieve articles in all journals with MSC: 73.35, 80.35

Additional Information

DOI: https://doi.org/10.1090/qam/408397
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society